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User's Manual for
*************************************************************
* MICROSCOPE: A Software System for Multivariate Analysis *
*************************************************************
by
Peter Alfeld and Bill Harris
Department of Mathematics
University of Utah
Salt Lake City
Utah 84112
Tel.: 801-581-6842 or 801-581-6851
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Abstract
This manual describes MICROSCOPE: a portable FORTRAN software
system for the analysis of multivariate functions. Given an interpolation
or approximation scheme, it allows the following questions, among others,
to be answered: Does the scheme interpolate? How often is it
differentiable? What functions does it reproduce exactly? If the scheme
is polynomial, what is its polynomial degree? Where is the smoothness of a
function reduced? Where are the bugs in a FORTRAN implementation?
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List of Contents
0. About This Manual .......................................6
1. Introduction ............................................8
1.1 Multivariate Interpolation and Approximation ............8
1.2 The Basic Approach ......................................9
1.3 The Alphanumerical Display ..............................13
1.4 Summary .................................................19
2. Applications ............................................21
2.1 Interpolation ...........................................22
2.2 Smoothness ..............................................27
2.3 Degree of Precision .....................................28
2.4 Degree of a Polynomial ..................................30
3. Using MICROSCOPE ........................................33
3.1 The MCRSCP Routine ......................................34
3.2 The Command Mode ........................................36
3.3 A Discontinuity in the Second Derivative ................38
4. A Detailed Description of Features and Commands .........44
4.1 Round-Off Effects .......................................44
4.2 Derivatives of Step Functions ...........................45
4.3 The Window ..............................................49
4.4 Reference Table of Commands .............................51
4.5 Commands to Control the Display .........................54
4.6 Commands to Control the Point of Examination ............58
4.7 Commands to Control the Direction of Investigation ......59
4.8 Commands to Control Accuracy, the Window,
and Round-Off Effects ...............................60
4.9 Commands to Control Execution ...........................62
4.10 Commands to Obtain On-Line Help .........................64
4.11 Commands to Keep a Record ...............................65
4.12 Commands to Analyze Cross Derivatives ...................65
4.13 Programming MICROSCOPE ..................................67
4.14 The PLOT command for obtaining a <PLOT79> Display .......69
4.15 User Intervention .......................................73
4.16 Default settings ........................................74
5. Installation Guide ......................................75
5.1 Description of the Package ..............................75
5.2 A Key to Selecting the Appropriate Files ................80
Appendix I: Numerical Differentiation .......................82
I.1 The Round-Off Threshold ........................85
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I.2 Round-off versus Truncation Errors .............87
I.3 Derivatives of Step Functions ..................89
Appendix II: Organization of MICROSCOPE .....................92
II.1 Organization of the Program Package ...........92
II.2 Sampling Scheme and Logic .....................95
II.3 Common Blocks .................................104
Appendix III: The Test Package ..............................110
Appendix IV: Examples for Graphical (<PLOT79>) Displays .....115
References ..................................................116
Acknowledgements ............................................117
Index .......................................................118
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List of Tables and Figures
Figure 1 The Alphanumerical Display ........................14
Figure 2 Interpolation to Position at Zero..................22
Figure 3 ...................................................23
Figure 4 Interpolation at x = 2 ............................24
Figure 5 Interpolation to the First Derivative .............25
Figure 6 ...................................................26
Figure 7 Free End Condition ................................27
Figure 8 Smoothness at x = 1 ...............................28
Figure 9 A Zero Error ......................................29
Figure 10 Round-Off with a Bias ............................30
Figure 11 A Cubic Function .................................31
Figure 12 ..................................................32
Figure 13 ..................................................38
Figure 14 ..................................................39
Figure 15 ..................................................40
Figure 16 ..................................................41
Figure 17 ..................................................42
Figure 18 At the limits of resolution ......................43
Figure 19 Derivatives of a Step Function ...................46
0
Figure 20 Derivatives of a C Function .....................47
1
Figure 21 Derivatives of a C Function .....................47
2
Figure 22 Derivatives of a C Function .....................48
3
Figure 23 Derivatives of a C Function .....................48
4
Figure 24 Derivatives of a C Function .....................49
Figure 25 An Invisible graph ...............................56
Figure 26 The Disentangled Graphs ..........................57
Figure 27 An Extended Discontinuity ........................67
Figure 28 A 3/2 times differentiable function ..............113
Figure 29 Another 3/2 times differentiable function ........114
Table: Efficiency of Window Widths .........................50
Quick Reference Table of Commands ..........................52
Table of Discretization Controling Commands ................60
Table: Relative Efforts ....................................62
Table 1: Constants for Differentiation Formulas ............84
Table 2: Threshold Values of h .............................86
Table 3: Optimum Values of h ...............................88
Table 4: The Superiority of Testing Higher Derivatives .....91
Epitome Flowchart ..........................................94
Outline of SGAMMA ..........................................104
Table of Test Parameters ...................................112
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0. About This Manual
This manual constitutes the comprehensive description of
MICROSCOPE, its operation, applications, foundations, and inner workings.
Section 1.1 describes what kind of problems can be addressed by
MICROSCOPE. Section 1.2 describes the basic approach and is fundamental.
Section 1.3 describes the alphanumerical screen display and is needed for
understanding the examples given throughout this manual. It also
introduces some terms and concepts and serves as a prelude to the
capabilities of MICROSCOPE.
Section 2 gives examples that illustrate four major applications
(interpolation, smoothness, degree of precision, and polynomial degree).
These do of course not exhaust the potential of MICROSCOPE. In section 3,
one other application (discovery or identification of a point where
smoothness is reduced) is used as a vehicle for introducing the mechanics
of using MICROSCOPE.
Section 4 gives a detailed description of all commands and
features, organized by type of feature or task to be accomplished. In
regular use, this will probably be the most frequently consulted section.
In reading sections 2 and 3, it is desirable that a working
version of MICROSCOPE is available so that hands on experience can be
gained. If this is not the case the installation guide (section 5) should
be consulted first. That section also contains a detailed discussion of
the portability features and possible variants of MICROSCOPE.
Appendices are included that describe some technical features in
detail: the differentiation formulas, the organization of the program
package, and the test package. There should also be an appendix IV
containing pen plots corresponding to the cruder alphanumerical graphics in
the body of the manual. However, that appendix is not machine legible, and
is distributed separately.
The manual itself is in machine readable form so that it can be
distributed together with the package and can be printed locally, and read
on line using for example the search features of a text editor. It was
generated by a MICROSCOPE program, with the text of the manual being
"notes" and the example screen displays being computed by MICROSCOPE and
automatically embedded into the text. (Actually, the MICROSCOPE program
generated a data file that was then fed into the text formatting program
DOCUMENT, see Beebe, 1980.) Machine legibility, however, accounts for the
somewhat awkward notation in some places. For example, greek letters have
been given abbreviations in terms of roman letters.
Definitions are denoted by writing the defined term in single
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quotation marks. Ordinary quotations are in double quotation marks.
The examples in this manual are reproducible since the set of
routines used for their generation is supplied with the package and can be
used for test purposes. To ensure uniformity of results obtained on
different machines, a facility has been added to simulate rounding, and all
examples have been computed in 10 digits arithmetic. More precisely, if x
is the quantity to be rounded to D digits, say, then x is replaced by
-D -D
z := (1+eps *10 )*x + eps *10
1 2
where eps and eps are random numbers between -1 and +1. The addition of
1 2
the eps term is not standard but appropriate in the present context,
2
because in investigations with MICROSCOPE small numbers are often due to
taking differences between very close large numbers, leading to a
cancellation of significant digits. The larger the magnitude of x, the
more insignificant the second term will become. If x is roughly 1 or
larger (in magnitude) our rounding is essentially equivalent to the
standard rounding (where eps = 0).
2
This manual will rarely be read sequentially from front to back.
Therefore, key ideas and concepts have sometimes been restated if they are
crucial in the given context. It is hoped that this redundancy will
contribute to the usefulness of this document.
Any crticisms, comments and suggestions about MICROSCOPE are
welcome and should be directed to Peter Alfeld, Department of Mathematics,
University of Utah, Salt Lake City, Utah 84112, 801-581-6842. We are
particularly interested in any applications of MICROSCOPE that are not
described in this manual.
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1. Introduction
1.1 Multivariate Interpolation and Approximation
This manual describes a FORTRAN software package, MICROSCOPE,
whose main applications are in the analysis of functions of several
variables (or 'surfaces') as they occur e.g. in the approximation of
multivariate data or in the design of geometric objects such as the body of
an automobile or an aircraft. This is a currently very active research
area with many unsolved and difficult problems. For a survey see Barnhill,
1983, and the references quoted therein.
MICROSCOPE is portable to the extent that it has passed the PFORT
verifier (Ryder, 1974) with the exception that it uses some non-FORTRAN
characters (namely ?,<,>,!,:,;). Some non-portable optional features are
supported by variants of the code. These include the use of lower case
letters in Input and Output, and the use of modifications of an
alphanumerical display on a CRT terminal. If the software (<PLOT79>, Beebe
1979) and hardware are available more sophisticated and pleasing graphical
displays can also be obtained. However, even if none of the additional
features are provided by a particular installation, a working, if somewhat
crude, version of MICROSCOPE can be constructed. See the installation
guide (section 5) for details.
The objects that can be examined by MICROSCOPE are functions of
from one to three independent variables. Allowing only up to three
variables enabled us to contain all numerical information on the CRT screen
and proved sufficent for all cases we have encountered. If a function of
more than three variables must be examined MICROSCOPE can be applied to a
suitable restriction of the function to a subdomain of dimension at most
three. If there is sufficent interest, future versions of MICROSCOPE
capable of handling more than three variables will be provided.
The functions of interest are usually obtained by applying an
interpolation or approximation scheme to discrete or transfinite (i.e.
infinitely many) data. As far as the use of MICROSCOPE is concerned it
matters little if a function is obtained by interpolation or approximation
(except that one will not need to test for interpolation in the latter
case), so we will use the two terms interchangeably.
We will refer to the function that is being examined as the
'trial function'. The trial function may be different from the
interpolating or approximating function in which the final user of a
numerical scheme is interested. For example, it is often useful to
generate data from a 'primitive function' and then use the difference
between the primitive function and its interpolant as the trial function.
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This makes discontinuities in derivatives large relative to the derivative
value and thereby amplifies discontinuities which are then more easily
detected.
Four types of questions occur most frequently in the analysis of
the trial function:
-1- Does the scheme interpolate?
-2- How often is the trial function differentiable? (I.e.
what is its 'degree of smoothness'?)
Note:Throughout almost all of this manual we will consider only functions
that possess a step discontinuity in some derivative, for this is the
type of smoothness limitation that occurs in multivariate
interpolation and approximation. There are of course other
4/3
possibilities. For example, the function f(x) = x is once
differentiable but its second derivative exhibits a pole rather than a
step. The test package (see appendix III) can be used to gain some
experience with such functions. In particular, the examples in
appendix III show how to identify and distinguish cusps and poles.
-3- Which functions are reproduced exactly by the scheme? In
particular, which is the maximum degree up to which all polynomials are
reproduced exactly? In other words, what is the 'degree of (polynomial)
precision' of the scheme?
-4- If the trial function (or some of its derivatives) are
polynomial what is their degree?
It should be obvious that only in the very simplest of cases
answers to these questions can be obtained from a plain display of the
surface of interest. Even given infinite display accuracy (although of
course any physical display is ultimately piecewise linear or even
discrete), it seems impossible to distinguish optically a once
differentiable function from one that is twice differentiable.
1.2 The Basic Approach
In the design of an approximation scheme one usually knows the
'critical sets' where the phenomena of interest such as discontinuities in
derivatives, or interpolation to certain data, may occur. For example, in
a bivariate interpolation scheme defined on a triangulation, critical sets
are usually edges of triangles, possibly internal edges, and vertices of
triangles. In a trivariate scheme defined on a tesselation into
tetrahedra, critical sets are faces, edges, maybe internal faces and edges,
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and internal and external vertices. If the critical set is not known it
can be discovered with MICROSCOPE (see section 3), but for the present we
will assume it is known.
The basic idea of MICROSCOPE is very simple: Pick a random point
(the 'point of examination') in a specific critical set. We will refer to
that set as 'the front'. Then determine a 'direction of investigation' and
consider a line through the point of examination in the direction of
investigation. This is the 'line of investigation'. Depending upon the
application, the line of investigation may be contained in the front, or it
may be at an angle (not necessarily a right one) to it. We will call any
derivative in the direction of investigation a 'tangential derivative'.
Any pure derivative in a direction other than the direction of
investigation is a 'cross derivative' and its direction is the 'cross
direction'. MICROSCOPE numerically approximates the relevant derivatives
(pure or mixed) and displays the approximations on the CRT terminal.
Depending on the resulting picture, one may then take further action, e.g.
pick another point of examination or another direction of investigation,
consider other derivatives, or improve the existing plot by altering the
underlying numerical parameters.
In using MICROSCOPE it is important to choose the underlying
domain and the primitive function so as to avoid artifacts due to the
regularity of the domain or peculiarities of the primitive function, and to
pick the points of examination randomly or arbitrarily within the critical
sets. Usually different types of critical sets will be present and of
course one will need to consider at least one representative of each set.
Numerical differentiation is a notoriously ill-conditioned
numerical process. However, in MICROSCOPE, the appearance of round-off
errors is immediately apparent because the points on the graph of the
relevant derivative are scattered across the screen in a random fashion.
Moreover, by plotting derivatives of a high degree with a large
discretization parameter the limitations imposed by round-off errors can
usually be overcome. For example, if the expected discontinuity in a
specific derivative is too unpronounced to be visible at the large value of
the discretization parameter necessitated by round-off errors, then higher
order derivatives can be computed (on an even coarser discretization).
These will be approximations of the Dirac Delta function and its
derivatives, and usually will pinpoint even weak discontinuities.
Numerically, we proceed as follows. Let F denote the trial
function, P the point of examination, and d the direction of investigation.
MICROSCOPE displays derivatives of F in the direction of d along the line
of investigation with P being the center point. A built-in facility allows
the replacement of F with a derivative of F in some other, independent,
direction. More complicated mixed partial derivatives can be built by
modifying the trial function, and MICROSCOPE offers some help with that as
well. However, for the present we will restrict our attention to a
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tangential derivative of F.
The directional derivatives are approximated at points
P(i) = P + i*s*d, where i = -N, -N+1, ...., N-1,N
and the quantity s > 0 determines the spacing of the points. It will be
refered to as the 'display interval' or 'interval' for short. The number N
determines the resolution of the display (as well as the numerical effort).
At each point P(i), we approximate the k-th derivative of F in the
(k)
direction d at the i-th point by a quantity Q .
i
The range of k in the present version of MICROSCOPE is from zero
to six. We have found this sufficient for all of our applications. The
(k)
quantities Q are determined by numerical differentiation (i.e. by
i
differentiating an interpolating polynomial). The formulas, given below,
were determined by the following criteria:
-1- Central Differences should be employed in order to avoid
artifact due to asymmetry.
-2- The discretization parameter (which does not necessarily
equal s) should be such that the number of points affected by the suspected
discontinuity (in some derivative) is independent of the order of the
derivative.
-3- Other than the above, the degree of the interpolating
polynomial should be as low as possible.
Furthermore, we picked discretization formulas that use points at
equally spaced intervals. This may be changed in future versions of
MICROSCOPE.
Denoting the 'discretization parameter' by h > 0, the above
criteria give rise to the following formulas:
(0)
Q = F5
i
(1)
Q = (-F1 + F9)/(2*h)
i
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(2) 2
Q = (F1 - 2F5 + F9)/h
i
(3) 3
Q = 4*(-F1 + 2F3 - 2F7 + F9)/h
i
(4) 4
Q = 16*(F1 - 4F3 + 6F5 - 4F7 + F9)/h
i
(5) 5
Q = 243*(-F1 + 4F2 - 5F4 + 5F6 - 4F8 + F9)/(2h )
i
(6) 6
Q = 729*(F1 - 6F2 + 15F4 - -20F5 + 15F6 - 6F8 + F9)/h
i
where
F1 = F(P(i) - hd)
2hd
F2 = F(P(i) - ---)
3
hd
F3 = F(P(i) - --)
2
hd
F4 = F(P(i) - --)
3
F5 = F(P(i))
hd
F6 = F(P(i) + --)
3
hd
F7 = F(P(i) + --)
2
2hd
F8 = F(P(i) + ---)
3
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F9 = F(P(i) + hd)
The first four of the above formulas can be found in Abramowitz
and Stegun, 1968, p. 914, and the last two can be derived by the
techniques described there. Higher order formulas can of course also be
derived.
(k)
It is interesting to note that Q considered as a function of x
i
(or a function of a continuously varying parameter i) is a function of the
same degree of smoothness as the trial function itself. Just the number of
points of reduced smoothness is increased, there being one corresponding to
each point in the differentiation formula passing through the point of
reduced smoothness. Thus numerical differentiation can be thought of as a
smoothing process. (This contrasts starkly with its numerical property of
a roughing procedure due to round-off effects.) The figures in section 4.2
illustrate the smoothing properties of numerical differentiation.
For a more detailed discussion of the truncation and round-off
properties of these formulas see Appendix I.
1.3 The Alphanumerical Display
In this subsection, the alphanumerical screen display generated
by MICROSCOPE is described in detail. The purpose of the description is
twofold: First, it is necessary for an understanding of the examples
sprinkled throughout this manual. Second, however, it also serves as an
illustration of some of the capabilities of MICROSCOPE and as a first
introduction to the power and versatility of the package.
During a MICROSCOPE session the CRT screen will usually be
occupied by the 'alphanumerical display' like the one illustrated in
figure 1. (More sophisticated graphical displays can also be obtained, see
sections 4.14 and 5.)
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222 I : **********************************
. 22 I : * I 22 .
. 222 I : * I 222 .
. 22 I : I 22 .
. 22 I : I 22 .
. 222 I :* I 222 .
.. 22 I : I 22 ..
7654321..876543212287654321.987654321+123456789.123456782212345678..1234567
. 222 I : I 222 .
.. 22 I *: I 22 ..
.. 22 I : I 22 ..
.. 222 I : I 222 ..
... 22I * : I22 ...
.... 22 * : 22 ....
**********************************22222222..........
===========================================================================
CD: deg = 1 dir = ( 0.000000D+00, 1.000000D+00) ch = 6.0000D-04
Point = ( 0.000000D+00, 1.000000D+00) s = 5.0000D-03
Direction = ( 1.000000D+00, 0.000000D+00) h = 3.0000D-02
F0 ( 9.39D-09, 1.27D-02) F2 ( 1.20D-01, 2.22D+00) F3 (-1.20D+01, 1.21D+01)
I/O: 25 6 6 1 2 3 27 28 29 30 NRML on current CALLS = 174
Figure 1 Demonstration of the Alphanumerical Display
2
In this example, the trial function is f(x,y) = x *y*abs(x*y).
We analyze a cross derivative of it in the direction (0,1). The direction
of investigation is (1,0), and the point of examination is (0,1). Because
of the absolute value term, the cross derivative is twice but not three
times tangentially differentiable at the point of examination.
We explain the display starting at the top and referring to lines
either by their position or by the first word contained in them.
The display is naturally divided into two parts: The first 15
lines constitute the 'graphical display', and lines 17 through 21 form the
'numerical display'. Line 16 separates the two displays and is present
only if there is sufficient space available.
-1- The Graphical Display
Every MICROSCOPE display shows the graphs of a set of tangential
derivatives of degrees between 0 and 6. The function that is being
differentiated tangentially is called the 'display function'. It may be
the trial function itself (the most frequently occuring case), or, as in
this example, a pure cross derivative of a degree between 1 and 6.
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Distance along the horizontal axis corresponds to displacement in
the direction of investigation from the point of examination (which
corresponds to the center of the horizontal axis). The scale on that axis
has the same meaning for all graphs shown. The vertical axis corresponds
to function values, but the vertical scale is different for each
derivative, and is chosen such that the graph fills the entire vertical
extent of the graphical display. This normalization is carried out and
modified automatically as the investigation proceeds. A period denotes the
graph of the display function, and the digit k, say, where k is between 1
and 6 and denotes the k-th tangential derivative. In this example, the
display function and its second and third tangential derivatives are being
shown.
Sometimes some derivative is more interesting than others, and
then it may be useful to accentuate it with asterisks. In the above
example, the third derivative has been so emphasized. Usually, the graph
of a given derivative overwrites the graph of any lower order derivative.
However, the graph of an accentuated derivative overwrites all other
graphs.
Both the number of lines and the number of columns in the display
can be set according to specific purposes. Throughout this manual we will
use 75 columns and 15 rows for the graphical display as this is appropriate
for the width of the printed output. A width of 79 might be preferable for
a standard CRT terminal displaying 80 columns; 135 columns and a larger
number of rows might be used for printed output; and a smaller number is
sometimes useful for preliminary investigations when the trial function is
expensive to evaluate. An even number of columns is normally undesirable
because then the point of examination does not lie in the precise center of
the display.
Line 8 in the example contains a horizontal scale counting from
the center to the left and right. This will usually be omitted so as not
to clutter the display, but it is sometimes useful for the identification
of points of interest. The distance from the center can be used to inquire
about numerical values of certain derivatives, or to shift the graph left
or right as required by the circumstances. The scale can be turned on or
off, or replaced with a line of hyphens. The graph of any function
overwrites the horizontal scale. The center of the display can optionally
be marked with a "+" sign. This is sometimes useful to pinpoint symmetry
properties. In the above example the graph of the third derivative passes
through the center of the display. The center mark overwrites the graph of
any function passing through it.
The central column of colons marks the column corresponding to
the point of examination. The two columns of I's outline 'the window',
i.e. that set of points at which the numerical differentiation is affected
by the behavior of the trial function at the point of examination. (This
is not always quite true when cross derivatives are being examined, see the
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discussion in section 4.11). For every point in the window, the interval
about the point spanned by the differentiation stencil contains the point
of examination. The differentiation formulas are chosen such that the
window is identical for all tangential derivatives. Obviously, for the
display function itself the window contains only the point of examination.
For its derivatives, the window contains all points that can be written as
P + td, where abs(t) does not exceed the value of h/s, (and, as before, P
is the point of examination, d is the direction of investigation, h is the
discretization parameter, and s is the interval). We define w:=2*h/s to be
the 'window width'. Obviously, by picking the window width < 1, it is
possible to generate displays in which the only point contained in the
window is P itself. However, for several reasons it is usually preferable
to use a larger window width. This aspect is discussed in more detail in
section 4.1. The default value of the window width is 12, as in the
example, but this can be changed as needed.
Without examining the numerical display, it is apparent from the
graphical display that some function is being investigated that is twice
but not three times differentiable since the (accentuated) third derivative
shows a clear step discontinuity. Note how that step is smeared out over
the window.
-2- The numerical display
The first line of the numerical display starting with "CD:"
indicates that the derivatives are mixed partials, i.e. they are
tangential derivatives of a cross derivative (of the trial function). This
line is absent if the trial function itself is being examined, and the line
carries only information that is pertinent to the cross derivative. In the
present example, the degree of the cross derivative ("deg") is 1, the cross
direction is (0,1), and the discretization parameter used for computing the
derivative (denoted by "ch" corresponding to "h" for the tangential
derivative) is 6D-4. The formulas for computing cross derivatives are
equivalent to those for computing tangential derivatives.
The line starting with "Point" describes the point of examination
(which is (0,1)) and the interval (which is 5D-3). This display implies
that the trial function is bivariate. Appropriate modifications take place
for functions of 1 or 3 variables. The next line starting with "direction"
describes the direction of investigation (which is (1,0)) and the
discretization parameter (which is 3D-2). Notice that the window width is
2h/s = 12 which is consistent with the graphical display
The "Point" and "Direction" descriptions are always present.
The next line, starting with "F0", gives the ranges of the
displayed derivatives. Ranges are labeled Fk where k is the order of the
tangential derivative. For example, F2 corresponds to the range of the
second (tangential) derivative (of a first order cross derivative) which
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here is (0.12,2.1). If more functions (up to seven) are displayed, then
their ranges are also given, adding up to two lines to the numerical
display as necessary.
The next and last line, starting with "I/O", contains information
about the present configuration of MICROSCOPE.
The 10 integers following "I/O" are the FORTRAN channel numbers
as they are employed by MICROSCOPE at the display time. They correspond to
the following:
-1- 'Input Device' (here 25), i.e. the device from which
commands are read. In an interactive session, this will usually be the
terminal. But it may also be a file of MICROSCOPE commands that generate a
specific setup, or a documentation like this manual.
-2- 'Output Device' (here 6), i.e. the device receiving the
command prompts and any help information. Usually, this will also be the
terminal, or, if MICROSCOPE is being driven by commands in a file, it may
be a Null or Dummy device.
-3- 'Graphics Device' (here also 6), i.e. the device receiving
the alphanumerical display illustrated in figure 1. This will again
usually be the terminal, but it may also be a separate display device or a
file receiving information for subsequent printing.
-4- 'Help Device' (here 1), i.e. the device containing the
MICROSCOPE help file, which is supplied with the package. MICROSCOPE has
several interactive help facilities and the information needed for running
them is read out of this file when MICROSCOPE is first called. Usually you
will not be concerned with precisely where MICROSCOPE receives its help
information. However, the channel number is printed in the display to help
prevent an accidental overwrite of the help file.
-5- 'Recording Device' (here 2), a device that can receive
selected information from the MICROSCOPE session, e.g. copies of the
screen display or 'notes' that explain the displays. This manual is the
result of reading a set of instructions, i.e. a MICROSCOPE 'program' from
the Input Device (a data file), processing it and writing the results onto
the Recording Device.
-6- 'Restart Device' (here 3). At any time during a session,
the current configuration of MICROSCOPE can be saved, allowing a later
restart from the current status of the investigation. This is accomplished
by writing the values of all relevant variables into a file pointed at by
the Restart device.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 18
-7- 'Logging Device' (here 27). It is possible to keep a diary
of th MICROSCOPE session listing the commands and data entered as well as
th results effected by them. This feature is activated by setting the
Logging Device number to a non-zero value.
-8- 'Point Loading Device' (here 28). A device allowing you to
read points from a file, rather than having to type them in.
-9- 'Direction Loading Device' (here 29). Similar to -8- for
loading the direction of investigation.
-10- 'Cross Direction Loading Device' (here 30). Similar to -8-
for loading the cross direction.
The first six I/O channel numbers are parameters of the master
MICROSCOPE routine (see section 3.1) and can each be changed during a
session. The last four channels are zero (i.e. unused) by default, and
can be set to non-zero (i.e. active) values during a MICROSCOPE session.
It is of course unnecessary to have all channel numbers distinct. For
example, all three loading devices might be identical, allowing for the
possibility of having just one list of points, directions, and cross
directions.
Following the device numbers is a flag "NRML on" or "NRML off"
(here on), indicating whether the direction of investigation entered into
MICROSCOPE has been normalized or not. The "Direction" line always gives
the direction as entered by the user. If NRML is off, the direction of
investigation, d, is identical to that printed in the display, otherwise it
is that printed in the display normalized to have unit Euclidean length.
With the normalization on, tangential derivatives are the standard
directional derivatives as they are defined in Calculus Textbooks. With
the normalization off, they are the more versatile Gateaux derivatives as
they are used frequently in multivariate interpolation and approximation.
The NRML flag applies similarly to the cross direction.
Following the NRML flag is the word "current". This indicates
that the numerical display corresponds to the graphics display. To explain
why this might not be so requires a brief excursion into the Organization
of MICROSCOPE. When drastic parameter changes are taking place it is often
more efficient to accumulate them before generating the new display.
Usually in that case the alphanumerical display is scrolled while commands
are being given, and vanishes from the screen. However, sometimes it is
preferable to view the old display rather than the commands given so far.
In that case, the display can be recalled on the screen, and the word
"current" is replaced by "GO pndg" (i.e. the GO command needs to be given
to incorporate into the graphical display changes of parameters that may
already be shown in the numerical display).
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 19
The last item in the last line is "CALLS = 174" where 174 is the
number of evaluations of the trial function expended up to this point in
the MICROSCOPE session. This allows monitoring the computational effort
spent. A typical multivariate surface will be expensive to evaluate, and
its cost will usually dominate the cost of running MICROSCOPE and thereby
determine how long you have to wait for a reponse from the terminal.
Therefore, a significant effort has been made to keep the number of
necessary function evaluations small. This feature contributes in no small
measure to the complexity of the MICROSCOPE source code. The example
display has 75 columns and we are examining a first order cross derivative
of the trial function. Each evaluation of the cross derivative requires
two evaluations of the trial function. Displaying the graph of the cross
derivative alone would therefore require a value of CALLS equal to
2*75 = 150. Note that computing and displaying the second and third
tangential derivatives of the cross derivative requires an additional
effort of only 16%. This is not a trivial accomplishment as may be
suggested by the fact that the same display with the slightly different
window width of 10 would require 330 evaluations. For a detailed
discussion of the numerical effort associated with various window widths
see section 4.3.
Notice that in the above example we identified a step
discontinuity in a mixed partial derivative of fourth order (a third order
tangential derivative of a first order cross derivative) without fuss or
doubt. If this was a real investigation we could state now that the
underlying scheme in any case is at most three times differentiable.
1.4 Summary
With MICROSCOPE, one answers questions like those in section 1.1
for specific examples. Thus only negative results (e.g. that a scheme is
not differentiable) can be proved strictly. However, positive results can
also be obtained. If the examples are chosen carefully to exclude
artifacts and make them representative, our program allows for answers to
the above questions with a degree of confidence that borders on certainty!
Naturally, a computational approach does not replace a rigorous proof.
However, in the authors experience it is often possible to confirm (or
shatter) fuzzy notions with MICROSCOPE, and to generate further insight
into the phenomena under consideration. Encouraging MICROSCOPE results may
help in building up the stamina needed for attempting a rigorous proof, and
detailed investigations may help in providing ideas for carrying out the
analysis.
Often, of course, MICROSCOPE will be employed in hindsight, when
the analyst already knows the answers to questions like the above from
theoretical reasoning. The examination with MICROSCOPE then serves to
corroborate the reasoning, and also, more mundanely but no less
importantly, to eliminate bugs in the implementation.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 20
In the authors experience, MICROSCOPE proved invaluable as a
debugging tool, often pinpointing within narrow limits that part of a code
that was at fault, but also sometimes uncovering flaws in the theoretical
approach, and, on a few exhilarating occasions, leading to the discovery of
new and unlooked for results that could then be proved theoretically. Thus
MICROSCOPE is useful in the identification as well as in the verification
of results.
In the development of MICROSCOPE we were occasionally confronted
with the objection that our alphanumerical displays are rather crude. Our
primary response to this is that the very purpose of MICROSCOPE consists of
translating very subtle properties into displays that make them glaringly
apparent. A large part of our interest centers around step functions, and
it seems fair to display them as such. Indeed, any curve fitting might
disguise the very features in which we are interested. However, we do
provide an interface to a sophisticated graphics package. See sections
4.14 and 5 for more information.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 21
2. Applications
In this section, we illustrate the main applications of
MICROSCOPE. Consider the simple example of interpolating to a primitive
function p by a cubic spline S, say, according to the conditions:
2 3
S(x) = a0 + a1*x + a2*x + a3*x if x < 1
and
2 3
S(x) = b0 + b1*x + b2*x + b3*x otherwise
where the coefficients are defined by
S(0) = p(0), S(1) = p(1), S(2) = p(2), S'(2) = p'(2), S"(0) = 0
and S is twice differentiable at x = 1. Thus we require interpolation to
'position' (i.e. function values) at 0,1,2, and interpolation to the first
derivative at 2 by a piecewise cubic function that is twice continuously
differentiable. At 0 we impose additionally a "free end" condition.
(There is an extensive literature on Splines, see e.g. de Boor, 1978, or
Schumaker, 1981)
For our present purposes we assume that p(x) is the exponential,
i.e.
x
p(x) = e
where e is the base of the natural logarithm. This function appears to be
sufficiently non-polynomial to avoid the introduction of artifacts.
A simple calculation shows that
a0 = 1
2
a1 = (-2*e +12*e-9)/7
a2 = 0
2
a3 = (2*e -5*e+2)/7
and
2
b0 = (5*e -16*e+12)/7
2
b1 = (-17*e +60*e-24)/7
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 22
2
b2 = (15*e -48*e+15)/7
2
b3 = (-3*e +11*e-3)/7
In the following examples we will verify that S with the above
coefficients does indeed possess the required properties.
2.1 Interpolation
The basic idea of verifying interpolation is to plot the
difference between the primitive function and its interpolant, and to check
that the error and its appropriate derivatives are indeed zero. So we
incorporate the exponential and S as defined above into MICROSCOPE (see
section 3.1 for details), and start an interactive session. After giving
the appropriate commands we obtain the following picture illustrating the
situation at x = 0:
... I : I
... I : I
.... I : I
... I : I
... I : I
.... I : I
.... I : I
.... I + I
..... : I
I ..... I
I :.....I
I : ......
I : I .......
I : I .........
I : I ..........
===========================================================================
Point = 0.000000D+00 s = 5.0000D-03
Direction = 1.000000D+00 h = 3.0000D-02
F0 (-3.33D-02, 6.76D-02)
I/O: 25 6 6 1 2 3 27 28 29 30 NRML on current CALLS = 249
Figure 2 Interpolation to Position at Zero
We notice that the range of the error is from -0.0333 to 0.0676.
Since the graph of the error function passes through the vertical axis
slightly below the center, this range is consistent with the error being
zero, i.e. with interpolation. A more reliable confirmation can be
obtained by asking for the value of the displayed function at the center
(i.e. 0) of the display. This procedure, which is not shown here, yields
a value of -9.4D-11, which is zero up to round-off errors. (If you try to
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 23
reproduce this example with your own version of MICROSCOPE you may get a
slightly different result because of the random nature of round-off
errors.) An alternative approach consists of halving the interval, and
thereby decreasing the range covered by the graph. In the numerical
display the range should shrink correspondingly, maintaining the value of
zero in its interior. Halving the interval yields the display:
.... I : I
.... I : I
.... I : I
.... I : I
.... I : I
..... I : I
.... I : I
..... + I
I ..... I
I : .....
I : I.....
I : I ......
I : I ......
I : I .......
I : I .......
===========================================================================
Point = 0.000000D+00 s = 2.5000D-03
Direction = 1.000000D+00 h = 1.5000D-02
F0 (-2.03D-02, 2.88D-02)
I/O: 25 6 6 1 2 3 27 28 29 30 NRML on current CALLS = 287
Figure 3
The range of the error has been reduced to from -0.0203 to 0.0288
and a zero value of the error at the point of examination is still
consistent with the graphical display. This procedure can be continued.
Proceeding similarly at the point x = 1 yields a similar result, and you
might try this for yourself. At the point x = 2, however, we obtain the
display:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 24
I : I ..
I : I .
I : I .
I : I .
I : I ..
I : I .
I : I ..
.. I + I .
... I : I ..
.... I : I ..
... I : I ..
.... I : I ...
..... I : I ...
..... I : I ....
......................
===========================================================================
Point = 2.000000D+00 s = 5.0000D-03
Direction = 1.000000D+00 h = 3.0000D-02
F0 ( 2.64D-11, 1.42D-02)
I/O: 25 6 6 1 2 3 27 28 29 30 NRML on current CALLS = 362
Figure 4 Interpolation at x = 2
The graph of the error function touches, but does not cross, the
x axis, which is consistent with our interpolating to the first derivative
as well as position. The value of the error at the center can be read
directly from the numerical display and is -2.4D-11 (you may get a slightly
different value because of the random round-off errors). This confirms
interpolation to position. Interpolation to the derivative is likely
because the graph appears to have a horizontal tangent. However,
confidence can be raised by computing the first derivative of the error as
well, yielding the display:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 25
I : I 111
I : I 11.
I : I 111 .
I : I 111 .
I : I 111 ..
I : I 111 .
I : I 111 ..
.. I + I 1111 .
... I : I 1111 ..
.... I : 1111 ..
... I :11111I ..
.... I111111 I ...
..... 1111111 : I ...
111111111. I : I ....
1111111111111111 ......................
===========================================================================
Point = 2.000000D+00 s = 5.0000D-03
Direction = 1.000000D+00 h = 3.0000D-02
F0 ( 2.64D-11, 1.42D-02) F1 (-6.11D-02, 1.77D-01)
I/O: 25 6 6 1 2 3 27 28 29 30 NRML on current CALLS = 374
Figure 5 Interpolation to the First Derivative
Inquiring about the value of the first derivative at the center
yields the number 5D-4. This is much larger than the value for the error
itself, and hence might cause some doubts. However, the derivative is
calculated numerically and hence not exactly. The value reported by
MICROSCOPE is actually the truncation error. This can be confirmed by
halving the discretization parameter yielding the display:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 26
I : I 1111
I : I 111 .
I : I 1111 .
I : I 111 ..
.. I : I 1111 .
.. I : I 1111 .
.. I : I 11111 ..
.. I + I1111 ..
... I : 11111 ..
.. I 11111 I ..
... 11111 : I ..
... 111111 I : I ..
1111111 I : I ....
11111111 .... I : I ....
11111111 ....................
===========================================================================
Point = 2.000000D+00 s = 2.5000D-03
Direction = 1.000000D+00 h = 1.5000D-02
F0 ( 2.64D-11, 3.03D-03) F1 (-4.21D-02, 7.09D-02)
I/O: 25 6 6 1 2 3 27 28 29 30 NRML on current CALLS = 418
Figure 6
Now the value of the derivative is 1.25D-4 which is a reduction
from the previous value by a factor 4. This is consistent with the
assumption that the reported error actually is the truncation error since
it decreases like the square of the discretization parameter h (see the
discussion in Appendix I).
The free end condition (S"(2) = 0) can be considered an
interpolation condition. However, it cannot be checked by examining the
error function, since the value of p"(2) is (in general) unknown. So one
has to consider the interpolant itself. We obtain the display:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 27
11 I : I 22222
1 I : I 22222 1
1 I : I 22222 1
11 I : I 22222 11
1 I : I 22222 1
11 I : I 22222 11
11 I : 22222 11
1 I 22+22 I 1
11 22222 : I 11
11 22222 I : I 11
111 22222 I : I 111
222221 I : I 11
22222 111 I : I 111
22222 1111 I : I 1111
22222 1111111111111111111
===========================================================================
Point = 0.000000D+00 s = 2.5000D-03
Direction = 1.000000D+00 h = 1.5000D-02
F0 ( 8.83D-01, 1.12D+00) F1 ( 1.26D+00, 1.27D+00) F2 (-2.53D-01, 2.53D-01)
I/O: 25 6 6 1 2 3 27 28 29 30 NRML on current CALLS = 505
Figure 7 Free End Condition
That the second derivative is indeed zero can now be verified
using the same techniques as above (i.e. inquiring directly or using a
decreased discretization parameter).
2.2 Smoothness
In the present example, the only critical set is the one
containing the point x = 1 (assuming we are confident that S has been
programmed to be polynomial elsewhere). When investigating smoothness it
is usually preferable to examine an error (if a primitive function is
available) rather than the interpolant itself. In the present example, the
error function at the single critical point yields the display:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 28
I : 3333333333333333333333333333333333
I : 3 I .2222
I : 3 I 12222
I : I 112222
I : I .112222
I :3 I .1112222
I : 11122222
I +1112222
I111112222 I
.1111222223: I
1111222222I : I
112222222 I : I
1222222 I 3 : I
1222222 I 3 : I
3333333333333333333333333333333333 : I
===========================================================================
Point = 1.000000D+00 s = 2.5000D-03
Direction = 1.000000D+00 h = 1.5000D-02
F0 ( 2.49D+00, 2.97D+00) F1 ( 2.39D+00, 2.90D+00) F2 ( 2.48D+00, 3.11D+00)
F3 ( 2.73D+00, 4.06D+00)
I/O: 25 6 6 1 2 3 27 28 29 30 NRML on current CALLS = 592
Figure 8 Soothness at x = 1
This display can be interpreted similarly as figure 1, and
clearly shows a function that is twice but not three times differentiable.
This is what we expect from our construction.
2.3 Degree of Precision
The 'precision class' of an approximation scheme is the set of
functions that are reproduced exactly by the scheme. Of particular
interest are the polynomials in the precision class. The largest number q,
say, such that all polynomials of degree up to q are in the precision class
is the degree of polynomial precision.
To decide if any particular function is in the precision class
one approximates that function and investigates if the error is zero within
round-off effects. This applies even to non-polynomial functions.
To answer the more narrow question of polynomial precision it is
possible to proceed more systematically and to examine polynomials of
increasing degree. The polynomials should be as general as possible, i.e.
they should be unlikely to introduce misleading artifacts. One possibility
is to choose the coefficients randomly. If really in doubt, one might
examine each polynomial in a suitable basis of the appropriate polynomial
space. The points of examination should be chosen such that each
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 29
appropriate part of the domain is covered. For instance, in the present
example (which, incidentally, has linear precision) one should choose one
point > 1, and one < 1, covering both cases in the definition of S.
The following figure illustrates the typical appearance of a zero
error.
. . I .: I .
. . I : I . .
. . I : I ..
. : . I . . .
.I . : I .
I . : I . . .
. . . I . . I.
. . I : I . .
. I : . I . . .
. I :. . I .
. I : I . . .
I : .
. . . . . . . . . I : .I . .
. . I. : I . . . ..
. . . . . I : I . .
===========================================================================
Point = 2.342300D-01 s = 2.5000D-03
Direction = 1.000000D+00 h = 1.5000D-02
F0 (-9.99D-11, 9.78D-11)
I/O: 25 6 6 1 2 3 27 28 29 30 NRML on current CALLS = 667
Figure 9 A Zero Error
Notice how the values of the function are spread randomly over
the entire display, accurately pinpointing the round-off nature of the
error. Sometimes, however, one has a situation in which the coefficients
of the interpolant are evaluated inaccurately, leading to an error that has
a scattered distribution with a bias like that in the following display:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 30
I : I . . .
I : I . .
I : I . .. .
I : I . . .
I : .I .. . .. . .
I ..: I . . . .
. .I : .. . . . .
. I . :. . I .
.. .. : . I . .
.. . . I . : . I
. .. . . . I . I
. . .. .. I : I
. . .. I : I
. . . I : I
. . . I : I
===========================================================================
Point = 2.342300D-01 s = 2.5000D-03
Direction = 1.000000D+00 h = 1.5000D-02
F0 ( 2.16D-10, 7.31D-10)
I/O: 25 6 6 1 2 3 27 28 29 30 NRML on current CALLS = 742
Figure 10 Round-off With a Bias
2.4 Degree of a Polynomial
Verifying that a certain function is a polynomial of a certain
degree is often useful when examining cross derivatives. For example, in
the bivariate Clough-Tocher scheme (Alfeld, 1984) which is piecewise cubic
on a triangle, differentiability between triangles is forced by the
requirement that the normal derivatives across edges be linear (rather than
quadratic) along the edges. In this illustration, let us verify that our
interpolant is indeed cubic. This can be accomplished by showing that the
third derivative is constant or that the fourth derivative is zero (within
round-off). One obtains for example the following display:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 31
3 I : I 22222
3 3 I : I 222221
3 I : I 3 3 32222211
3I : I 22222111
3 3 3 I 3 : 3 3I3 222221111 3 3
3 33 3 3 : 3 32322 1111 3 3
3 3 3 3 3 I : 22222. 11113 3 3
33 3 3 I3 23222. 11113 3 3 3 3 3 3 3
22222. 11111 I 3
3 3 3 3 3 22222.I111113 3 3 I 3
22223. 11111 33 :3 I 3 3 3 3 3
3 22223 311111 I : I
22222111111 I : I 3 3 3
222221111 I : I
2222211 3 I : I
===========================================================================
Point = 2.342300D-01 s = 2.5000D-03
Direction = 1.000000D+00 h = 1.5000D-02
F0 ( 1.18D+00, 1.43D+00) F1 ( 1.29D+00, 1.41D+00) F2 ( 3.87D-01, 8.92D-01)
F3 ( 2.73D+00, 2.73D+00)
I/O: 25 6 6 1 2 3 27 28 29 30 NRML on current CALLS = 841
Figure 11 A Cubic Function
Notice that the third derivative is clearly affected by
round-off, without any apparent bias. On the other hand, since the
displayed limits of its range are identical the size of the range is less
than 1/273th of the derivative value. A closer examination reveals that
the third derivative ranges from 2.7306 to 2.7323. A telltale sign of
round-off errors is that they can be decreased even further by increasing
the discretization parameter. Doubling h yields the display:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 32
I : I 3 322222
3 3 I 3 : I 2222231
33 I : I 22222 111
3 3 33 I : 333I 233223 111
3 3 3 3 : I 3 22322. 31111
3 3 I : I 22222. 111 3
3 3 3 3 I 3 33322222.33 33 1111 3 3 3 3
3 3 3 I 22222. I 1111 3 3
3 33 3 32322. : I1111 3 3 3
22222.I 3 : 1111333 3 3
3 322222. I 11311 I 3
3 3 22222. 11111 : I 3 3
22222. 1111111 I : I
23222. 111111111 I : I
2222211111111 3 I : I 3
===========================================================================
Point = 2.342300D-01 s = 5.0000D-03
Direction = 1.000000D+00 h = 3.0000D-02
F0 ( 1.06D+00, 1.56D+00) F1 ( 1.27D+00, 1.50D+00) F2 ( 1.34D-01, 1.15D+00)
F3 ( 2.73D+00, 2.73D+00)
I/O: 25 6 6 1 2 3 27 28 29 30 NRML on current CALLS = 885
Figure 12
The range is now from 2.731361 to 2.731569 which is a reduction
by a factor 8. This is consistent with the third derivative being constant
since the round-off errors in the third derivative increase like the third
power of h, see Appendix I. Further increases do not alter the basic
scattered structure of the third derivative (until the range covered in the
display includes the point x = 1, where the third derivative has a step
discontinuity). Increasing the value of h should eventually decrease the
round-off errors below a level at which the true change in the derivative
becomes visible in the display if the third derivative was not constant.
The fact that this does not happen indicates that there is no such change.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 33
3. Using MICROSCOPE
This section contains a description of the master MICROSCOPE
routine MCRSCP and a first introduction to the command language of
MICROSCOPE. For a more detailed description consult section 4.
During an interactive session the terminal screen may be either
'active' or 'inactive'. On an active screen, changes in parameters are
incorporated immediately into the alphanumerical display which is
maintained on the screen. On an inactive screen, commands are accumulated
and carried out only when the screen is activated by either the command GO
or FORCE. An inactive screen shows the last few commands and the
corresponding data whereas an active screen shows at most the current
command and its data, in addition to the alphanumerical display. (However,
an incomplete alphanumerical display may be called on an inactive screen,
see the description of the command RSCREEN in section 4.5 and the
discussion of the alphanumerical display in section 1.3.)
A second distinction applies to the way in which screen changes
are implemented on an active screen. In one version of MICROSCOPE, which
we call the 'screen version', the alphanumerical display is kept stationary
and only modified rather than replaced. This version requires you or your
system manager to provide four simple screen editing routines that are
specific to each computer/terminal combination (see chapter 5.) If this
facility is not available, or if you use a hardcopy terminal, then the
'scroll version' of MICROSCOPE may be employed, which scrolls the old
screen and replaces it by a new version after each change.
The screen version is much preferable, and in this manual we
usually assume it is available.
Another distinction pertains to the utilization of lower case
letters. The use of lower case letters in a FORTRAN program is not
portable, but now most installations do provide this facility. In this
manual we assume lower case letters are available. If so, any lower case
letters occuring in the input to the program are treated as if they were
upper case. Thus commands can be given in lower case letters. Output from
MICROSCOPE consists of a mixture of lower and upper case letters. However,
to maintain portability we provide a version of MICROSCOPE that works
solely with upper case letters. Section 5 contains more information.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 34
3.1 The MCRSCP Routine
To put MICROSCOPE into operation, you will have have to write a
FORTRAN main program that calls the routine
MCRSCP(F, INPUT, OUTPUT, GRAPHC, HELP, RECORD, RESTART,
X LINES, WIDTH, PLOT, NUMRCL, PROMPT)
The first parameter is the name of a DOUBLE PRECISION function
(the trial function). It has the form
DOUBLE PRECISION FUNCTION F(X)
DOUBLE PRECISION X
DIMENSION X(1)
.
.
.
F = ...
RETURN
END
and must be declared EXTERNAL in the calling program. MICROSCOPE will set
at most the first three components of X before calling F (i.e. it can
handle only functions of up to three variables). Notice that F does not
have any parameters other than the values of the independent variables.
Usually, however, it will also depend on parameters describing e.g. the
domain data structure. These are most conveniently passed to F in a common
block.
All of the remaining parameters of MCRSCP are input integers.
MCRSCP will not change them, and they may therefore be integer values
rather than variables. The first six of them (i.e. INPUT, OUTPUT, GRAPHC,
HELP, RECORD, RESTART) are FORTRAN device numbers corresponding to the
first six I/O numbers listed in the numerical display (see section 1.3).
It is your responsibility to set up the necessary correspondences between
device numbers and physical devices or disk files.
The last five parameters describe the physical dimensions of the
screen display. Their meaning is as follows: (Numbers in parentheses give
the value used for the examples in this manual, square brackets contain the
allowable ranges for the parameters).
LINES (24): The total number of lines used on the terminal
screen [10 < LINES < 58].
WIDTH (75): The number of columns used for the graphical display
[0 < WIDTH < 136]. Usually one will choose as large a value as possible
for maximum resolution. However, small values may sometimes be preferable
for increased speed. For a symmetrically placed vertical axis, the value
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 35
of WIDTH should be odd. The numerical display occupies up to 75 columns.
This is therefore the minimum physical width of the output device. WIDTH
is also the number of points at which derivatives are sampled in the
<PLOT79> display.
PLOT (15): The number of lines allocated to the graphical
display [0 < PLOT < 49]. Again, a value of PLOT as large as possible
should be supplied. The numerical effort is independent of PLOT.
NUMRCL (8): The number of lines above the screen bottom at which
the numerical display begins. [5 + PROMPT < NUMRCL < LINES - PLOT].
PROMPT (2): The number of lines above the screen bottom at which
the MICROSCOPE prompts are given. [1 < PROMPT < LINES - PLOT - 6].
Usually, MCRSCP will be called only once during a session.
However, it is possible to call your own routines from MCRSCP. You have to
supply a subroutine SUBUSR without any parameters which can be called by
MCRSCP A detailed discussion of this procedure is given in section 4.15 on
user intervention. For now we may assume that SUBUSR is a dummy routine
SUBROUTINE SUBUSR
RETURN
END
(which is supplied with the MICROSCOPE package and which you can of course
write yourself).
When called, MCRSCP first prints an identifying stencil on the
OUTPUT device, containing its version number and the last modification
date. Next, it checks to see if the screen parameters satisfy the
restrictions listed above. If not then MCRSCP exits, rather than
terminates, in order to give you a chance to correct your mistake.
On the first call to MCRSCP it computes the machine round-off
unit (see section 4.1) and prints its common logarithm to give you an idea
of how many decimal digits are carried on the computer. The actual number
relevant to your investigation may of course be less depending on the
details of the arithmetic involved in evaluating the trial function.
Also on the first call only, MCRSCP then prints the first line of
any available news (see section 4.10) to help you decide if you wish to
read them.
After setting some defaults MCRSCP then enters the command mode,
in which it accepts, interprets and carries out the commands entered on the
Input Device.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 36
3.2 The Command Mode
The MICROSCOPE prompt is of the form:
n >>
where n is the number of the command to be given. It starts at 1 and is
incremented by 1 after each command.
A MICROSCOPE command is a string of at least two letters and
digits. However, only the first two characters are significant and
recognized by the parser. They must be entered in the first two columns
immediately following the prompt. On an active screen, this will be in the
same line as the prompt, on an inactive screen, or in the scrolling
version, on the next line. Blank spaces are significant. (MICROSCOPE
simply reads the command using a 2A1 format. The remaining columns can be
filled with arbitrary characters. They are not read or processed by
MICROSCOPE. This is sometimes useful for including comments in a program.
A command line is terminated by pressing the RETURN button.
After a command you will be prompted for the appropriate data (if
any are needed). The prompt describes what data are needed and is specific
to the command. There are seven types of data requests:
-1- No Data
-2- A command name (to obtain help on the requested command)
-3- 1 Integer
-4- 2 Integers
-5- 1 Double Precision variable
-6- 1 Double Precision Vector
-7- 2 Double Precision Vectors
The dimension of the vectors is identical to the number of
variables of the trial function. (The default is 2, see section 4.18 for a
listing of all defaults). Numbers are processed by an internal parser and
are essentially format free. The following distinctions and restrictions
apply, however:
- Blanks are ignored. For example "1 2" is considered the single
integer 12.
- Numbers on the same line must be separated by commas.
- Numbers can also be separated by writing them on separate lines
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 37
(omiting the comma).
- If two vectors are requested the second vector must start on a
new line .
- An empty string (obtained by just pushing the RETURN button) is
considered to mean zero whenever a single number (Integer or Real) is
requested. This is particularly useful when a zero entry implies that a
submode (see the descriptions of the PLOT and the USER commands in sections
4.14 and 4.15 respectively) is to be exited.
- When two integers are requested either one or both can be
represented by an empty string. (The only command for which this is useful
is TYPE.)
- Floating Point numbers (like 1.0E-12 or 1D-4) are processed
correctly.
- Floating Point numbers may also be represented as integers.
After receiving the data, MICROSCOPE processes the command and
the data. If the screen is active the command is carried out and the
alphanumerical display is updated, otherwise the next command is prompted
for.
If a command is not recognized a message is printed to that
effect and a new command is prompted for. If data are requested, and a
number is not recognized, then new data will be requested. However, if
data are syntactically correct, but otherwise meaningless (e.g. you
request the graph of the 117th derivative), then the command will be
aborted and a new command will be prompted for.
There are currently 63 commands which divide into 11 major groups
corresponding to sections 4.5 through 4.15. Rather than describing the
groups in this section, we illustrate some of them in the course of a
search for a point of reduced smoothness (the 'active point'), giving the
names of the commands used, and further illustrating the capabilities of
MICROSCOPE.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 38
3.3 A Discontinuity in the second derivative
The trial function underlying this example is
2 x x
f(x) = (eta*x )/2 + e if x > 0 and f(x) = e otherwise
(where eta = -0.005). Clearly f is once but not twice differentiable at
the origin. The exponential term serves to conceal the discontinuity in
the second derivative. (Similar functions that are 0 through 5 times
differentiable can also be examined for arbitrary values of eta.) The
parameters are changed by giving the user intervention command USER
(assuming the test package has been loaded, see appendix III and section
5).
Let us pretend we do not know the the degree of smoothness but we
wish to examine it. We first inform MICROSCOPE that the trial function has
one independent variable, using the DMNSN command. Then we enter a search
interval (here [-1,3], say) using the IINTVL command. At this stage, the
screen is still inactive. This is changed, and the first display is
obtained, by giving the GO command. The display will remain active
throughout the remainder of this section. We obtain:
I : I ..
I : I .
I : I ..
I : I .
I : I ..
I : I ..
I : I ..
I : I ..
I : I ...
I : I ...
I : I ....
I : I .....
I : .......
............ I
........................... I : I
===========================================================================
Point = 1.000000D+00 s = 5.4054D-02
Direction = 4.000000D+00 h = 3.2432D-01
F0 ( 3.68D-01, 2.01D+01)
I/O: 25 6 6 1 2 3 27 28 29 30 NRML on current CALLS = 960
Figure 13
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 39
This display shows no lack of smoothness at all. So, not knowing
about the nature of the suspected discontinuity at all we turn on the
horizontal scale (using DSCALE) to locate any phenomena of interest and
activate (using DGRAPH) the graph of the highest possible (i.e. sixth)
derivative. These changes are incorporated one by one (not shown here)
into an active screen. The final result is this:
6 I : I ..
I : I .
6 I : I ..
I : I .
6 I : I ..
I : I .666666666
6 6 I : I 666666666666666666
66666666666666321.987654666666666666666666666666123456789.123..6789.1234567
6 6 I : I ...
I : I ...
6 I : I ....
I : I .....
6 I : .......
............ I
....................6...... I : I
===========================================================================
Point = 1.000000D+00 s = 5.4054D-02
Direction = 4.000000D+00 h = 3.2432D-01
F0 ( 3.68D-01, 2.01D+01) F6 (-6.07D+01, 6.27D+01)
I/O: 25 6 6 1 2 3 27 28 29 30 NRML on current CALLS = 972
Figure 14
Now it is clear that there is a phenomenon of some kind occuring
at the point -17 measured on the horizontal scale. To examine this
further, we shift the graph by 17 (using SHIFT). This centers the
interesting parts of the graph, yielding:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 40
I 6 : I ..
I : I .
I 6 : I ..
I : I .
I : 6 I ..
I : I ..
I 6 : 6 I ..6666666666
6666666666666666666666666666666654321.12366666666666666666666666689.1234567
6 :6 I ...
I : I ...
I6 : I ....
I : I .....
I 6: .......
............ I
........................... I 6 I
===========================================================================
Point = 8.108108D-02 s = 5.4054D-02
Direction = 4.000000D+00 h = 3.2432D-01
F0 ( 1.47D-01, 8.00D+00) F6 (-6.07D+01, 6.27D+01)
I/O: 25 6 6 1 2 3 27 28 29 30 NRML on current CALLS = 989
Figure 15
Already at this stage it is possible to tell the nature of the
reduced smoothness. The sixth derivative clearly shows four distinct local
extrema. This qualifies it as the third (numerical) derivative of a delta
function. The second derivative of the trial function must therefore be a
step function. The trial function is hence once but not twice
differentiable. Derivatives of step functions are discussed in detail in
section 4.2 and in appendix I.
To confirm our diagnosis directly we examine the second
derivative. Turning on its graph and erasing the distracting graph of the
trial function (using EGRAPH), however, yields the following display which
shows no lack of smoothness at all in the second derivative:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 41
I 6 : I 22
I : I 2
I 6 : I 22
I : I 2
I : 6 I 22
I : I 22
I 6 : 6 I 226666666666
6666666666666666666666666666666654321.12366666666666666666666666689.1234567
6 :6 I 222
I : I 222
I6 : I 2222
I : I 22222
I 6: 2222222
222222222222 I
222222222222222222222222222 I 6 I
===========================================================================
Point = 8.108108D-02 s = 5.4054D-02
Direction = 4.000000D+00 h = 3.2432D-01
F2 ( 1.48D-01, 8.08D+00) F6 (-6.07D+01, 6.27D+01)
I/O: 25 6 6 1 2 3 27 28 29 30 NRML on current CALLS = 989
Figure 16
The second derivative appears smooth because the discretization
is too crude to show its discontinuity. So we keep decreasing the value of
the discretization parameter (and simultaneously the interval, keeping the
window width fixed). Since the active point is only approximately in the
center of the display, this will cause it to drift to one side which has to
be counteracted by more shift commands. Because of the organization of
MICROSCOPE it is particularly efficient and convenient to divide (or
multiply) with powers of two. Halving h four times (using HALVE) followed
by a shift of 24 and four more halvings yields, after erasing the scale and
the graph of the sixth derivative:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 42
I : I 2222
I : I 2222
I : I 222
I : I 222
I : I 222
222222 : I 222
22 I 2: I 2222
2 222 I 2 I 222
22 2 I :22 I 222
2222 I : 22222
2222 I : I
22 I : I
2222 I : I
222 I : I
2222 I : I
===========================================================================
Point = -1.040834D-17 s = 2.1115D-04
Direction = 4.000000D+00 h = 1.2669D-03
F2 ( 9.92D-01, 1.00D+00)
I/O: 25 6 6 1 2 3 27 28 29 30 NRML on current CALLS = 1353
Figure 17
Now it is clear that there is a jump discontinuity in the second
derivative of the trial function. The location of the active point can be
seen to be 0 +/- about s = 0.0002. Even the size of the jump can be
estimated. Inquiring (using TYPE) for the values of the derivative at the
left and right end points of the window (where in this crude display it
assumes its extrema) yields values of 9.98846D-1 and 9.96522D-1
respectively, corresponding to a jump of -2.32D-3. The real jump is of
course eta = -5.0D-3. The discrepancy is due to a truncation error, not
just in approximating the derivative, but also in picking the points of
evaluation. The estimate can be improved by extrapolating (linearly) the
left and right parts of the second derivative to the point of examination,
using an estimate of the third derivative which can be obtained from one of
the earlier displays. With the third derivative set equal to 1, this
procedure yields an estimate of -2.32D-3 - 12s = -4.86D-3 which is a very
good result considering the difficulties of numerical differentiation.
In the present setting, it is not possible to decrease the
discretization parameter further without succumbing to round-off errors in
the second derivative (you may wish to try this yourself). However, the
above example illustrates how to alleviate the limitations of round-off
errors. We already recognized the true nature of the reduced smoothness
when examining the sixth rather than the second derivative. With the
accuracy used for the examples in this manual (10 digits), steps in the
second derivative 300 times as small as the one above can be detected. The
following display shows the graph of the sixth derivative for the same
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 43
function as above with eta = -1.7D-5 and h = 0.24. These values have been
taken from table 4 in appendix I (tau = 1.0D-10).
I : I 66
I : I 66
I : I 66
I : I 66
I : I 66
I : I 666
I : I 66
I : I 666
I : I 6666
I 6: I 6666
I 6 : 6 I 6666
I : 6 666
666 6 I
666666666666 I 66 : 6 I
666666666666666666 I :6 I
===========================================================================
Point = -1.040834D-17 s = 4.0000D-02
Direction = 4.000000D+00 h = 2.4000D-01
F6 ( 2.23D-01, 4.40D+00)
I/O: 25 6 6 1 2 3 27 28 29 30 NRML on current CALLS = 1440
Figure 18 At the limits of resolution
The sixth derivative shows 4 extrema, identifying the step in the
second derivative.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 44
4. A Detailed Description of Features and Commands
4.1 Round-Off Effects
The fact that all arithmetic operations are carried out with
finite accuracy is fundamental and all pervasive in working with
MICROSCOPE. Numerical Differentiation is particularly sensitive to
rounding since it involves the subtraction of nearly equal quantities,
leading to a cancellation of significant digits. Consequentially, the
error in numerical differentiation does not decrease indefinitely with the
discretization parameter, but rather starts to behave erratically beyond a
certain minimum value and tends to increase as the discretization parameter
is decreased further. However, it is its very randomness that makes
round-off errors readily apparent in a MICROSCOPE display so that you will
not be misled into accepting meaningless results at their face value.
The precise effect of rounded arithmetic depends on several
factors:
-1- The number of significant (binary) digits carried by a
particular computer. Most users will prefer an approximation of the
equivalent number of decimal digits, and so MICROSCOPE computes that number
and prints it out at the beginning of a session. More precisely,
MICROSCOPE computes the smallest power of 2, tau, say, such that within the
given computer arithmetic the quantity 1 + tau satisfies a test for greater
than 1, and then prints the rounded value of the negative common logarithm
of tau.
-2- The accuracy with which the trial function is being
evaluated. That accuracy will always be less than or at most equal to the
accuracy indicated in -1-. Usually it will be much less, for any of the
following reasons:
-2.1- Typical multivariate interpolants are very complicated
algebraically, involving many operations which tend to increase round-off
errors.
-2.2- If, as is frequently the case, the trial function is the
difference between a primitive function and its interpolant, then a
cancellation of significant digits occurs.
-2.3- If the display function is a cross derivative of the trial
function it is already contaminated by round-off errors due to numerical
differentiation, thus exhibiting effects that one would normally expect to
occur only for higher order tangential derivatives.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 45
-3- The precise nature of round-off effects is also very
sensitive to the relative magnitudes of various tangential derivatives
involved. For example, a constant function will exhibit round-off errors
at any value of the discretization parameter.
Of the above three sources of round-off errors, you will usually
have a firm knowledge only of the first. Knowledge about the other two may
be gained by using MICROSCOPE itself. This is usually more effective than
attempting a theoretical analysis.
The precise nature of round-off errors is discussed in Appendix
I. Detailed formulas and tables are given. However, the usual approach
will be to experiment with various values of the discretization parameter,
choosing it as small as possible without generating the telltale random
pattern that indicates round-off errors.
4.2 Derivatives of Step Functions
One perhaps surprising feature of numerical differentiation is
that doubtful discontinuities in some derivatives can often be displayed
very clearly by plotting higher order derivatives, even though a large
discretization parameter is necessary to keep them free of round-off
errors. This fact becomes quickly apparent empirically, and is analyzed in
Appendix I.
The conceptual basis for the approach is provided by a Dirac
Delta function (see Courant and Hilbert, 1962, p.456). A step function can
be thought of as rising at the discontinuity with infinite slope from one
discrete value to another. If it is piecewise constant its derivative can
be thought of as a function that is zero everywhere, except at the
discontinuity, where it is infinite such that the integral over the
derivative equals the jump in the step function. That derivative is a
delta function.
More precisely, a delta function about a point P, say, is defined
as the limit of a sequence of non-negative functions that are different
from zero only in neighborhoods of P. The diameter of those neighborhoods
tends to zero, and the value of the integral over the functions remains
constant (at 1 in the classical definition).
Derivatives of a delta function can be visualized by considering
a function in the above sequence. Suppose it is bell shaped. Then its
derivative will have two local extrema of opposite signs. The second
derivative will have three extrema, etc. Taken to the limit, the k-th
derivative of a step function has k local extrema of alternating signs, all
located at the same point.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 46
Numerical derivatives of a step function exhibit the same
phenomena, if the discontinuity is smeared out over a window containing
sufficiently many points. This is illustrated in the pictures below. The
step discontinuities occur in the m-th derivatives, where m = 0,1, ..., 5.
The discretization parameter is h = 1, and the size of the jump is 1. More
precisely, the m-th function is defined by
m
s(x) = 0 if x < 0 and x /m! otherwise,
where the exclamation mark denotes the factorial.
Notice that as the degree of the discontinuous derivative
increases, the smoothness of the graphs of higher derivatives increases.
This is consistent with the interpretation of numerical differentiation as
a smoothing process. (The numerical derivatives have the same degree of
smoothness as that of the display function, just the number of the points
of reduced smoothness is increased.)
1111111..........5555555555444444444466666666665555555555222222222211111111
I : I
I : I
I 6666666666 : I
I : 444444444444444I
I : I
I : 6666666666I
6666666I : 66666666
55555556666666666 : 55555555
444444444444444 : I
I : I
5555555555 : 66666666665555555555I
I : I
I : I
333333333333333222226666666666555555555544444333333333333333........
Point = 0.000000D+00 s = 3.3333D-02
Direction = 1.000000D+00 h = 1.0000D+00
F0 (-1.00D+00, 9.98D-11) F1 (-5.00D-01, 7.68D-11) F2 (-1.00D+00, 1.00D+00)
F3 (-4.00D+00, 4.00D+00) F4 (-4.80D+01, 4.80D+01) F5 (-2.43D+02, 3.65D+02)
F6 (-7.29D+03, 7.29D+03)
I/O: 25 6 6 1 2 3 27 28 29 30 NRML on current CALLS = 1662
Figure 19 Derivatives of a Step Function (m = 0)
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 47
222222222211 665 444 66 33333 2222222222
I 221111 65 65 4 : 4 6 36 33 22 I
I 22 1111 665 655 4 : 4 633 66 33 22 I
I 22 11615 54 : 4 33 6 5 33 I
I 22 6 51111 6 45 : 436 6 5255533 I
66666666 266 5 1116 55 : 33 6 662 5553 66666666
I666666 6 25 4 11115:33 4 22 6 666666I
555555555 6666 5 22 4 6 3511 6 4 22 5 6666 555555555
I 555 5 22 4 633 :551611 4 22 5 I
44444444 355 24 33 : 5 11142 5 44444444
I444 3555 5 4 2233 6 : 65 22 4115 444I
I 444 335 4 3322 6 : 6 552 45 1111 444 I
I 44433 4 33 22 : 22 5 54 11444 I
I 4443 433 226:622 5 5 4 444 1111 I
I 4443 262 55 444 11111111111
===========================================================================
Point = 0.000000D+00 s = 3.3333D-02
Direction = 1.000000D+00 h = 1.0000D+00
F1 (-1.00D+00, 4.28D-11) F2 (-1.00D+00, 2.85D-10) F3 (-2.00D+00, 2.00D+00)
F4 (-8.00D+00, 1.60D+01) F5 (-8.10D+01, 8.10D+01) F6 (-1.46D+03, 9.72D+02)
I/O: 25 6 6 1 2 3 27 28 29 30 NRML on current CALLS = 1797
0
Figure 20 Derivatives of a C Function (m = 1)
333333333333322222 666665555555 4444444 3333333333333
I 33 22222 6 56 : 544 44 33 I
I 33 2222 5 : 45 4 33 I
I 3 62 5 6 : 4 5 44 3 I
I 33 6 252 : 4 5 644 I
I 3 5 22 6: 5 666 6664 I
I 3 6 222:4 6 3 6666 I
6666666666666 3 6 5 62 5 6 3 666666666666
555555555 4666 6 5 4: 22 5 6 3 555555555
I 555 46666666 3 :6 22 3 555 I
I 55 4 35 4 : 222 63 55 I
I 55 44 5 3 4 : 6 2625 55 I
I 5 4 33 4 : 33 222 5 I
I 5 455 344 : 6 33 6 55222225 I
I 55555 4444444 33333333366666 55555222222222222222222
===========================================================================
Point = 0.000000D+00 s = 3.3333D-02
Direction = 1.000000D+00 h = 1.0000D+00
F2 (-1.00D+00, 1.99D-10) F3 (-1.00D+00, 1.74D-09) F4 (-2.67D+00, 2.67D+00)
F5 (-8.98D+00, 1.35D+01) F6 (-1.46D+02, 1.46D+02)
I/O: 25 6 6 1 2 3 27 28 29 30 NRML on current CALLS = 1932
1
Figure 21 Derivatives of a C Function (m = 2)
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 48
444444444444444443333 666 5555555 44444444444444444
I 444 3333 6 : 6 5 5 444 I
I 4 333 6 : 6 5 554 I
I 44 33 6 : 6 445 I
I 4 33 : 5 4 55 I
I 4 633 : 5 6 4 5 I
I 4 33:5 4 555 I
55555555555555 4 6 53 6 4 55555555555555
I 555 4 5: 33 4 I
I 55 4 6 5 : 33 6 4 I
6666666666666666 5 6 5 : 336 6666666666666666
I 66 5 4 5 : 433 66 I
I 66 55 6 45 : 44 6333 66 I
I 6 5 6 55 4 : 4 6 33336 I
I 666666555 4444444 666666333333333333333333333
===========================================================================
Point = 0.000000D+00 s = 3.3333D-02
Direction = 1.000000D+00 h = 1.0000D+00
F3 (-1.00D+00, 1.60D-09) F4 (-1.33D+00, 1.14D-08) F5 (-3.00D+00, 3.00D+00)
F6 (-1.28D+01, 2.70D+01)
I/O: 25 6 6 1 2 3 27 28 29 30 NRML on current CALLS = 2067
2
Figure 22 Derivatives of a C Function (m = 3)
555555555555555555544444 : 666666 5555555555555555555
I 55 4444 : 6 6 55 I
I 5 44 : 6 66 5 I
I 5 44 : 65 I
I 5 44 : 6 566 I
I 5 4 :6 5 6 I
I 5 44: 5 666 I
66666666666666666 5 6 5 66666666666666666
I 666 5 :44 5 I
I 66 5 6: 4 5 I
I 6 5 6 : 44 5 I
I 6 5 6 : 45 I
I 66 55 : 55 444 I
I 6 66 : 5 444 I
I 66666 5555555 444444444444444444444444
===========================================================================
Point = 0.000000D+00 s = 3.3333D-02
Direction = 1.000000D+00 h = 1.0000D+00
F4 (-1.00D+00, 9.60D-09) F5 (-1.38D+00, 6.61D-08) F6 (-4.13D+00, 4.13D+00)
I/O: 25 6 6 1 2 3 27 28 29 30 NRML on current CALLS = 2202
3
Figure 23 Derivatives of a C Function (m = 4)
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 49
6666666666666666666665555 : 666666666666666666666
I 66 555 : 66 I
I 66 55 : 66 I
I 6 55 : 6 I
I 6 55 : 6 I
I 6 5 : 6 I
I 55: I
I 6 5 6 I
I 6 :55 6 I
I 6 : 5 6 I
I 6 : 556 I
I 6 : 655 I
I 6 : 6 55 I
I 6 : 6 555 I
I 66666 5555555555555555555555555
===========================================================================
Point = 0.000000D+00 s = 3.3333D-02
Direction = 1.000000D+00 h = 1.0000D+00
F5 (-1.00D+00, 5.09D-08) F6 (-1.65D+00, 2.67D-06)
I/O: 25 6 6 1 2 3 27 28 29 30 NRML on current CALLS = 2337
4
Figure 24 Derivatives of a C Function (m = 5)
4.3 The Window
Sometimes it is desirable to use very small window widths or even
completely separate differentiation stencils, for example when searching a
large domain for a point of reduced smoothness. Then steps in a derivative
will appear as such rather than as gradual slopes that might be obscured by
other features of the relevant graphs. Also, cusps and poles are best
displayed using small windows (see appendix III for some examples).
Usually, however, it will be preferable to use larger windows.
There are two main reasons to use overlapping differentiation
stencils: first, as illustrated in the preceding subsection, this is
necessary to resolve all the extrema of a derivative of a delta function,
and, second, it allows the use of one evaluation of a trial function for
several values of a derivative, thereby increasing efficiency and
accelerating your interaction with MICROSCOPE. Using overlapping stencils
is equivalent to having a window width (w = 2h/s) that is greater than 1.
Due to the internal organization of MICROSCOPE, for any given
window width, the number of evaluations (of the display function) required
is determined solely by the largest degree of the plotted derivatives. The
following table lists formulas for computing the number N of evaluations
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required given the window width (w), the highest degree of a plotted
derivative (k) and the number of columns (C) in the graphical display.
max k: 0 1 or 2 3 or 4 5 or 6
w=2h/s
< 1 C 3C 5C 9C
2 C C + 2 2C + 3 4C + 5
4 C C + 4 C + 4 3C + 8
6 C C + 6 2C + 9 2C + 9
8 C C + 8 C + 9 3C + 16
10 C C + 10 2C + 15 4C + 25
12 C C + 12 C + 12 C + 12
14 C C + 14 2C + 21 4C + 35
16 C C + 16 C + 16 3C + 32
18 C C + 18 2C + 27 2C + 27
20 C C + 20 C + 20 3C + 40
22 C C + 22 2C + 33 4C + 55
24 C C + 24 C + 24 C + 24
Table: Efficiency of Window Widths
The special role played by w = 12 is obvious: the numerical
effort is independent of the degree of the plotted derivatives, and exceeds
the number of columns in the display only by a constant. This is true
whenever w is a multiple of 12. The reason for this is, of course, that
the display function is sampled at multiples of 1/2 and 1/3 of the
discretization parameter. For efficiency, the default value of w is 12.
This is also sufficient for most examinations involving delta functions.
Sometimes, however, a larger multiple, like 24 or 36 may be desirable.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 51
4.4 Reference Table of Commands
This subsection contains a quick reference table of all available
commands. For each command, a one-line description is given. Often, the
information contained in these short descriptions will be all that is
needed. Therefore, the descriptions are used in several other places An
appropriate selection of them occurs at the beginning of each of the
following subsections. The HSUMRY command lists them ordered by topics.
The more detailed HELP command gives a documentation of an individual
command headed by the one-line description.
Name Description Data Error Screen
ACCENT mark derivative with asterisks TOGGLE k -1<k<7 immediate
C1CROSS read 1st component of cross direction C1 C><0 deactiv.
C2CROSS read 2nd component of cross direction C2 C><0;1 deactiv.
C3CROSS read 3rd component of cross direction C3 C><0;1 deactiv.
CCHANNL change channel number ic,nc 4 ---
CDIRCTN read direction of cross direction C C><0 deactiv.
CHVALUE read ch for cross differentiation ch 5 deactiv.
CORDER read order of cross derivative k -1<k<7 deactiv.
CWINDOW change window width by factor SIGN m m=0 immediate
D1CHDIR read 1st component of dir. of inv. D1 D><0 deactiv.
D2CHDIR read 2nd component of dir. of inv. D2 D><0;1 deactiv.
D3CHDIR read 3rd component of dir. of inv. D3 D><0;1 deactiv.
DCENTER draw center of graph. display TOGGLE --- --- immediate
DGRAPH draw graph k -1<k<7 immediate
DIVIDE divide h by integer m SIGN m d><0 immediate
DMNSN read dimension v of domain v 0<v<4 deactiv.
DOUBLE double h --- --- immediate
DSCALE draw scale in graphical display TOGGLE --- --- immediate
DXAXIS draw horizontal axis TOGGLE --- --- immediate
EGRAPH erase graph k -1<k<7 immediate
EXIT return to calling program --- --- deactiv.
FLIP replace d by -d TOGGLE --- --- immediate
FORCE recompute everything --- --- activ
GO compute whatever is needed --- --- activ
HALVE halve h --- --- immediate
HELP obtain detailed help on command cmd cmd 2 ---
HSUMRY obtain a summary of all commands --- --- ---
IDIRCTN read the direction of investigation D D><0 deactiv.
IHVALUE read value of h h h>0 deactiv.
IINTVL read endpoints of line of investig. A,B A><B deactiv.
IPOINT read the new point of examination P --- deactiv.
LCROSS load a new cross direction kc 3 deactiv
LDRCTN load a new direction of investigation kd 3 deactiv
LIST print all currently available commands --- --- ---
LOG give channel number for logging nc --- ---
LPOINT load a new point of examination kp 3 deactiv.
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MULTPLY multiply h by factor m SIGN m m><0 immediate
NEWS print the news --- --- ---
NORMAL turn normalization of D on/off TOGGLE --- --- deactiv.
OUTPUT write screen immage on recording device --- --- ---
P1CHPNT read first component of P P1 --- deactiv.
P2CHPNT read second component of P P2 1 deactiv.
P3CHPNT read third component of P P3 1 deactiv.
PAUSE pause until RETURN from channel nc nc --- ---
PLOT enter plotting (<PLOT79>) mode --- --- ---
QUIT terminate MICROSCOPE session --- --- ---
RCROSS read channel number for cross list nc --- ---
RDIRCTN read channel number for direction list nc --- ---
RESTART reestablish earlier parameter settings --- --- ---
ROTATE change direction of investigation DD D><0 deactiv.
RPOINT read channel number for point list nc 1 ---
RSCREEN refresh screen --- --- pending
RWIND rewind device nc nc --- ---
SETDF reset defaults --- --- deactiv.
SHIFT shift point of examination is --- immediate
STORE store current parameter settings --- --- ---
TCENTER type value of k-th derivative at center k -1<k<7 ---
TNOTE take a note on recording device --- --- ---
TYPE Type value of derivative at some point k,is -1<k<7 immediate
UNDO undo previous parameter changes --- --- ---
USER enter user mode --- --- ---
WAIT make program wait for GO or FORCE com. --- --- deactiv.
ZOOM change window width by factor SIGN m m><0 immediate
Quick Reference Table of commands
Key to table of commands:
-- Description --
TOGGLE the command reverses itself if called again
SIGN means that reversing the sign of the input parameter
generates the opposite effect (e.g. multiply instead of
divide).
-- Data --
A one endpoint of the line of investigation
B the other endpoint of the line of investigation (P = (A +
B)/2)
C cross direction
C1 first component of cross direction
C2 second component of cross direction
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C3 third component of cross direction
ch discretization parameter for cross derivatives
cmd command name
D direction of investigation
D1 first component of direction of investigation
D2 second component of direction of investigation
D3 third component of direction of investigation
DD correction to be added to D
ic type of device (1 = input, 2 = output, 3 = graphic, 4 =
record, 5 = restart)
is integer shift in number of columns on display, +ve shifts
graph to right, point to left, -ve is reverse
k degree of a derivative (to be plotted, erased, etc.)
kc position of the cross direction in a list
kd position of the direction of investigation in a list
kp position of the point of examination in a list
m integer multiplier.
n integer divider
nc channel (device) number
P point of examination
P1 first component of the point of examination
P2 second component of the point of examination
P3 third component of the point of examination
--- no data are required
-- Error --
Restrictions on the data are listed in this column. The entry
"---" means that no check is made for errors. The symbol "><" means "not
equal". If the error conditions cannot be given explicitly then the
following reference numbers have been used:
1. Components to be changed must not have an index that is
greater than the current dimension.
2. The command must be a valid MICROSCOPE command. If it is not
recognized you will be prompted for another command to be helped on.
Typing any valid command will get you some information on it and will then
return you to command mode.
3. There must be one vector in legal format in each line of the
list. If illegal characters are encountered, an error termination of the
command will occur. Also, the index number must not exceed the number of
lines in the list. If it does, the program will reach the end of the file
containing the list, and the precise action after that depends upon the
computing system.
-- Screen --
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 54
activ the command activates the screen
deactiv. the command deactivates the screen - i.e. the GO or FORCE
command must be given before the computation proceeds. For
some commands, there is a check if the newly input datum is
identical with the old one, in which case the screen remains
active.
immediate If the screen is active then the effects of the command are
incorporated before the next command is requested
pending If the screen is inactive, this command will put the
graphical display on the screen, even if computations are
pending.
--- The graphical and numerical display, and the screen status
are not changed. Notice, however, that the actual screen
display may be destroyed by error messages or help
information. The next display changing command (or RSCREEN)
will then refresh the screen.
4.5 Commands to Control the Display
In this and the following subsections, input to MICROSCOPE is via
the assigned input channel, unless otherwise stated. This will usually be
the terminal, but may also be a file for a batch type session.
ACCENT mark derivative with asterisks TOGGLE k -1<k<7 immediate
DCENTER draw center of graph. display TOGGLE --- --- immediate
DGRAPH draw graph k -1<k<7 immediate
DMNSN read dimension v of domain v 0<v<4 deactiv.
DSCALE draw scale in graphical display TOGGLE --- --- immediate
DXAXIS draw horizontal axis TOGGLE --- --- immediate
EGRAPH erase graph k -1<k<7 immediate
FLIP replace d by -d TOGGLE --- --- immediate
RSCREEN refresh screen --- --- pending
TCENTER type value of k-th derivative at center k -1<k<7 ---
TYPE type value of derivative at some point k,is -1<k<7 immediate
We now turn to the descriptions of individual commands. Screen
display commands control what appears on the terminal screen. (For the
effects of these commands on the <PLOT79> drawing see section 4.14.) There
are several subgroups of commands:
-1- Deciding on the Tangential Derivatives to be Displayed.
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By default, only the display function is displayed. The graph of
the display function itself is marked by periods, the graph of its k-th
tangential derivative (k = 1,2, ..., 6) by the digit k. Using DGRAPH or
EGRAPH respectively, you can turn on or off the graphs of the zeroth
through sixth derivatives. It is also possible to display no functions at
all. In any case, the display function is evaluated only at the points
where it is needed.
-2- Accentuating a Graph.
If you wish to emphasize the graph of a particular derivative
then you can plot its graph using asterisks rather than a period or the
digit k. The command ACCENT turns this option on or off. Note that the
numerical display contains no direct information on which derivatives have
been accentuated. Also, if you accentuate several derivatives at once
their graphs cannot be distinguished from each other. The graph of an
accented derivative overwrites anything else in the display, except for the
center mark. The ACCENT command also has the effect of turning the graph
of a derivative on, i.e. it does not need to be preceded by a DGRAPH
command. By default no derivative is accented.
-3- Drawing a Horizontal Axis
A horizontal axis can be drawn through the center of the
graphical display, using the command DXAXIS. If desired, a scale can be
obtained by using DSCALE. Both commands are on/off switches. The point at
which the axis or scale intersects the central column of the graphical
display does not in general correspond to the value zero. (That point is
different for each of the tangential derivatives plotted, and may not even
be contained in the graphical display.) The purpose of the axis is to give
help in examining symmetry properties, and the scale is useful in locating
points of interest for more detailed examination. Both the scale and the
axis are off by default.
-4- Marking the Center
The on/off command DCENTER may be used to mark the center of the
graphical display with a plus sign. This is sometimes useful for
confirming symmetry properties. The center mark is off by default.
-5- Flipping the Direction of Investigation
The FLIP command has the effect of replacing the direction of
investigation d by -d. This turns the graph of any derivative about the
vertical axis, and it also turns the graph of any odd degree derivative
about the horizontal axis, thereby possibly disentangling graphs that have
been overwritten by others.
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Consider for example the exponential function and its first
derivative. Ordinarily the two graphs would be identical, yielding a
display in which the display function is completely covered by the graph of
its first derivative:
I : I 111
I : I 111
I : I 1111
I : I 111
I : I 1111
I : I 1111
I : I 1111
I : I 1111
I : 11111
I 11111 I
111111 : I
111111 I : I
1111111 I : I
11111111 I : I
111111111 I : I
===========================================================================
Point = 0.000000D+00 s = 1.6667D-02
Direction = 1.000000D+00 h = 1.0000D-01
F0 ( 5.40D-01, 1.85D+00) F1 ( 5.41D-01, 1.86D+00)
I/O: 25 6 6 1 2 3 27 28 29 30 NRML on current CALLS = 2424
Figure 25 An Invisible Graph
After using the FLIP command one obtains:
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... I : I 1111111111
... I : I 1111111
.... I : I 1111111
... I : I1111111
.... I : 11111
.... I 11111 I
.... 11111 : I
11111 I : I
1111 ..... : I
1111 I ..... I
111 I : ......
1111 I : I ......
111 I : I .......
111 I : I ........
111 I : I .........
===========================================================================
Point = 0.000000D+00 s = 1.6667D-02
Direction = -1.000000D+00 h = 1.0000D-01
F0 ( 5.40D-01, 1.85D+00) F1 (-1.86D+00,-5.41D-01)
I/O: 25 6 6 1 2 3 27 28 29 30 NRML on current CALLS = 2424
Figure 26 The Disentangled Graphs
where the two graphs are now clearly visible.
-6- Obtaining Numerical Values
The alphanumerical display does not provide precise numerical
values. Values of any computed derivative to 21 digits may be obtained by
using the TCENTER or the TYPE command. TCENTER provides values at the
center of the display, TYPE does so for any point in the display. For
TYPE, measure the distance in columns form the center of the graphical
display, positive to the right and negative to the left. For clarity, if
the screen version of MICROSCOPE is employed, the column in which the value
is read is briefly marked with exclamation marks. Returning to command
mode requires typing a carriage return. This is necessary so that the
prompt for the next command does not overwrite the numerical information
before it can be read.
Only derivative values that have been previously computed can be
read. This includes all derivatives through [k] where k is the highest
degree of a derivative that has been previously displayed on the current
active screen, and [k] is k+1 if k is odd or k if k is even.
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-7- Refreshing the screen
The command RSCREEN has two distinct applications:
On an active screen, it serves to refresh the screen image after
it has been destroyed e.g. by a system message or by HELP information.
On an inactive screen, the last few commands instead of the
alphanumerical display are shown. In that situation, RSCREEN calls the
display onto the screen without carrying out any computations. Thus
changed parameters may be shown in the numerical display without having
been incorporated into the graphical display. This condition is indicated
by the flag "GO pndg" in the numerical display. (If the screen is active
then the flag "current" is displayed instead). The second application is
useful if the alphanumerical display has to be viewed in order to decide on
the next command.
When entering MCRSCP the screen is inactive.
-8- Setting the Domain Dimension
The DMNSN command informs MICROSCOPE of the number of independent
variables (1,2 or 3). It is listed here because it does affect the
numerical display. However, this command is usually given at most once at
the beginning of the MICROSCOPE session. The default dimension is 2.
DMNSN also resets the point of examination, the direction of investigation,
the cross direction, and the order of the cross derivative to their default
values.
4.6 Commands to Control the Point of Examination
IINTVL read endpoints of line of investig. A,B A><B deactiv.
IPOINT read the new point of examination P --- deactiv.
LPOINT load a new point of examination kp 3 deactiv.
P1CHPNT read first component of P P1 --- deactiv.
P2CHPNT read second component of P P2 1 deactiv.
P3CHPNT read third component of P P3 1 deactiv.
RPOINT read channel number for point list nc --- ---
SHIFT shift point of examination is --- immediate
The simplest way of defining the point of examination is to read
it from the input device by using the IPOINT command. Individual
components may be defined similarly by using the commands P1CHPNT, P2CHPNT
or P3CHPNT.
Sometimes it is useful to generate a list of possible points of
examination before the MICROSCOPE session, and then refer to them by their
position (i.e. line number) in the list, rather than by typing the
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 59
individual components. This is facilitated by the LPOINT command, which
prompts for the position of the point and then reads it from a device
(usually a file) with a device number that has been set by the RPOINT
command (the default is zero). All point loading devices must contain one
vector per line.
The SHIFT command can be used to shift the graph by a specified
number of columns to the right (if the number is positive) or left (it is
negative). There is no restriction on the number of columns by which to
shift the graphs. In particular, the number may exceed the number of
columns in the display.
It is also possible to define the part of the domain that is to
be shown in the display by defining its two endpoints. This is
accomplished by the IINTVL command, which will prompt for the left and
right endpoint (in that order) of the graphical display. These have to be
separated by at least one line feed. Notice that IINTVL defines
simultaneously the point of examination, the direction of investigation,
the discretization parameter, and the display interval.
4.7 Commands to Control the Direction of Investigation
D1CHDIR read 1st component of dir. of inv. D1 D><0 deactiv.
D2CHDIR read 2nd component of dir. of inv. D2 D><0;1 deactiv.
D3CHDIR read 3rd component of dir. of inv. D3 D><0;1 deactiv.
IDIRCTN read the direction of investigation D D><0 deactiv.
LDRCTN load a new direction of investigation kd 3 deactiv
NORMAL turn normalization of D on/off TOGGLE --- --- deactiv.
RDIRCTN read channel number for direction list nc --- ---
ROTATE change direction of investigation DD D><0 deactiv.
The direction of investigation can be read from the input device
by using the command IDIRCTN. Individual components can be changed by the
commands D1CHDIR, D2CHDIR or D3CHDIR.
Similarly to the point of examination, the direction of
investigation may also be defined by giving its position in a list. The
command LDRCTN loads the direction from the list (usually a file) whose
channel number has been defined by the RDIRCTN command.
A new direction may also be defined by rotating the old one. The
command ROTATE will read the difference between the old and the new
direction, and update the direction. This is sometimes useful for
examining the effects of deviating slightly from the previous direction of
investigation.
The numerical display will always contain the direction of
investigation as it has been input to MICROSCOPE. Internally the direction
of the derivative (i.e. the vector d in the differentiation formulas) is
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usually the given direction divided by its Euclidean length. This leads to
the standard definition of a directional derivative. However, sometimes it
is more useful to use the unnormalized direction for the differentiation
(i.e. to have d identical to the displayed numerical value). The
normalization can be turned on or off using the command NORMAL, and is on
by default. Normalization changes the values of the directional
derivatives, and the domain that is covered by the graphical display. An
identical graphical display (but with different numerical values) can be
obtained by dividing the display interval instead of the specified
direction of investigation by the same factor.
4.8 Commands to Control Accuracy, the Window, and Round-Off Effects
CWINDOW change window width by factor SIGN m m=0 immediate
DIVIDE divide h by integer m SIGN m d><0 immediate
DOUBLE double h --- --- immediate
HALVE halve h --- --- immediate
IHVALUE read value of h h h>0 deactiv.
MULTPLY multiply h by factor m SIGN m m><0 immediate
ZOOM enlarge central portion of displ. SIGN m m><0 immediate
This group of commands controls the three parameters h
(discretization parameter), s (the display interval), and w (the window
width) which are related by w = 2h/s (w must lie in the interval [1/16,576]
and can assume only certain rational values). Each of the commands changes
two of the three parameters, leaving the third constant, according to the
following table:
Parameter left constant
w s h
DIVIDE CWINDOW ZOOM
DOUBLE
HALVE
IHVALUE
MULTPLY
Table of Discretization Controling Commands
Most frequently, the window width is left unchanged, often
throughout the entire MICROSCOPE session. This accounts for the abundance
of commands covering that case.
The command IHVALUE serves to define an entirely new value of the
discretization parameter (and the display interval).
Often you will want to modify the current value of h, e.g. to
get rid of the influence of round-off errors, or perhaps to increase the
accuracy of numerical values. Due to the organization of MICROSCOPE it is
particularly efficient (i.e. requires few evaluations of the trial
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function) if h is multiplied or divided by powers of 2. The factor 2 is
used so frequently that two specific commands (HALVE and DOUBLE) have been
provided for this case, which require no arguments.
Multiplications or divisions with other integer factors can be
accomplished by using the commands MULTPLY and DIVIDE respectively. It is
possible to reverse the meaning of the commands by entering negative
factors (e.g. MULTPLY followed by -3 is equivalent to DIVIDE followed by
+3). This is due more to the internal workings of MICROSCOPE than to any
attempt to provide a truely meaningful convenience.
The command CWINDOW changes the current window width by the
specified factor. A positive factor corresponds to an increase of the
window width, a negative one to a decrease. The change is obtained by
altering the value of the discretization parameter while leaving the
display interval fixed.
The command ZOOM has a similar effect. However, it alters the
display interval and leaves the discretization parameter unchanged. This
is useful when operating close to the round-off level with a value of h
that must not be decreased. On the other hand, if round-off errors are
irrelevant then the CWINDOW command should be used because it is more
efficent.
The primary purpose of the commands described here (other than
IHVALUE is to alter the display while the screen is active. If several
commands are given while the screen is inactive only the last one will
actually be implemented. If several changes are to be affected before the
trial function is evaluated, then this can be accomplished by erasing the
graphs of all derivatives, and then activating the screen (using GO or
FORCE), and making the desired changes.
For the following illustration it is assumed that the screen is
activ derivatives through second order have just been displayed, 79 columns
are being carried in the graphical display, and the current window width is
12. The table below gives the additional number of evaluations of the
display function needed to accomodate changes by factors 2,3,...,8. An
entry "--" indicates that the change is impossible (e.g. repeated
applications of the HALVE command can only accomplish changes by powers of
2). Entries in parentheses indicate that the change has led to a
non-integer half window width and therefore is unlikely to be employed.
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2 3 4 5 6 7 8
DIVIDE 46 91 68 91 76 91 80
MULTPLY 46 60 68 91 76 91 80
DOUBLE 46 -- 92 -- -- -- 138
HALVE 46 -- 92 -- -- -- 138
IHVALUE 91 91 91 91 91 91 19
CWINDOW 12 24 36 48 60 163 84
(increasing)
CWINDOW 0 0 (82) (237) 0 (237) (237)
(decreasing)
ZOOM 52 76 96 139 151 163 154
(increasing)
ZOOM 40 52 (116) (182) 66 (198) (204)
(decreasing)
Table: Relative Efforts
4.9 Commands to Control Execution
EXIT return to calling program --- --- deactiv.
FORCE recompute everything --- --- activ
GO compute whatever is needed --- --- activ
QUIT terminate MICROSCOPE session --- --- ---
RESTART reestablish earlier parameter settings --- --- ---
SETDF reset defaults --- --- deactiv.
STORE store current parameter settings --- --- ---
UNDO undo previous parameter changes --- --- ---
WAIT make program wait for GO or FORCE com. --- --- deactiv.
To activate an inactive screen the GO or the FORCE command must
be used. GO assumes that previous function evaluations are still valid and
FORCE reevaluates the trial function at all needed points regardless of
prior evaluations. Ordinarily GO will be used for the sake of efficiency,
but the FORCE command is necessary if changes have been applied to the
trial function (either after leaving MCRSCP or by using the USER command)
that invalidate the earlier evaluations.
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Usually, the command QUIT is used to terminate a MICROSCOPE
session. That command leads to a simple FORTRAN STOP statement. However,
it is possible to leave the routine MCRSCP via the EXIT command (leading to
a RETURN statement), and then call MCRSCP again, perhaps after changing
some screen parameters. After reentering MCRSCP, these will again be
checked for consistency, and the screen will be marked inactive. However,
the defaults will not be set, so that it is possible to return to MCRSCP
and find everything as it has been left with the exception of what you may
have reset yourself between MCRSCP calls. After reentering MCRSCP it may
be necessary to use FORCE instead of GO to avoid peculiar displays due to
altered screen parameters.
The recommended way of changing problem defining parameters (e.g.
selecting a new trial function) is to use the command USER (see section
4.15).
Usually the screen is set inactive by some drastic change of key
parameters such as the point of examination or the direction of
investigation. It can also be set inactive by using the command WAIT. This
option is particularly useful in the scrolling version when it may be
convenient to effect several changes before scrolling the display just
once, rather than having it scrolled after each individual command.
The state of a MICROSCOPE investigation can be stored using the
command STORE, and later restored using the command RESTART. STORE writes
the values of all relevant parameters onto the Restart Device and then
rewinds that device. Thus, if the command STORE is given several times
only the status during the last issue of the command can be restored. If
it is necessary to store several data settings, then the restart channel
number has to be changed by using the CCHANNL command (see section 4.13),
or a different file has to be assigned to it by the command USER.
If desired, some of the default settings can be reestablished by
using the SETDF command. More precisely, the effects of SETDF are:
-1- The screen is deactivated
-2- All previous function evaluations are discarded
-3- The window width w is set to 12
-4- Only the display function itself is flagged to be plotted
-5- Any cross derivatives are turned off
-6- The discretization parameter h is set to its default value (1D-4)
-7- The x-axis, center mark, and horizontal scale are all turned off
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-8- If the test package has been loaded (see Appendix III) then
the test function is reset to its defaults (N = 1, D = 10,
no exponential is added).
While the screen is inactive, it may be possible to recover from
an erroneous command using the command UNDO. This has an effect only on
quantities set after the last GO or FORCE command, and resets them to their
values prior to that command. The following parameters are reset (i.e.
any change is undone):
-1- The point of examination
-2- The direction of investigation
-3- The left and right endpoints of the investigation interval
-4- The value of the discretization parameter h
-5- The window width w
-6- The number of independent variables
-7- Marking the center point
-8- Drawing a horizontal scale
-9- Drawing a horizontal axis
-10- Selection of graphs to be plotted
Note that any changes regarding a cross derivative or the
normalization of the direction of investigation are not undone.
4.10 Commands to Obtain On-Line Help
HELP obtain detailed help on command cmd cmd 2 ---
HSUMRY obtain a summary of all commands --- --- ---
LIST print all currently available commands --- --- ---
NEWS print the news --- --- ---
The quickest way of obtaining information on possible commands is
the command LIST which will print a list of all currently available
commands on the screen. More detailed information on all commands can be
obtained through the HSUMRY command which prints the table in section 4.3
reorganized by topics.
Information on a specific command can be obtained by typing HELP,
followed by the name of the command you want help on. MICROSCOPE will
print not only a short description of the command, but also the current
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 65
values of those parameters that are affected by the command. Thus if, for
example, you wish to find out what the current number of independent
variables is you can request HELP on DMNSN.
The command NEWS will print a text that may change from time to
time and will also depend on your particular installation. At the time of
this writing the "news" are a very brief primer on MICROSCOPE.
Finally, if you type a question mark to MICROSCOPE, it will
invite you to enter a command, and will suggest HSUMRY and LIST as
possibilities.
4.11 Commands to Keep a Record
LOG give channel number for logging nc --- ---
OUTPUT write screen immage on recording device --- --- ---
TNOTE take a note on recording device text --- ---
Results obtained with MICROSCOPE can be recorded permanently on
the Record Device. The command TNOTE will copy any text you type on the
Input Device until it encounters the string 'EC' (End Comments, both E and
C must be capital) in columns 1 and 2. It then returns to command mode.
The length of text lines may be up to 80 columns or the number of columns
specified for the graphical display, whichever is larger. Trailing blanks
are stripped, to make the record file more compact.
The command OUTPUT causes MICROSCOPE to write a copy of the
graphic and the numerical display onto the recording device. This manual
is generated by using TNOTE and OUTPUT throughout a lengthy string of
MICROSCOPE commands.
There is also a logging device which is active if its channel
number is different from zero (it is inactive by default). The command LOG
sets the logging device number. If the device is active all commands,
data, displays, and notes are recorded on it.
4.12 Commands to Analyze Cross Derivatives
C1CROSS read 1st component of cross direction C1 C><0 deactiv.
C2CROSS read 2nd component of cross direction C2 C><0;1 deactiv.
C3CROSS read 3rd component of cross direction C3 C><0;1 deactiv.
CDIRCTN read direction of cross direction C C><0 deactiv.
CHVALUE read ch for cross differentiation ch 5 deactiv.
CORDER read order of cross derivative k -1<k<7 deactiv.
LCROSS load a new cross direction kc 3 deactiv
RCROSS read channel number for cross list nc --- ---
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 66
Cross derivatives, like all derivatives in MICROSCOPE, are
calculated using numerical differentiation. The formulas are analogous to
those given for tangential derivatives. In general, the cross stencils
will have no points in common. Therefore, one evaluation of a k-th order
cross derivative requires k+1 evaluations of the trial function.
By choosing the cross direction to be parallel to the direction
of investigation the range of tangential derivatives can be shifted from
[0,6] to [k,6+k]. This situation, however, is not treated separately by
MICROSCOPE and does not lead to savings in function evaluations.
The commands controlling the cross derivative (if any) are
similar to those controling the direction of investigation. By default, no
cross derivative is computed. It is activated by reading a non-zero order
of the derivative, using CORDER, or by defining a cross direction using
CDIRCTN (which reads the cross direction), C1CROSS, C2CROSS, or C3CROSS
(which read individual components), or LCROSS which loads a cross direction
from a channel defined by RCROSS (the default channel number is zero). The
default order is 1 if the cross derivative is activated by setting a cross
direction. If it is activated by setting the order its default direction
is the vector with all of its components equal to 1.
The cross discretization parameter ch is either set explicitly
using CHVALUE, or by default, in which case it assumes the same value as
the tangential parameter h at the time when the cross derivative is
activated. For efficiency, later changes of h have no effect on the value
of the cross parameter. Note that any change of ch requires a reevaluation
of the display function at all needed points.
Cross derivatives are turned off by setting their order to zero
(using CORDER). The NORMAL command has the same effect on cross
derivatives as it does on tangential derivatives.
When plotting cross derivatives it is important to keep in mind
that the significance of the window may be blurred if a stencil used for
the cross differentiation, and centered outside of the window, nonetheless
crosses the front. This can be avoided either by keeping the cross
discretization parameter much smaller than the display interval, or by
choosing the cross direction parallel to the front.
In the following example, the trial function is
2
f(x,y) = x *abs(x)*y*abs(y)
the point of examination is [0,1], the direction of investigation is [1,0],
and the cross direction is [1,-1]. The value of h = 3D-3 and ch = 6D-3.
Obviously the effects of the discontinuity in the second tangential
derivative are carried beyond the window.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 67
111 I33333333333I 22222222222222222222222222222
11 3 : I22 ..111
11 I : 3 ...11
11 3I : 22I3 .... 11
11 I : 2 I .... 11
11 I : 2 I ..... 111
11 3 I :2 I.3....... 11
111 .........2...... 11
11 .......3 I 2: I 3 11
....11 I 22 : I 111
.... 11 3 I 2 : I 11
... 11 I2 : I 3 11
... 311 2 : I 113
... 3 221 : 111
33333333333333333333333332222 I11111111111I 33333333333333333333333333
CD: deg = 1 dir = ( 1.000000D+00, -1.000000D+00) ch = 6.0000D-03
Point = ( 0.000000D+00, 1.000000D+00) s = 5.0000D-04
Direction = ( 1.000000D+00, 0.000000D+00) h = 3.0000D-03
F0 (-7.49D-04, 7.28D-04) F1 ( 1.05D-02, 8.00D-02) F2 (-4.40D+00, 4.18D+00)
F3 (-1.34D+01, 1.00D+03)
I/O: 25 6 6 1 2 3 27 28 29 30 NRML on current CALLS = 2598
Figure 27 An Extended Discontinuity
4.13 Programming MICROSCOPE
CCHANNL change channel number ic,nc 4 ---
PAUSE pause until RETURN from channel nc nc --- ---
RWIND rewind device nc nc --- ---
Some rudimentary programming of MICROSCOPE is possible. The
basic approach consists of writing a sequence of commands into a file and
assigning it to the input device. The various devices can be switched by
using the CCHANNL command. It will prompt for two integers, the first
defining the type of channel to be switched, the second giving the new
channel number. The types are
1: Input Device
2: Output Device
3: Graphics Device
4: Recording Device
5: Restart Device
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 68
For an explanation of these devices see section 1.3. Their
channel numbers are defined originally by parameters passed on to the
routine MCRSCP (see section 3.1). Upon entering MCRSCP these are copied
into variables that may be changed by CCHANNL. Thus the input parameters of
MCRSCP are not changed by that command and assume their old values should
MCRSCP be exited (using EXIT) and later reentered without changing its
parameters.
The loading devices for the point of examination, the direction
of investigation, and the cross direction can be switched by using the
commands RPOINT, RDIRCTN and RCROSS, respectively. The Help Device cannot
be switched as the information contained in it is an integral part of the
MICROSCOPE package and is only read during a session. Indeed, it is
advisable to protect the help file against an accidental overwrite if
possible, e.g. by using a READONLY switch in an OPEN statement.
Any channel can be rewound by the RWIND command. This facility
can be used e.g. to regenerate a specific parameter configuration or to
overwrite previously generated information that is no longer needed. An
infinite loop can be created by putting a rewind command at the end of the
input file.
The PAUSE command can be used to view the output of a MICROSCOPE
program at leisure during run time. It will ask for a channel number and
then suspend computation until a dummy character is received from that
channel. The channel will usually be the terminal.
Computation by a program can be suspended by changing the input
device to the terminal (entering interactive mode) and later continued by
switching the input channel back to the appropriate device.
Note that for all of the device handling commands no precaution
has been taken to ensure that only legal channel numbers are given, and
that appropriate devices are assigned to the channel numbers.
Comments can be included in a program in two ways. Firstly, a
'comment line' replaces a command line and is marked by a "+" character in
column 1 or 2. Secondly, any characters contained in the space starting
with column 3 of a command line are ignored by MICROSCOPE. This facility
does not exist for lines containing data. Indeed, any spurious characters
may lead to the rejection of the data by the parser.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 69
4.14 The PLOT command for obtaining a <PLOT79> Display
PLOT enter plotting (<PLOT79>) mode --- --- ---
The alphanumerical displays usually generated by MICROSCOPE are
adequate to obtain the desired answers to your questions, but they may not
be suitable e.g. for publications, reports, etc. Depending upon the
availability of suitable hard- and software (<PLOT79>, see Beebe, 1979) you
may be able to generate more presentable plots. This is accomplished by
the command PLOT which generates a drawing of the current alphanumerical
display, augmented by some special options.
Details about the link between MICROSCOPE and <PLOT79> are given
in section 5 and your system manager may have put out suitable local
information in this respect. For the remainder of this subsection, we will
assume that the <PLOT79> package is available, that you have linked it with
MICROSCOPE and your own program, and that you have some suitable plotting
device available. Appendix IV containes copies of black and white pen
plots corresponding to the alphanumerical plots scattered throughout this
manual.
The command PLOT puts you into the plot submode. The precise
kind of plot you will eventually generate depends upon the MICROSCOPE
commands given so far, and the specifications you are going to make in plot
mode.
We first discuss the PLOT specifications. After entering the
PLOT submode you will see a list of current options on the screen. You can
change these options one by one. After each change an updated list will be
displayed. To enter an option change enter the reference number of the
option followed by the appropriate action (as prompted by the program), to
activate the actual plotting process enter 0, and to leave the plotting
mode and return to command mode without plotting anything enter any
negative number. The list of PLOT options is left unchanged between calls
to PLOT.
There are 12 PLOT options:
-1- Title (Default: blank)
A title printed above the alphanumerical display, consisting of
up to 72 characters can be drawn.
-2- Legend (Default: blank)
A string of text printed below the alphanumerical display,
consisting of up to 72 characters can also be drawn.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 70
For both the title and the legend, a default character height is
provided which will be decreased automatically if insufficient space is
available.
-3- Labeling of graphs (Default: ON)
If ON, labels are drawn giving the degree of the derivative of
each graph. On a crowded display, these are sometimes awkwardly placed,
and marking the graphs (option 4) may be preferable.
-4- Marking of Graphs (Default: OFF)
If ON, each point at which a derivative has been evaluated is
marked with a special symbol, similarly to the graphical display on the
termina The symbols for the k-th derivative are
k: 0 1 2 3 4 5 6
Symbol: dot plus asterisk circle x square triangle
Marking the graphs instead of labeling them has the advantage
that it is possible to tell from any small part of the graph to which
derivative it corresponds, making it unnecessary to trace a graph from the
point of interest to its label. It has the disadvantage that it slows down
significantly the speed with which the graphs can be drawn. Also,
accentuating a graph is only possible with the marking on (see below). The
marking symbol is repeated in the numerical display if the marking option
is on, enabling easy association of marks and degrees of tangential
derivatives.
-5- Drawing the plot in color (Default: ON)
The graph of the tangential derivatives can each be drawn in a
different color, if supported by the existing hardware. In the most
extreme case, when all derivatives are to be plotted in different colors,
seven pens for their graphs plus an eighth pen for the numerical display,
scales, axes, etc. must be supplied.
MICROSCOPE selects a pen by calling a routine SETPEN(I) where
I = 1 corresponds to the pen drawing the axes etc., and I > 1 corresponds
to the graph of the (I-2)th tangential derivative (I = 2,3,...,8). This
range of I has to be translated into the appropriate <PLOT79> range. On
some installations the pens are numbered 0 through 7, on others from 1 to
8. The appropriate SETPEN routines are contained in the files PEN07 and
PEN18, respectively (see section 5). If only a smaller range of pens is
available you may want to write your own pen selection routine, tailoring
it perhaps to the specific range of derivatives in which you are
interested. Your routine must be of the following form:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 71
SUBROUTINE SETPEN(I)
INTEGER I,J
C I is in the range from 1 to 8
J = some function of I
C J is in the range of pens available
C on the local plotting device
CALL STPEN(J)
RETURN
END
The subroutine STPEN is a MICROSCOPE routine that calls the
<PLOT79 routine SETPN(J) if the value of J has changed since the last call
to STPEN. SETPEN is not referenced if the color option is turned off.
In the numerical display, the ranges of the tangential
derivatives are drawn in the same colors as their graphs.
Using color makes for very nice drawings, but for reproduction
purposes it may be preferable to use black and white (like the <PLOT79>
drawings in Appendix IV).
-6- Drawing a page number (Default: None)
A large and prominent integer can be drawn below the legend. We
have found this useful, for example, for associating a drawing with the
reference number of a point in a list (e.g. the point of examination), and
associating or defining a figure number (like in this manual).
No page number is drawn if its value is zero.
-7- Left bottom label (Default: none)
A label of up to 20 characters of small type can be drawn in the
left bottom corner of the plot.
-8- Right bottom label (Default: none)
A similar label can be drawn in the right bottom corner of the
plot.
-9- Time (Default: off)
The time at which the plot was generated can be drawn in the top
left corner of the plot.
-10- Date (Default: OFF)
Similarly, the date of the plot generation can be drawn in the
top right corner.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 72
Both time and date are printed in the same size as the bottom
labels. They are sometimes useful for identification and record keeping.
-11- Numerical Display (Default: ON)
If desired, the numerical display can be turned off. This is
sometimes useful to obtain quick drawings.
-12- Frame (Default: ON)
If desired, the frame that is normally drawn around the entire
<PLOT79> drawing can be turned off.
We now turn to the MICROSCOPE options. Essentially everything
that appears in the alphanumerical screen display is also contained in the
<PLOT79> drawing, although it may assume a slightly different form.
-1- Horizontal scale
If a horizontal scale is activated (by the DSCALE command) then a
standard labeled horizontal scale is drawn on the bottom of the graphical
<PLOT79> drawing. The labels on the scale give the distance from the point
of examination in units of the direction of investigation. Notice that
this may be the normalized version of the direction given in the numerical
display, depending on whether the normalization option is on. If it is,
then the scale in the <PLOT79> drawing gives the Euclidean distance from
the point of examination.
-2- x Axis
A horizontal line is drawn through the center of the graphical
display if the x axis is turned on (using DXAXIS). This is sometimes
useful for illustrating symmetry properties.
-3- Marking the center of the display
If this option is turned on (using DCENTER) the center of the
graphical display is marked with a cross.
-4- Accentuating a graph
If a graph is to be accentuated its marks (as discussed above)
are drawn twice their normal size. However, if color is available, a
better emphasis can usually be obtained by using a special color.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 73
4.15 User Intervention
USER enter user mode --- --- ---
Upon receiving the command USER, MICROSCOPE calls a subroutine
SUBUSR which is often a dummy routine, but which can be written to suit
special purposes. SUBUSR has no parameters, but it can communicate with
MICROSCOPE routines or your own routines via COMMON blocks. In utilizing
this facility it is important to keep in mind that ordinarily MICROSCOPE
has no means of determining that the trial function has changed and that
previous evaluations of it are no longer valid. Therefore the command
FORCE may have to be used rather than GO.
Five examples of possible applications follow:
-1- Mixed partial derivatives
If derivatives other than pure tangential derivatives of pure
cross derivatives are to be examined this can be accomplished by defining
the trial function to be the derivative that is to be displayed, or to be
further differentiated, and make it dependent upon parameters that are
modified by SUBUSR as needed. In this context, it may be convenient to use
the routine
DOUBLE PRECISION FUNCTION DERIV(F,NDIM,P,IORDER,DIR,H)
DOUBLE PRECISION F,P(1),DIR(1),H
INTEGER NDIM,IORDER
...
RETURN
END
which computes at the point P the IORDERth derivative in the direction DIR
of the double precision function F, using the formulas given in section 1.2
with the discretization parameter H. F must be declared EXTERNAL in the
calling program, and is of the same form as the trial function described in
section 3.1. NDIM is the number of independent variables F depends on and
must not exceed 3. DERIV is used within MICROSCOPE (to compute any cross
derivatives) and can therefore be readily called by your own programs.
Obviously, the modified trial function may itself be a tangential
derivative. Then the range of derivatives that can be displayed is shifted
from the normal 0th through 6th order to a higher one.
-2- Switching Trial Functions
If several trial functions are to be investigated in a single
session (or with a single linking and loading process) they can all be
written into one DOUBLE PRECISION FUNCTION, and distinguished by a
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 74
parameter that is passed in a common block, and that may be altered by
SUBUSR. The examples in this manual are handled in this manner (see
appendix III for details).
-3- Switching Data or Parameters
Often the trial function depends upon data or parameters that can
be changed by SUBUSR without having to leave MCRSCP.
-4- File Handling
File handling is system dependent. You may find it convenient to
handle the assignments of files or devices to channel numbers via USER.
4.16 Default settings
In this subsection, E denotes a vector of 1 to 3 components that
are all equal to 1. The default settings for MICROSCOPE are as follows:
Number of independent variables: 2
Point of examination: 0 (the origin)
Direction of Investigation: E
Discretization parameter: 1D-4
Window width: 12
Cross direction: E
Order of Cross Derivative: 0 (no cross derivative)
Order of Tangential Derivatives to be displayed: 0
No scale or x axis is drawn, the center of the display is unmarked, and
no derivative is accentuated.
PLOT options: see section 4.14
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 75
5. Installation Guide
5.1 Description of the Package
In this section, which is directed mainly to the local system
manager, we describe the files supplied with the MICROSCOPE package, and
how they can be put together. Usually, it will be the best procedure to
select or write files that generate the most powerful and convenient local
version of MICROSCOPE, to put them into one file or a library and to make
the appropriate information available to your local users.
MICROSCOPE is a portable set of FORTRAN routines. The user must
supply a main program and one or two subroutines that are specific to the
application. Thus users must have access to at least the relocatable
versions of a suitable subset of MICROSCOPE, if not the source codes.
You should have obtained a magnetic tape containing 21 files in
the following sequence:
1. HELPLC (902)
2. HELPUC (902)
3. LC (1889)
4. MAIN (48)
5. MANLC (5150)
6. MANUAL (7456)
7. MANUC (5150)
8. PEN07 (9)
9. PEN18 (8)
10. PENDMY (3)
11. PLTDMY (169)
12. PLTLC (978)
13. PLTUC (980)
14. POS (14)
15. SCREEN (89)
16. SCROLL (86)
17. SUPPRT (3894)
18. UC (1862)
19. USRDMY (8)
20. USRLC (271)
21. USRUC (270)
Numbers in parentheses give the number of lines (of up to 80
characters each) in the files.
Below, the files are divided into several groups. One or two
files have to be picked from each of the groups 1 through 6. Within a
group, files are listed by decreasing convenience, and increasing
portability. If the bottom files are chosen from each group a completely
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 76
portable version of MICROSCOPE (except for the use of a few non-FORTRAN
characters, namely ?,<,>,!,:,;) is obtained.
-1- Case Dependent Support Routines
LC: Contains those subroutines that require the support of
lower case letters in order to work properly. This feature is not
portable, but widely available, and greatly increases the convenience in
using MICROSCOPE.
UC: To be used in case of LC, if lower case letters are not
available. UC is supplied as part of the package, but it can also be
generated from LC by replacing all lower case letters with their upper case
equivalents, and by replacing the subroutine LCUC by a dummy routine.
(Upper case versions of all other files can be generated from their lower
case counter parts by replacing lower case letters by uppercase ones
everywhere. The leading comments describing the file will of course no
longer apply.)
If UC is used in case of LC all output will be in upper case
letters, and lower case input will not be recognized.
LC and UC contain their respective versions of the master routine
MCRSCP, which is the first routines in each of the two files. All other
routines then follow alphabetically.
-2- Case Independent Routines
SUPPRT: This file contains the bulk of the MICROSCOPE package,
consisting of completely portable routines.
-3- The File Access Routine
POS: This file contains a
SUBROUTINE POS(IDEV,N)
which positions the device IDEV at the N-th record, such that subsequent
reading from IDEV yields the N-th record. This is accomplished by
rewinding IDEV and then reading a dummy character from the first N-1 lines
(records) in IDEV. If this at all time consuming it may be worthwhile to
replace POS by a faster system dependent routine.
-4- The Screen Display Package
SCREEN: To be used if screen editing is locally available, and
highly recommended over its alternative, SCROLL, described below. To take
advantage of screen editing, the following four small interface routines
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 77
must be written for each possible computer/terminal combination (in all
cases, IDEVCE is the channel number of the relevant terminal):
SUBROUTINE CLSCRN(IDEVCE,I,J)
INTEGER IDEVCE,I,J
C Clear the screen of the device with number IDEVCE below the point with
C the coordinates (I,J) where I is the horizontal coordinate measured
C from the left of the screen starting at 1, and J is the vertical
C coordinate measured from the top of the screen starting at 1.
...
RETURN
END
SUBROUTINE PCURSR(IDEVCE,I,J)
INTEGER IDEVCE,I,J
C Position the cursor at the position (I,J) of the device with number
C IDEVCE, where I,J are as for CLRSCRN
...
RETURN
END
SUBROUTINE PLCHRS(IDEVCE,I,J,N,CHARS)
INTEGER IDEVCE,I,J,N,CHARS(135)
C Write the string of N characters contained in A1 format in the integer
C array CHARS, starting at the location (I,J). I,J, and IDEVCE are as
C in CLRSCRN.
...
RETURN
END
SUBROUTINE BLSCRN(IDEVCE)
C Blank the screen of the device with number IDEVCE
...
RETURN
END
SCROLL: SCREEN can be replaced with SCROLL if the SCREEN package
cannot be used. This will have the consequence that the entire screen
display is regenerated after each change, rather than just modified.
-5- The PLOT package:
PLTLC: This file contains the interface between MICROSCOPE and
<PLOT79>. The routines use and recognize lower case letters.
PLTUC: The equivalent of PLTLC, but without the lower case
letters. Note, however, that <PLOT79> supports Hershey Fonts that include
lower case letters regardless of the computing system. Hence the <PLOT7
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 78
drawings generated by PLTUC do contain lower case letters. The
communication with the terminal via the command PLOT is completely in terms
of upper case letters, however.
With both the PLTLC or PLTUC package a pen selection routine also
has to be supplied (see section 4.14). The MICROSCOPE package contains
three routines in separate files: PEN18 corresponding to 8 pens labeled 1
to 8, PEN07 corresponding to 8 pens labeled 0 to 7, and PENDMY to be used
if color is not available (e.g. on dot matrix printer). However, you may
wish to leave it up to the user to select or write his own pen selection
routines.
PLTDMY: To be used if <PLOT79> is not available. In this case
only alphanumerical screen displays like those shown throughout this manual
can be obtained. However, the dummy routine simulates the interaction with
<PLOT79> up to the point at which the actual drawing takes place. This
feature is useful for the development of MICROSCOPE programs, and makes it
possible to run programs without <PLOT79> even if they call the package.
There is only the upper case version of PLTDMY.
-6- The Test package
Ordinarily, users will not be interested in the specific routine
SUBUSR employed in the generation of this manual. They will want to write
their own. However, the manual version of SUBUSR, and the routines called
by it, are contained in the files:
USRLC: contains the lower case version.
USRUC: contains the upper case version, to be used if lower
case letters are not available.
USRDMY: contains a dummy SUBROUTINE SUBUSR, which disables the
USER command. If the MICROSCOPE routines are held in a library it may be
possible to keep this dummy routine in it as a standby if the user does not
supply his own version.
-7- The <PLOT79> package
Obviously the <PLOT79> package must be linked with MICROSCOPE if
it is to be utilized. This is a large graphics package that can be easily
interfaced with almost any graphics device. It can be obtained for a
nominal charge from Dr. Nelson H.F. Beebe, Department of Physics,
University of Utah, Salt Lake City, Utah 84112, Tel.: 801-581-5254.
-8- Data files
Sample Driver Program
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 79
The file MAIN contains the driver program as it was used in the
generation of this manual. It is listed as a data file because the program
needs to be modified to suit local circumstances. In particular, the OPEN
statement is not portable.
Interactive Help Files
The package includes two help files that contain relevant help
information (but no code). Again, there are the lower and upper case
versions:
HELPLC
HELPUC
One of these files (preferably HELPLC) must be readable by the
user programs, and the users must be informed how to access them. It is
possible to design a sequence of MICROSCOPE commands that overwrites the
help file, and, if possible, the user should be denied write access to it.
You may also want to keep easily accessible backup copies of these files.
The help files each contain the following items:
1. On the first line there are three integers I,M,N in 3I3
format. These have the following meanings: I is the number of lines
occupied by the table of commands that can be accessed by the command
HSUMRY. M is the total number of lines occupied by the detailed information
accessed by HELP. N is the number of available commands.
2. The next four lines are occupied by N numbers written in 20I2
format. The k-th such number gives the length (in lines) of the detailed
HELP information on the k-th command. These N numbers must add up to M.
3. The next I lines contain the HSUMRY information
4. The next M lines contain the detailed HELP information.
5. The next line contains an integer J in I3 format that
contains the number of lines occupied by the news.
6. The last J lines contain the news.
Manual files
MANUAL: containes a verbatim machine readable copy of this
manual. It is written with FORTRAN carriage control and ready for printing
(at 60 lines per page and up to 75 characters per line). Underlining is
accomplished by having the underlines in a separate line with a "+"
character in the first column, indicating that the line is to overwrite the
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 80
preceding one. There is no upper case version of this file. If lower case
letters are not available a xeroxed copy of the manual rather than a
printed version should be distributed.
This installation guide is chapter five of the manual.
MANLC: This file contains a version of the manual that is
suitable for on-line reading and editing. Page breaks and underlines have
been eliminated, otherwise the text is identical to that in MANUAL.
MANUC: The upper case equivalent of MANLC.
5.2 A Key to Selecting the Appropriate Files
The following is a quick key to selecting the appropriate subset
of the above files. Proceed by answering the questions and following
instructions until you reach a line giving the relevant file names or
descriptions. The pen selection routine is described in section 4.14 (and
can be supplied by the user if necessary), all other ingredients are
described in this installation guide.
1. Are lower case letters available on your machine? Yes GO TO 2
No GO TO 3
2. Have you supplied an interface to screen editing? Yes GO TO 4
No GO TO 5
3. Have you supplied an interface to screen editing? Yes GO TO 6
No GO TO 7
4. Is <PLOT79> available? Yes GO TO 8
No GO TO 9
5. Is <PLOT79> available? Yes GO TO 10
No GO TO 11
6. Is <PLOT79> available? Yes GO TO 12
No GO TO 13
7. Is <PLOT79> available? Yes GO TO 14
No GO TO 15
8. LC, SUPPRT, POS or local equivalent, SCREEN, PLTLC, pen selection
routine, possibly USRLC, screen interface, HELPLC, MANUAL, MANLC
9. LC, SUPPRT, POS or local equivalent, SCREEN, PLTDMY, possibly USRLC,
screen interface, HELPLC, MANUAL, MANLC
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 81
10. LC, SUPPRT, POS or local equivalent, SCROLL, PLTLC, pen selection
routine, possibly USRLC, HELPLC, MANUAL, MANLC
11. LC, SUPPRT, POS or local equivalent, SCROLL, PLTDMY, possibly USRLC,
HELPLC, MANUAL, MANLC
12. UC, SUPPRT, POS or local equivalent, SCREEN, PLTUC, pen selection
routine, possibly USRUC, screen interface, HELPUC, MANUC
13. UC, SUPPRT, POS or local equivalent, SCREEN, PLTDMY, possibly USRUC,
screen interface, HELPUC, MANUC
14. UC, SUPPRT, POS or local equivalent, SCROLL, PLTUC, pen selection
routine, possibly USRUC, HELPUC, MANUC
15. UC, SUPPRT, POS or local equivalent, SCROLL, PLTDMY, possibly USRUC,
HELPUC, MANUC
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 82
Appendix I: Numerical Differentiation
In this appendix, we will discuss in some detail the properties
of the formulas listed in section 1.2. Without loss of generality we
assume the point of examination is the origin. We also restrict our
attention to purely tangential derivatives. Thus we may assume there is
only one independent variable. Of particular interest are step
discontinuities in certain derivatives (for other types of reduced
smoothness see the examples at the end of Appendix III). Thus we consider
the trial function
F(x) = PHI(x) + S(m,eta,x)
where PHI(x) is a smooth function (i.e. all its relevant derivatives are
continuous) and S is determined by the requirements that its m-th
derivative be piecewise constant and have a jump discontinuity of magnitude
eta at x = 0. Explicitly, S is defined by
m
S(m,eta,x) = 0 if x < 0 and eta*x /m! otherwise
where the exclamation mark denotes the factorial.
To simplify matters we will assume S is evaluated exactly, and
all round-off errors occur in PHI. The properties of the numerical
differentiation formulas depend very critically on the relative sizes of
the derivatives of PHI, about which little can be assumed in general. Our
approach will be to derive general formulas incorporating the derivatives,
but to illustrate the relevant phenomena by using
x
PHI(x) = e
since the derivatives of the exponential are all identical (assuming the
normalization option is on).
We will denote the degree of a derivative by parenthesized
superscripts.
A simple calculation shows that, for k >= m, the approximation
(k) (k)
Q of f at x = 0 is given by the fundamental equation
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 83
(k) (k)
Q = PHI true value
2 (k+2)
+ h *gamma *PHI truncation error
k
(1)
-k
+ tau*h *(1+abs(PHI))*alpha round-off term
k
m-k
+ eta*delta *h delta term
m,k
If k < m the delta term is absent. We discuss the individual
terms in turn:
- The true value
This is the expression which we are trying to approximate.
(k)
PHI is evaluated at zero.
- The truncation error.
This term can be obtained by expanding the error into a Taylor
series about h = 0, as is done in any textbook on numerical analysis. The
(k+2)
function PHI is to be thought of as having been evaluated at a
suitable (unknown) point in the domain of the display. The gamma are
k
constants given in table 1.
- The round-off term
We assume rounding is taking place as described in section 0,
i.e. if x is the quantity to be rounded to D digits, say, then x is
replaced by
-D -D
z := (1+eps *10 )*x + eps *10 (2)
1 2
where eps and eps are random numbers between -1 and +1. The quantity tau
1 2
in (1) is a suitable average of the eps and eps terms in (2). We will
1 2
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 84
-D
set it to 10 . The function PHI is again evaluated at a suitable point in
the display, and the alpha are the sum of the absolute values of the
k
coefficients in the k-th differentiation formula. Their values are listed
in table 1.
- The delta term.
We use the expression "delta term" because it arises from
differentiating the step function s(m,eta,x), and the first derivative of a
step function is a Dirac delta function. The delta are constants that
m,k
can be read off the MICROSCOPE plots in section 4.2. They are also given
in table 1.
k alpha gamma delta
k k m,k
m = 0 1 2 3 4 5 6
1 2 1/6 1/2 1 - - - - -
2 4 1/12 1 1 1 - - - -
3 24 1/16 4 2 1 1 - - -
4 256 1/24 48 16 3 1 1 - -
5 2430 1/27 365 81 14 3 1 1 -
6 46656 1/36 7290 1458 146 27 4 2 1
Table 1: Constants for Differentiation Formulas
We now address three questions:
1. When do round-off effects become visible?
2. When do round-off effects exceed the truncation error?
3. How small a discontinuity can be detected with MICROSCOPE?
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 85
I.1 The Round-Off Threshold
We are concerned with the question of when round-off effects
begin to obscure the graph of a derivative. More precisely, we ask when
the round-off term becomes larger than a certain fraction of the rise in
(k)
the graph of the derivative PHI over an interval of length s. An
equation for the critical step size, which constitutes a limit on h below
which no clear graphs can be obtained, is generated by ignoring the delta
(k)
term and the truncation error, and approximating the rise of PHI by the
mean value theorem. This leads to:
-k (k+1)
psi *h *(1+abs(PHI))*tau*alpha = 2*h*abs(PHI )/w
1 k
(k+1)
where w = 2h/s is the window width, PHI again is evaluated at a
suitable point, and psi is a safety factor. The larger psi , the clearer
1 1
(k)
will be the graph of PHI .
Solving for h yields:
(k+1) (
h = [psi (1+abs(PHI))*tau*alpha *w /(2*abs(PHI ))] 1/(k+
1 k
Table 2 contains threshold values for various values of D (the
number of digits carried) and psi = 5, for the case that PHI is the
1
exponential. All derivatives have been evaluated at x = 0.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 86
Digits (D) Degree of Derivative (k)
1 2 3 4 5 6
1 3.46D+00 2.88D+00 3.46D+00 4.34D+00 4.94D+00 6.00D+00
2 1.10D+00 1.34D+00 1.95D+00 2.74D+00 3.37D+00 4.32D+00
3 3.46D-01 6.21D-01 1.10D+00 1.73D+00 2.29D+00 3.11D+00
4 1.10D-01 2.88D-01 6.16D-01 1.09D+00 1.56D+00 2.24D+00
5 3.46D-02 1.34D-01 3.46D-01 6.88D-01 1.06D+00 1.61D+00
6 1.10D-02 6.21D-02 1.95D-01 4.34D-01 7.25D-01 1.16D+00
7 3.46D-03 2.88D-02 1.10D-01 2.74D-01 4.94D-01 8.34D-01
8 1.10D-03 1.34D-02 6.16D-02 1.73D-01 3.37D-01 6.00D-01
9 3.46D-04 6.21D-03 3.46D-02 1.09D-01 2.29D-01 4.32D-01
10 1.10D-04 2.88D-03 1.95D-02 6.88D-02 1.56D-01 3.11D-01
11 3.46D-05 1.34D-03 1.10D-02 4.34D-02 1.06D-01 2.24D-01
12 1.10D-05 6.21D-04 6.16D-03 2.74D-02 7.25D-02 1.61D-01
13 3.46D-06 2.88D-04 3.46D-03 1.73D-02 4.94D-02 1.16D-01
14 1.10D-06 1.34D-04 1.95D-03 1.09D-02 3.37D-02 8.34D-02
15 3.46D-07 6.21D-05 1.10D-03 6.88D-03 2.29D-02 6.00D-02
16 1.10D-07 2.88D-05 6.16D-04 4.34D-03 1.56D-02 4.32D-02
17 3.46D-08 1.34D-05 3.46D-04 2.74D-03 1.06D-02 3.11D-02
18 1.10D-08 6.21D-06 1.95D-04 1.73D-03 7.25D-03 2.24D-02
19 3.46D-09 2.88D-06 1.10D-04 1.09D-03 4.94D-03 1.61D-02
20 1.10D-09 1.34D-06 6.16D-05 6.88D-04 3.37D-03 1.16D-02
21 3.46D-10 6.21D-07 3.46D-05 4.34D-04 2.29D-03 8.34D-03
22 1.10D-10 2.88D-07 1.95D-05 2.74D-04 1.56D-03 6.00D-03
23 3.46D-11 1.34D-07 1.10D-05 1.73D-04 1.06D-03 4.32D-03
24 1.10D-11 6.21D-08 6.16D-06 1.09D-04 7.25D-04 3.11D-03
25 3.46D-12 2.88D-08 3.46D-06 6.88D-05 4.94D-04 2.24D-03
26 1.10D-12 1.34D-08 1.95D-06 4.34D-05 3.37D-04 1.61D-03
27 3.46D-13 6.21D-09 1.10D-06 2.74D-05 2.29D-04 1.16D-03
28 1.10D-13 2.88D-09 6.16D-07 1.73D-05 1.56D-04 8.34D-04
29 3.46D-14 1.34D-09 3.46D-07 1.09D-05 1.06D-04 6.00D-04
30 1.10D-14 6.21D-10 1.95D-07 6.88D-06 7.25D-05 4.32D-04
31 3.46D-15 2.88D-10 1.10D-07 4.34D-06 4.94D-05 3.11D-04
32 1.10D-15 1.34D-10 6.16D-08 2.74D-06 3.37D-05 2.24D-04
33 3.46D-16 6.21D-11 3.46D-08 1.73D-06 2.29D-05 1.61D-04
34 1.10D-16 2.88D-11 1.95D-08 1.09D-06 1.56D-05 1.16D-04
35 3.46D-17 1.34D-11 1.10D-08 6.88D-07 1.06D-05 8.34D-05
36 1.10D-17 6.21D-12 6.16D-09 4.34D-07 7.25D-06 6.00D-05
37 3.46D-18 2.88D-12 3.46D-09 2.74D-07 4.94D-06 4.32D-05
38 1.10D-18 1.34D-12 1.95D-09 1.73D-07 3.37D-06 3.11D-05
39 3.46D-19 6.21D-13 1.10D-09 1.09D-07 2.29D-06 2.24D-05
40 1.10D-19 2.88D-13 6.16D-10 6.88D-08 1.56D-06 1.61D-05
Table 2: Guide to selecting a value of the discretization paramete
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 87
I.2 Round-off versus Truncation Errors
In working with MICROSCOPE, one will rarely be interested in
obtaining accurate numerical values of derivatives. Rather, qualitative
features will be of primary concern. However, if accuracy is paramount,
then the discretization parameter should be picked as small as possible
without the round-off term swamping the truncation error. The optimal step
size is obtained by equating the round-off term with the truncation error
and solving for h. This yields:
(k+2) 1/(k+2)
h = [(1+abs(PHI))*tau*alpha / (gamma *PHI )]
k k
Table 3 lists values of h for the exponential.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 88
Digits (D) Degree of Derivative (k)
1 2 3 4 5 6
1 1.69D+00 2.49D+00 4.50D+00 8.25D+00 1.18D+01 1.88D+01
2 7.83D-01 1.40D+00 2.84D+00 5.62D+00 8.49D+00 1.41D+01
3 3.63D-01 7.87D-01 1.79D+00 3.83D+00 6.11D+00 1.06D+01
4 1.69D-01 4.43D-01 1.13D+00 2.61D+00 4.40D+00 7.93D+00
5 7.83D-02 2.49D-01 7.13D-01 1.78D+00 3.17D+00 5.95D+00
6 3.63D-02 1.40D-01 4.50D-01 1.21D+00 2.28D+00 4.46D+00
7 1.69D-02 7.87D-02 2.84D-01 8.25D-01 1.64D+00 3.34D+00
8 7.83D-03 4.43D-02 1.79D-01 5.62D-01 1.18D+00 2.51D+00
9 3.63D-03 2.49D-02 1.13D-01 3.83D-01 8.49D-01 1.88D+00
10 1.69D-03 1.40D-02 7.13D-02 2.61D-01 6.11D-01 1.41D+00
11 7.83D-04 7.87D-03 4.50D-02 1.78D-01 4.40D-01 1.06D+00
12 3.63D-04 4.43D-03 2.84D-02 1.21D-01 3.17D-01 7.93D-01
13 1.69D-04 2.49D-03 1.79D-02 8.25D-02 2.28D-01 5.95D-01
14 7.83D-05 1.40D-03 1.13D-02 5.62D-02 1.64D-01 4.46D-01
15 3.63D-05 7.87D-04 7.13D-03 3.83D-02 1.18D-01 3.34D-01
16 1.69D-05 4.43D-04 4.50D-03 2.61D-02 8.49D-02 2.51D-01
17 7.83D-06 2.49D-04 2.84D-03 1.78D-02 6.11D-02 1.88D-01
18 3.63D-06 1.40D-04 1.79D-03 1.21D-02 4.40D-02 1.41D-01
19 1.69D-06 7.87D-05 1.13D-03 8.25D-03 3.17D-02 1.06D-01
20 7.83D-07 4.43D-05 7.13D-04 5.62D-03 2.28D-02 7.93D-02
21 3.63D-07 2.49D-05 4.50D-04 3.83D-03 1.64D-02 5.95D-02
22 1.69D-07 1.40D-05 2.84D-04 2.61D-03 1.18D-02 4.46D-02
23 7.83D-08 7.87D-06 1.79D-04 1.78D-03 8.49D-03 3.34D-02
24 3.63D-08 4.43D-06 1.13D-04 1.21D-03 6.11D-03 2.51D-02
25 1.69D-08 2.49D-06 7.13D-05 8.25D-04 4.40D-03 1.88D-02
26 7.83D-09 1.40D-06 4.50D-05 5.62D-04 3.17D-03 1.41D-02
27 3.63D-09 7.87D-07 2.84D-05 3.83D-04 2.28D-03 1.06D-02
28 1.69D-09 4.43D-07 1.79D-05 2.61D-04 1.64D-03 7.93D-03
29 7.83D-10 2.49D-07 1.13D-05 1.78D-04 1.18D-03 5.95D-03
30 3.63D-10 1.40D-07 7.13D-06 1.21D-04 8.49D-04 4.46D-03
31 1.69D-10 7.87D-08 4.50D-06 8.25D-05 6.11D-04 3.34D-03
32 7.83D-11 4.43D-08 2.84D-06 5.62D-05 4.40D-04 2.51D-03
33 3.63D-11 2.49D-08 1.79D-06 3.83D-05 3.17D-04 1.88D-03
34 1.69D-11 1.40D-08 1.13D-06 2.61D-05 2.28D-04 1.41D-03
35 7.83D-12 7.87D-09 7.13D-07 1.78D-05 1.64D-04 1.06D-03
36 3.63D-12 4.43D-09 4.50D-07 1.21D-05 1.18D-04 7.93D-04
37 1.69D-12 2.49D-09 2.84D-07 8.25D-06 8.49D-05 5.95D-04
38 7.83D-13 1.40D-09 1.79D-07 5.62D-06 6.11D-05 4.46D-04
39 3.63D-13 7.87D-10 1.13D-07 3.83D-06 4.40D-05 3.34D-04
40 1.69D-13 4.43D-10 7.13D-08 2.61D-06 3.17D-05 2.51D-04
Table 3: Values of h for maximum accuracy
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 89
I.3 Derivatives of Step Functions
It turns out that very small jumps (of magnitude eta, say) in
some derivative can be detected only by examining a higher order derivative
rather than the discontinuous derivative itself. We determine optimal
values of h and minimum values of eta by imposing two requirements. First,
the delta term should be significantly larger than round-off errors, and,
second, the delta term should be significantly larger than the rise of
(k)
PHI over half the window. This leads to the equations
m-k -k
eta*delta *h = psi *h *(1+abs(phi))*tau*alpha
m,k 2 k
and
m-k (k+1)
eta*delta *h = psi *h*PHI
m,k 3
where psi and psi are safety factors.
2 3
We consider two cases. If m = 0 (i.e. there is a step in the
trial function) equation (4) is independent of h. Solving for eta yields
the minimum value
eta = psi *(1+abs(PHI))*tau*alpha / delta
2 k 0,k
The maximum value of h (above which the delta term will be
(k)
obscured by the growth in PHI ) is obtained by solving (5) for h:
(k+1) 1/(k+1)
h = [eta*delta / (psi *PHI )]
0,k 3
If m > 0 (i.e. the discontinuity occurs in a derivative) the
minimum value of eta can be computed by eliminating h between (4) and (5),
and solving for eta. This gives
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 90
(k+1) m
eta = [(psi *abs(PHI ))
3
k+1-m 1/(k+1)
*(psi *(1+abs(PHI))*tau*alpha ) )] / delta m,
2 k (
The corresponding optimal value of h (above which the delta term
(k)
is too small to be visible in the general growth of PHI and below which
it is obscured by round-off errors) can be obtained by eliminating eta
between (4) and (5) and solving for h, giving
1/(k+1)
h = (psi *(1+abs(PHI))*alpha *tau / psi )
2 k 3
Notice that this expression is independent of m, the degree of the
derivative in which the jump occurs.
Table 4 lists minimum values of eta (in the last seven columns)
and the corresponding optimal value of h (in the second column) for the
case of the exponential, and for a few values of tau. Notice how the size
of the jump that can be detected is sometimes decreased by several orders
of magnitude if higher order derivatives are examined.
tau = 1.0D-10
Degree of discontinuous derivative
k optimal h 0 1 2 3 4 5 6
0 5.0D-10 1.5D-09
1 4.5D-05 1.2D-08 1.3D-04
2 1.6D-03 1.2D-08 7.6D-06 4.8D-03
3 1.2D-02 1.8D-08 2.9D-06 4.6D-04 3.7D-02
4 4.8D-02 1.6D-08 1.0D-06 1.1D-04 6.9D-03 1.4D-01
5 1.2D-01 2.0D-08 7.8D-07 3.9D-05 1.6D-03 4.0D-02 3.5D-01
6 2.4D-01 1.9D-08 4.0D-07 1.7D-05 3.7D-04 1.0D-02 8.7D-02 7.2D-01
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 91
tau = 1.0D-17
Degree of discontinuous derivative
k optimal h 0 1 2 3 4 5 6
0 5.0D-17 1.5D-16
1 1.4D-08 1.2D-15 4.2D-08
2 7.4D-06 1.2D-15 1.6D-10 2.2D-05
3 2.2D-04 1.8D-15 1.6D-11 1.5D-07 6.6D-04
4 1.9D-03 1.6D-15 2.5D-12 7.0D-09 1.1D-05 5.7D-03
5 7.9D-03 2.0D-15 1.1D-12 8.3D-10 4.9D-07 1.9D-04 2.4D-02
6 2.4D-02 1.9D-15 4.0D-13 1.7D-10 3.7D-08 1.0D-05 8.7D-04 7.2D-02
tau = 1.0D-24
Degree of discontinuous derivative
k optimal h 0 1 2 3 4 5 6
0 5.0D-24 1.5D-23
1 4.5D-12 1.2D-22 1.3D-11
2 3.4D-08 1.2D-22 3.5D-15 1.0D-07
3 3.9D-06 1.8D-22 9.1D-17 4.6D-11 1.2D-05
4 7.6D-05 1.6D-22 6.3D-18 4.4D-13 1.7D-08 2.3D-04
5 5.4D-04 2.0D-22 1.7D-18 1.8D-14 1.6D-10 8.7D-07 1.6D-03
6 2.4D-03 1.9D-22 4.0D-19 1.7D-15 3.7D-12 1.0D-08 8.7D-06 7.2D-03
tau = 1.0D-31
Degree of discontinuous derivative
k optimal h 0 1 2 3 4 5 6
0 5.0D-31 1.5D-30
1 1.4D-15 1.2D-29 4.2D-15
2 1.6D-10 1.2D-29 7.6D-20 4.8D-10
3 7.0D-08 1.8D-29 5.1D-22 1.5D-14 2.1D-07
4 3.0D-06 1.6D-29 1.6D-23 2.8D-17 2.8D-11 9.1D-06
5 3.7D-05 2.0D-29 2.5D-24 3.9D-19 4.9D-14 4.0D-09 1.1D-04
6 2.4D-04 1.9D-29 4.0D-25 1.7D-20 3.7D-16 1.0D-11 8.7D-08 7.2D-04
Table 4: The superiority of testing Delta Functions and their derivatives
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 92
Appendix II: Organization of MICROSCOPE
This appendix contains (1) a brief discussion of the
organizational framework upon which MICROSCOPE is built, (2) a detailed
explanation of the sampling scheme and logic which MICROSCOPE uses to
calculate the tangential derivatives, and (3) a brief description of the
common blocks used.
MICROSCOPE has evolved over a period of several years,
progressing through several preliminary versions. The core of the present
version was designed and written by Bill Harris during the summer of 1983.
MICROSCOPE then grew, under the hands of Peter Alfeld, to a substantially
larger package which is the current version. This explains lack of
uniformity in the commenting of the source code. It also accounts for the
inconsistency in the notation used within the program versus that employed
in this manual. However, only very serious users interested in
understanding and possibly manipulating the code (for whome this appendix
is written) should suffer from such idiosyncrasies of the package.
II.1 Organization of the Program Package
MICROSCOPE follows a common and well established format for
interactive programs in FORTRAN. The user must provide a short main
program, which assigns FORTRAN Input/Output device numbers and screen
parameters, and then calls a master routine (MCRSCP). MCRSCP is a driver
program from which all other routines are called and to which execution
returns after each command has been completed. This is accomplished by (i)
prompting for a new command, (ii) reading and recognizing the new command,
(iii) using a computed GO TO to branch to the appropriate routine, (iv) a
bit of logic and computation (i.e. sampling the function and calculating
the derivatives), (v) displaying the results in some graphic form, and (vi)
updating an options table and returning to step (i).
To help understanding the following flowchart several ideas and
their representative variables need to be introduced. The logical variable
LGO indicates whether the screen is active (LGO = .TRUE) or inactive (LGO =
.FALSE.). It is set and reset at a number of places in the package. The
only place where the value of LGO has an effect on the program flow is in
the routine MCRSCP, between statements 7900 and 8300.
There are 4 basic types of commands. These correspond to the
integer variable ITYPE which is set appropriately by each command.
Commands of ITYPE = 1 imply that the function must be sampled from scratch.
Since other changes may be desired before computation begins, these
commands deactivate the screen and this is accomplished by setting
LGO = .FALSE.. Commands which are of ITYPE = 2 require some computation.
Two special ITYPE = 2 commands are FORCE and GO. These cause LGO to be set
to .TRUE. and are the only commands that activate an inactive screen. The
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 93
remaining commands of ITYPE = 2 only make modifications of the current
setup and usually only require small amounts of additional computation.
When LGO = .TRUE. any ITYPE = 2 command is executed before the next command
is prompted for. ITYPE = 3 commands do not require computation but cause
the alphanumerical display to be altered. Finally, commands of ITYPE = 4
do not alter the display and do not require computation.
The other major logical control variable is LSCRN. Its main use
is in screen-editing versions of MICROSCOPE. LSCRN = .FALSE. means that
the alphanumerical display has been overwritten by text and must be
regenerated. LSCRN = .TRUE. means that the display is intact from the
previous command and only selected pixels need to be updated. LSCRN is
used by the display routine SGRAPH in the file SCREEN .
Computation time for evaluating display functions can be sizable.
The program logic and sampling scheme were designed to utilize previous
computations and eliminate unnecessary recomputations to the extent
possible. For example, shifts and magnification changes often require only
a few additional evaluations.
The flowchart that follows gives an overview of the logical flow
in the MCRSCP driver routine. It concentrates on the portion between
statement 1500 and 8300 where the logic is somewhat involved.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 94
Epitome Flowchart
!
! from Main Program
!
V
* output message identifying MICROSCOPE
* check screen parameters
* copy channel numbers, screen parameters, and set flags
* set LGO = .FALSE.
* compute round-off characteristics (NUMDIG)
* set default values for some of the variables contained in
the various common blocks
* input some of the help documentation as well as news (INHELP)
1500 * prompt/input/parse command (DIALOG)
* reset error flags
* branch to the appropriate subroutine (using a computed GO TO)
This corresponds to statements 1600 through 7800. Many
of these commands alter variables, with subscript 2, in
the common block OPTION. These will then be used by
later routines.
7900 * if command is of ITYPE = 1 (i.e. requires complete recomputation)
then set: LGO = .FALSE.
* if ITYPE = 2 and LGO = .TRUE.
then calculate values for NUM and DEN corresponding to
sampling changes. (See documentation in part 2
of this appendix) (SGAMMA)
* check for errors and output appropriate messages if any occurred
* if error = .TRUE.
<------------then go to 1500
* if LGO = .FALSE. and the Refresh Screen (RSCREEN) command is given
then display the old graphical information (SGRAPH)
* if LGO = .FALSE.
<------------then go to 1500
* branch to the appropriate section of the program depending on
the value of ITYPE (using a computed GO TO)
8000 * (if ITYPE = 2)
* check to see which aspects of the sampling have been altered
by the command and set the appropriate logical flags (CHKCMP)
* call the routines flagged by the CHKCMP routine.
(the actual sampling and computation of derivatives occurs
only in the routine SCMUPD)
8100 * (if ITYPE = 3)
* set up the graphics data (i.e. update plot array) (SGRUPD)
* graph the data (SGRAPH)
* if logging is required
then output a copy of the graphical display in scroll mode
* copy current set of parameter values from new to old. (OPCOPY)
<-------* go to 1500
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 95
II.2 Sampling Scheme and Logic
In the remainder of this appendix an explanation of the sampling
scheme and computational logic is given. The variable h, as used in the
source code, is not the same as the variable labeled as h on output. To
reduce confusion we will use the following notation in this appendix (but
nowhere else in this manual):
Variable in Variable in Variable in Output
Source Code Appendix and elsewhere in
the manual
h ss (step size) s
sw sw (stencil width) h
The output names correspond to the common usage while the names
used in the program and appendix have the meanings defined below.
The basic idea of the sampling scheme consists of representing
the relevant part of the line of investigation in two linear arrays, one
corresponding to points in the domain, and one containing corresponding
values of the display function (if it has been evaluated). Any changes in
the stencil width and step size, as well as small shifts of the point of
examination are then implemented by shuffling the information in these two
arrays. The size of the arrays has been chosen to accomodate fairly large
changes in parameter values with little computation. The key variables and
arrays used in the program are as follows:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 96
XS(IXSIZE): the sampling array for the domain space, giving the
distance from the point of examination in units of the direction
of investigation.
FS(IXSIZE): the sampling array containing the value of the display
function f at the corresponding point in XS().
IXSIZE: the dimension of the XS() and FS() arrays. (the current
value of IXSIZE is 5377)
ICENTR: the index of the center element of the XS() and FS() arrays.
(the current value of ICENTR is 2689)
ROPSTS(2): this is the "step size" corresponding to 8 points in the
XS() array. (see the illustration below) It appears in the common
block OPTION. The reason for introducing it here is that for most
computations it will remain fixed while the other parameters,
such as sw and ss, will be doubled, halved, etc.
ss: (step size) the spacing between the points plotted along the
line of investigation. This is h in the source code.
sw: (stencil width) the width of the interval over which the
finite-difference formulas are calculated. This is h on output.
nh: the spacing between plotted points in the index space. It must
be a positive integer multiple of 4.
nw: half the stencil width in index space. It must also be a
positive integer multiple of 4. The number of points in index
space corresponding to the stencil width is 1 + 2*nw.
The sampling points are spaced unevenly along the line of
investigation, but evenly in index space. In the following figures,
asterisks denote the points at which the display function needs to be
evaluated. A hyphen marks a point for which space for a function value has
been provided in the arrays XS and FS, but where no evaluation is
necessary. Consider a point p at which we wish to evaluate the display
function and its derivatives up to order 6. The stencil is centered about
p.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 97
p
-.......-...-...-.......*.......-...-...-.......- f(p)
.
. (1)
*.......-...-...-.......-.......-...-...-.......* f (p)
.
. (2)
*.......-...-...-.......*.......-...-...-.......* f (p)
.
. (3)
*.......-...*...-.......-.......-...*...-.......* f (p)
.
. (4)
*.......-...*...-.......*.......-...*...-.......* f (p)
.
. (5)
*.......*...-...*.......-.......*...-...*.......* f (p)
.
. (6)
*.......*...-...*.......*.......*...-...*.......* f (p)
sw sw sw sw sw sw sw sw
- -- - -- - -- - -- 0 -- -- -- --
2 3 4 6 6 4 3 2
<---------------------- sw --------------------->
(h on output)
The figure describes the actual sampling structure along the line
of investigation. Space has been provided only for points at which the
display function may have to be evaluated. Providing space for sampling at
increments of sw/12 would be simpler but wasteful since at most only 9
points (not 13) are needed in each stencil.
Next, consider the problem of extending the sampling over an
entire line segment. The particular extension used in MICROSCOPE was
chosen so that changing the step size and stencil width by powers of two
would require minimal computation. Other factors are allowed but require
greater computation.
Considering a portion of the line segment containing a sequence
of three plotted points: p1,p2, and p3, the spacing (for the ratio sw/ss =
2/1, nw = 4, and assuming all tangential derivatives are to be plotted)
looks like:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 98
..*.....*..*..*.....*.....*..*..*.....*.....*..*..*.....*.....*..*..*.....*..
. p1 p2 p3 .
. . . . .
*.....*..*..*.....*.....*..*..*.....* (stencil for p1). .
-4 -3 -2 -1 0 1 2 3 4 . .
. . . .
*.....*..*..*.....*.....*..*..*.....* (stencil for p2).
-4 -3 -2 -1 0 1 2 3 4 .
. . . .
.(stencil for p3) *.....*..*..*.....*.....*..*..*.....*
-4 -3 -2 -1 0 1 2 3 4
<---------------- sw --------------->
<------ ss ------> <------ ss ------>
Note how the stencils overlap, allowing the use of one value of
the display function for more than one (here two or three) values of a
derivative. If the stencil width is doubled while the step size is kept
constant (corresponding to an increase in the window width by a factor 2)
we obtain:
..*.....*..*..*.....*.....*..*..*.....*.....*..*..*.....*.....*..*..*.....*..
. p1 p2 p3 .
..*.....*..-..-.....*.....-..-..*.....*.....*.....-.....* (stencil for p1).
-4 -3 0 3 4 5 8 .
. . . . .
*........-..*.....*.....*..-..-.....*.....-.....*.....*.....*...........*
-8 -5 -4 -3 0 3 4 5 8
. . . .
.(stencil for p3) *.....-..-..*.....*.....*...........*.....-.....*.....*..
-8 -5 -4 -3 0 3 4
. . . . .
<---------------------------------- sw --------------------------------->
<------ ss ------> <------ ss ------>
The above operation requires additional evaluation of the display
function only at the boundary of the interval of investigation. The
reverse process of halving the window width would require no additional
evaluations at all.
So that all derivatives, up to and including the sixth, can be
evaluated, the minimal number of points between plotted points (such as
p1,p2,p3) is 4. Furthermore, the distance is always a multiple of 4.
Similarly, the minimal number of points needed in a stencil is 9 (i.e.
-4,...,0,...,4), and it is always of the form 1+(an integer multiple of 8),
although evaluation takes place at at most 9 points. These features are
illustrated in the following progression of increasing stencils:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 99
..*.....*..*..*.....*.....*..*..*.....*.....*..*..*.....*.....*..*..*.....*..
. . p2 . .
. *.....*..*..*.....*.....*..*..*.....* .
. -4 -3 -2 -1 0 1 2 3 4 .
. . . . .
*.....-..-..*.....*.....*..-..-.....*.....-..-..*.....*.....*.....-.....*
-8 -5 -4 -3 0 3 4 5 8
. . . . .
..*.....-..*..-.....*.....-..-..-.....*.....-..-..-.....*.....-..*..-.....*..
-8 -6 -4 0 4 6 8
These 3 stencils span index sets of 9,17, and 25 points,
respectively (although the third set would not fit on this page). The
variable nw for these is 4,8, and 12, respectively.
The relation between the quantities ss and ROPSTS(2) is given by
nh
ss = -- * ROPSTS(2)
8
The program strives to maintain a value of nh = 8. For a more
detailed discussion see the description of the routine SGAMMA below.
The relation between the point of examination, P, a point of
evaluation, PS, the distance t of ps from P, the direction of examination,
d, and the quantities in MICROSCOPE, is given by:
.
.
.
XS(ICENTR-2) = -ROPSTS(2)/4
XS(ICENTR-1) = -ROPSTS(2)/6
XS(ICENTR) = 0
XS(ICENTR+1) = ROPSTS(2)/6
XS(ICENTR+2) = ROPSTS(2)/4
XS(ICENTR+3) = ROPSTS(2)/3
XS(ICENTR+4) = ROPSTS(2)/2
XS(ICENTR+5) = 2*ROPSTS(2)/3
XS(ICENTR+6) = 3*ROPSTS(2)/4
XS(ICENTR+7) = 5*ROPSTS(2)/6
XS(ICENTR+8) = ROPSTS(2)
.
.
.
and
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 100
P = ROPPNT(*,2)
d = ROPUDI(*,2)
t = XS(i)
PS = P + t * d
FS(i) = f(PS)
LDEF(I) = .TRUE. <===> FS(I) has been evaluated
To explain the calculation of the derivatives, consider a point p
at which we wish to evaluate the function and its derivatives. To evaluate
all derivatives up to the sixth we need 9 points. We number them as
follows:
1 2 3 4 5 6 7 8 9
*.......*...*...*.......*.......*...*...*.......*
. . . . p . . . .
. . . . . . . . .
sw sw sw sw 0 sw sw sw sw
- -- - -- - -- - -- -- -- -- --
2 3 4 6 6 4 3 2
Instead of using the formulas given in section 1.2 directly, the
following equivalent but more efficient formulas are used:
g0 = f5
g1 = f1 + f9
g2 = - ( f1 - f9 )
g3 = 4 * ( f3 + f7 )
g4 = 4 * ( f3 - f7 )
g5 = 4 * ( f2 - f8 ) + 5 * ( f6 - f4 )
g6 = - 6 * ( f2 + f8 ) + 15 * ( f4 + f6 )
(0)
Q = g0
i
(1)
Q = (1/sw) * g2
i
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 101
(2) 2
Q = (2/sw) * ( g1 - 2*g0 )
i
(3) 3
Q = (4/sw) /4 * ( 2*g2 + g4 )
i
(4) 4
Q = (4/sw) * ( g1 - g3 + 6*g0 )
i
(5) 5
Q = (6/sw) /2 * ( g2 + g5 )
i
(6) 6
Q = (6/sw) * ( g1 + g6 - 20*g0 )
i
The sequence in which MCRSCP calls the main routines is:
SGAMMA - calculates gamma (see following discussion)
CHKCMP - checks to see what changes in computations are needed
SSMPUD - shift and copy
SCMUPD - compute new values
When a step size/stencil width changing command is given, the
program will attempt to change nh and nw without having to change ROPSTS(2)
(and hence copy or throw out data). Otherwise, MCRSCP attempts to choose a
change in scaling (ROPSTS(2)) which saves as much data as possible. This
is the purpose of the routine SGAMMA which we now discuss in some detail.
First we introduce additional notation.
width: the number of columns on the terminal screen used for plotting
m: scaling factor for step size (either a positive integer or 1/(positive
integer) )
v: scaling factor for stencil width (or window), (either positive integer
or 1/(positive integer) )
rho: lowest common multiple of the denominators of m and v.
s: = ROPSTS(2). (base step size when nh = 8), (this differs from the s
that denotes the interval elsewhere in this manual)
Primes will denote the new values for nh,nw,ss,and s. For
example: h' denotes the step size after changes have been made.
Define: gamma = s/s' = num/den > 0
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 102
Constraints: The commands set the values of m and v. There are 5
constraints which are then used to choose the value of gamma. They are not
all independent of each other.
(1) 4 <= nh' <= IXSIZE/width
(2) 4 <= nw' <= IXSIZE/2
(3) nh'/4 must be integer <===> mod(nh',4) = 0
(4) nw'/4 must be integer <===> mod(nw',4) = 0
(5) effective sampling width = nh' * width + 2 * nw' <= IXSIZE
in the index space
We have:
ss = (nh/8) * s and ss' = (nh'/8) * s'
Also:
ss' = m * h
Hence:
ss' = (nh'/8)*s' = m * h = m * (nh/8) * s
===> (nh'/8) * s' = m * (nh/8) * s
===> nh' = m * nh * s/s' = m * nh * gamma
Similarly:
sw = (nw/4) * s, sw' = (nw'/4) * s', and sw' = v * sw
===> sw' = (nw'/4) * s' = v * sw = v * (nw/4) * s
===> nw' = v * nw * s/s' = v * nw * gamma
In summary:
nh' = m * nh * gamma
nw' = v * nw * gamma
Referring to the conditions numbered above, divide (1) by m * nh and (2) by
v * nw to get:
4 IXSIZE 1
-------- <= gamma <= ------- * --------
m * nh width m * nh
and
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 103
4 IXSIZE 1
-------- <= gamma <= ------- * --------
v * nw 2 v * nw
Let
4 4
a = max( ------- , ------- )
m * nh v * nw
and
IXSIZE 1 IXSIZE 1
b = min( -------- * ------- , -------- * -------- )
width m * nh 2 v * nw
From above, gamma must satisfy:
a <= gamma <= b
Furthermore:
nh' = m * nh * gamma and mod(nh',4) = 0
===> (*) mod( m * nh * gamma , 4 ) = 0
nw' = v * nw * gamma and mod( v * nw * gamma , 4 ) = 0
===> (**) mod( v * nw * gamma , 4 ) = 0
The routine SGAMMA uses the above two conditions (*) and (**) to
choose acceptable values for gamma (and hence num and den since gamma =
num/den).
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 104
Outline of SGAMMA
* check to see if the changes m and v can be accomodated by changing
nh and nw, but not ss (h in code).
* if no changes in ss are needed
<---------- then num = den = 1 ==> gamma = 1 and return
* calculate a and b
* if m = v
<---------- then set gamma = m, ROPSTS(2) = m * ROPSTS(2), and return
* calculate rho
m' = rho * m (m' and v' are positive integers)
v' = rho * v
* determine "all" possible numerators and denominators for gamma
(up tp 32) which satisfy:
a <= num/den <= b
mod( m * nh * gamma , 4 ) = 0
mod( v * nw * gamma , 4 ) = 0
* rank the candidate values of gamma according to 4 criteria
(1) nh' * width + 2 * nw' <= IXSIZE
(2) 8 * width <= nh' * width + 2 * nw' <= ICENTR
(this allows for halving and doubling without rescaling)
(3) gamma a power of 2 (positive or negative)
(4) nh' > 4 and nw' > 4
* choose the combination of num and den which has the highest cor-
responding rank and set the variables accordingly.
<--* return
II.3 Common Blocks
Here we list the COMMON blocks used in MICROSCOPE. This
information may be useful in modifying the code and in accessing the data
structures of the package via the command USER (see section 4.15). For
each common block we list in parentheses the number of words (i.e. 2 words
for each DOUBLE precision variable and 1 word for each REAL, INTEGER, or
LOGICAL variable) occupied by the block. The blocks are listed in the same
sequence as they appear in the routine MCRSCP.
CB (16) controls cross differentiation.
CBDSW : width of differentiation stencil
CBD(3) : normalized cross direction
CBDU(3) : unnormalized cross direction
ICBD : order of cross differentiation
LCBD : .TRUE. if cross differentiation is on, .FALSE.
otherwise
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 105
IO (5) contains the input/output device numbers.
RECORD : Record device number. (Output data file for recording
screen images and comments)
GRAPHD : Graphics device number. (terminal or graphics display)
HELPD : Help file device number.
RSTRTD : Restart file device number. (unformatted output data
file for use in restarting at a later date)
INPUTD : Input device number. (usually the terminal)
LOG (1) controls logging
LCHN : Logging device number (= 0 <===> logiing is off
LOGCOM (5391) contains logical arrays used primarily by SCMUPD
to keep track of what has already been computed.
LDF(7) : Denotes whether the (i-1)'th derivative has been calcu-
lated. LDF(i) = .FALSE. means the (i-1)'th derivative
has not been calculated. LDF(i) = .TRUE. means it has.
LPLT(7) : Denotes whether the (i-1)'th derivative is currently
being displayed. LPLT(i) = .FALSE. means the (i-1)'th
derivative is not being displayed and LPLT(i) = .TRUE.
means it is.
LDEF(5377) : Denotes whether or not the interpolation function has
been calculated at the coordinates corresponding to
XS(i). (i.e. whether FS(i) has been assigned)
LDEF(i) = .FALSE. means the i'th value has not been
sampled and LDEF(i) = .TRUE. means it has.
FUNCOM (23,426) contains arrays pertaining to the values of the
interpolation function at its sampled points.
XS(5377) : The distance array, always centered about 0. Its defini-
tion and usage is given in the second part of this
appendix.
FS(5377) : The array containing the values of the interpolation
function corresponding to the points specified by the
XS() array. See the second part of this appendix for
discussion.
DF(135,7) : The finite differences. DF(i,j) corresponds to the i'th
plotted point and (j-1)'th derivative. These are the
p1,p2,p3's of the discussion in the second part of
this appendix.
DFMNMX(2,7): These are the minimums/maximums of the finite differences
in the DF() array.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 106
ERRCOM (2) used to handle error checking and reporting.
When errors are detected in any of the sub routines
or computation routines, the errors are flagged and
then reported by the CHKERR routine.
ERRCOD : This is an integer code number for the type of error
detected.
ERR : This is the logical flag which denotes whether or not
an error has occurred. ERR = .FALSE. means no error
has occurred and ERR = .TRUE. means an error has
occurred.
PLTCOM (16342) contains display versions of the derivatives.
IPLOT(135,7): IPLOT(i,j) is the "screen coordinate" (integer) of the
(j-1)'th derivative in the i'th column.
ISCRN1(135,57): The previous screen image. ISCRN1(i,j) contains the
Hollerith representation of the character to be
displayed at the (i,j)'th pixel on the screen.
(where indexing starts at the upper left corner
of the screen). i is the horizontal component and
j is the vertical component.
ISCRN2(135,57): The current screen image. This has the same description
as ISCRN1(,) above.
SCALE(7) : The scale factors used to individually scale, up or down,
the finite difference "curves" (in the DF(,) array)
so that the curve fills up the entire screen in the
sense that the maximum point lies on the top row and
minimum point lies on the bottom row of the screen.
LOADIO (3) contains channel numbers of the loading devices.
PLOAD : point loading device
DLOAD : direction loading device
CLOAD : cross direction loading device
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 107
SCREEN (7) contains the values of parameters describing the
device on which most interaction occurs (i.e. the
screen of the terminal in most cases)
OUTPUT : The device number of the graphics device (usually the
terminal)
LINES : The number of usable lines on the screen. Note that
this may have to be one less than the number of
displayable lines.
WIDTH : The number of columns to be displayed. It is best to
choose an odd number for this so that the center
can be plotted. It must be less than or equal to
135. 75 is a common value.
ILP : The number of lines in the curve plotting portion of the
display. It must be less than or equal to 57.
IPRMPT : The number of lines above the screen bottom at which
the prompting commands are given. It must be at
least 2.
IDSPLA : The number of lines above the screen bottom at which the
numerical data region begins. It must be greater than
IPRMPT by 6 or more.
LSCRN : The logical variable denoting whether the curves are
currently being displayed or they have been over-
written by text. This is to allow the program to
take advantage of terminals which allow screen
editing. LSCRN = .FALSE. means that the screen has
been overwritten since the previous command and
LSCRN = .FALSE. means that the screen is intact from
the previous command.
OPTION (122) contains most of the variables, other than the
sampling arrays in FUNCOM, which may change due to
commands. There are 2 copies of each variable.
(1) is the previous value and (2) is the current
value. If mistakes are made in the inactive (deactiv)
mode (i.e. LGO = .FALSE.) then the UNDO command will
erase the changes and copy the old set (1) to (2).
ROPDI(3,2) : The vector specifying the direction of investigation.
Note that for all vector valued quantities, the
dimension is specified by IOPDM() below.
ROPPNT(3,2): The vector specifying the center of the line of
investigation. If the shift is 0 (i.e. IOPSH(2) = 0)
then this is the point which is at the center of the
screen.
ROPDR1(3,2): Currently unused vector.
ROPDR2(3,2): Currently unused vector.
ROPSTS(2) : The nominal spacing between the points at which the
finite differences are calculated. These will be
the plotted points. "Nominal" refers to the base
step size when nh(2) = 8, as discussed in the second
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 108
part of this appendix. As noted earlier, this is
not the h which appears on output.
ROPSTW(2) : The stencil width (sw). This corresponds to the h which
appears on output. It is discussed in the second
part of this appendix.
ROPUDI(3,2): The normalized (unit) vector specifying the direction
of investigation. This is the normalized version
of the ROPDI() vector.
NH(2) : The step size in index space. See the discussion in the
second part of this appendix.
NW(2) : Half of the stencil width in index space. See the
discussion in the second part of this appendix.
ILEFT(2) : The index of the left-most element of the FS() array
which has been assigned a value.
IRIGHT(2) : The index of the right-most element of the FS() array
which has been assigned a value.
IOPDM(2) : The dimension of the domain space. It can be 1,2 or 3.
IOPSH(2) : The shift, in integer increments, of the plotted points
from the center defined by ROPSTS(*,2).
IOPSS(2) : The scaling factor for the step size. It is set by
the commands.
IOPWN(2) : The scaling factor for the stencil width. This
multiplies/divides the current window, as it appears
on the screen. It is set by the commands.
OPDC(2) : The logical flag determining whether or not the center
of the screen is to be marked.
OPDS(2) : The logical flag determining whether or not a scale is
to be displayed along the horizontal axis.
OPDX(2) : The logical flag determining whether or not a line of
dashes will be displayed to represent the x-axis.
OPCMP(2) : The logical flag denoting whether or not any computations
have yet been done for the current set of ROPPNT and
ROPDI.
OPSPL(2) : Currently not in use.
OPDF1(7,2) : The logical array specifying which derivatives are to
be displayed. It is set by the DGRAPH and EGRAPH
commands.
OPDF2(7,2) : The logical array specifying which derivatives are to
to be accentuated as they are displayed. It is set
by the ACCENT command.
NRMLZE (1) controls normalization of derivatives
NORMAL : = .TRUE. <===> normalization is on
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 109
ROOM (2) controls whether one line of the graphical
display is transferred to the numerical display.
This is only necessary if all six tangential
derivatives of a cross derivative are plotted.
ILPUSR : User specified space for the graphical display
IDSUSR : User specified space for the numerical display
The corresponding variables ILP and IDSPLA in
the common block SCREEN are modified as
necessary.
HELPER (7330) contains help information
HELP : Help device channel number
JHELP1 : Length of the HSUMRY information (in lines)
JHELP2 : Length of the HELP information (in lines)
JHELP3 : Number of available commands
JHELP(99,2): (I,1) line in which detailed help for I-th
command begins
(I,2) length of detailed help
IHELP(72,99): (*,I) prompt text for I-th command
USER (5) see appendix III
PLOWN (296) contains current PLOT options
ITITLE(72) : Title
IBOTTM(72) : Legend
NUMBR : page number
BL(20) : left bottom label
BR(20) : right bottom label
FRAME : = .TRUE. <===> frame is drawn
COLOR : = .TRUE. <===> plot is in color
NUMRCL : = .TRUE. <===> numerical display is drawn
LMARK : = .TRUE. <===> graphs are marked
LBLS : = .TRUE. <===> graphs are labeled
DATE : = .TRUE. <===> date is drawn
TIME : = .TRUE. <===> time is drawn
FOOWN (1) controls revaluation after the command FORCE has
been issued.
LFO : = .TRUE. <===> reevaluate
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 110
Appendix III: The Test Package
The examples in this manual can be reproduced, and experience
with MICROSCOPE can be gained, by using the test routines (a special choice
of the trial function F and the user intervention routine SUBUSR, plus a
few support routines) supplied with the package. The test code is
contained in the files USRLC and USRUC (see chapter 5).
Several different trial functions are available for examination,
and rounding to any number of digits up to the number supported on your
local installation can be simulated.
All options are selected using the command USER, which puts you
into user mode. You can change options similarly as in the PLOT mode until
you are satisfied, and then return to the command mode.
Options are chosen by passing numbers to the program according to
the prompts given. A current list of options is maintained on the screen.
The trial function is selected by giving its reference number.
The parameter eta is 1 by default and can be changed by typing -3. By
default rounding to 10 digits is simulated. Rounding can be changed by
typing -1 and giving the desired number of significant digits. It can be
turned off by giving 0 as the number of digits. The following trial
functions are available:
x
1: F(x) = s(0,eta,x) or F(x) = s(0,eta,x) + e
x
2: F(x) = s(1,eta,x) or F(x) = s(1,eta,x) + e
x
3: F(x) = s(2,eta,x) or F(x) = s(2,eta,x) + e
x
4: F(x) = s(3,eta,x) or F(x) = s(3,eta,x) + e
x
5: F(x) = s(4,eta,x) or F(x) = s(4,eta,x) + e
x
6: F(x) = s(5,eta,x) or F(x) = s(5,eta,x) + e
The addition of the exponential in the above functions is off by
default, and is toggled on or off by typing -2.
eta*x
7: F(x) = e
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 111
2
8: F(x,y) = eta*abs(x)*x + (1-eta)*abs(y)*y
2
9: F((x,y) = x *abs(x)*y*abs(y)
x
10: F(x) = the spline interpolant to e used in section 2
x
11: F(x) = e - function 10
(i.e. F(x) is the error in the spline 10)
12: F(x) = 0
13: F(x) = eta*x
e
14: F(x) = x ta
e
15: F(x) = x ta
The user mode is left by typing zero (or just RETURN).
The commands STORE and RESTART are effective on the test
parameters which are stored in the COMMON block USER. The parameters have
the following meanings:
ETA: DOUBLE PRECISION variable containing the value of eta.
IROUND: Number of digits to which function values are rounded (IROUND = 0
<===> no rounding is taking place).
N: reference number of the trial function.
ADD: LOGICAL variable that indicates whether an exponential term is to be
added to the trial functions defined by N = 1,...,6.
The following table gives the USER defaults and the parameter
settings necessary to generate the figures in this manual. N is the
reference number of the trial function, D is the number of digits rounded
to. The entry "---" means "does not apply". The default settings are
effective on the first USER and after giving the command SETDF. Other
parameters necessary for reproducing the examples in this manual (point,
direction, dimension, interval etc.) can be read off the alphanumerical
display in each figure.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 112
Figure N D ETA exponential added
Default 1 10 1 NO
1 9 10 --- ---
2-6 11 10 --- ---
7-8 10 10 --- ---
9 12 10 --- ---
10 13 10 2D-9 ---
11-12 10 10 --- ---
13-18 3 10 -5D-3 YES
19 1 10 -1 NO
20 2 10 -1 NO
21 3 10 -1 NO
22 4 10 -1 NO
23 5 10 -1 NO
24 6 10 -1 NO
25-26 7 10 1 ---
27 9 10 --- ---
28 14 10 1.5 ---
29 15 10 1.5 ---
Table of test parameters
Because of the random nature of round-off errors, somewhat
different displays may be obtained even with the above parameter settings.
Some of the above test functions so far have not been used in
this manual. They illustrate more subtle phenomena that will not normally
occur in multivariate interpolation and approximation. In a typical
application, the trial function will have a finite degree of
differentiability across the front, and be infinitely often differentiable
in any direction contained in the front. Function 8 illustrates a
situation where the degree of differentiability is finite in any direction,
but is dependent upon the direction of investigation. The function is once
differentiable in the y direction and twice differentiable in the x
direction. The relative sizes of the jumps in the appropriate derivatives
can be controlled by the parameter eta.
The functions 14 and 15 can be used to illustrate discontinuities
other than step discontinuities. The functions differ in the way they are
defined for negative values of x, for which the value of function 14 is
negative, and that of function 15 positive, if the integer part of abs(eta)
is even, and of the opposite sign if it is odd.
For both functions 14 and 15, the value at x = 0 is 0 if eta is
negative. This prevents a floating point overflow and a meaningless
display.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 113
The following two figures illustrate functions 14 and 15 for
eta = 1.5 (i.e. a function that is 1.5 times differentiable). The first
derivative is continuous, but the second derivative exhibits a pole. The
illustration of these features is enhanced by choosing the window width to
be less than 1.
.. 2 1111111111
.. I 11111111 ..
.. I 111111 ..
. I 111111 .
.. I 111 ..
.. I 111 ..
.. I1 ..
.. 1 ..
... 1I ...
.. 111 I ..
.. 1111 I ..
..11111 2I2 ...
1111111 ... 2 I 2 ...
11111111 ...2222 I 2222...
2222222222222222222222222222222 ........... 2222222222222222222222222222222
===========================================================================
Point = 0.000000D+00 s = 4.0000D-04
Direction = 1.000000D+00 h = 2.0000D-04
F0 ( 2.14D-11, 1.80D-03) F1 (-1.82D-01, 1.82D-01) F2 ( 6.17D+00, 1.41D+02)
I/O: 25 6 6 1 2 3 27 28 29 30 NRML on current CALLS = 2901
Figure 28 A 3/2 times differentiable function
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 114
11111 I2 11111
1111 I 1111.
1111 I 2 1111..
1111 I 22 1111..
111 I 2222 111..
111 I 22222222222222222
111 I ......111 222222222222
11 ......2..... 11
222222222222222 ...11. I 11
22222222222222 11 I 11
.... 22221 I 11
.... 22 I 1
.... 2 I 1
... 1I1
.... 21
===========================================================================
Point = 0.000000D+00 s = 4.0000D-04
Direction = 1.000000D+00 h = 2.0000D-04
F0 (-1.80D-03, 1.80D-03) F1 ( 1.41D-02, 1.82D-01) F2 (-3.81D+01, 3.81D+01)
I/O: 25 6 6 1 2 3 27 28 29 30 NRML on current CALLS = 3052
Figure 29 Another 3/2 times differentiable function
Figure 29 shows a cusp in the first derivative whereas in figure
28 the first derivative passes vertically through the origin. Both
derivatives are continuous at the point of examination and have a vertical
tangent there. The fact that the second derivative has a pole can be
verified by observing that the extreme function value increases as the
discretization parameter is decreased. For a cusp, the extreme function
value approaches a limit as h decreases. This phenomenon can be utilized
to distinguish between a cusp and a pole.
It is of course straightforward to program other or additional
examples for a trial function by modifying the routines in the test package
appropriately.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 115
Appendix IV: Examples for Graphical (<PLOT79>) Displays
The following pages contain <PLOT79> Equivalents of the figures in this
manual
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 116
References
1. Abramowitz, M., and Stegun, I., 1968, Handbook of Mathematical
Functions, Dover Publication
2. Alfeld, Peter, A Trivariate Clough-Tocher Scheme for Tetrahedral Data
to appear in Computer Aided Geometric Design, An International
Journal, North Holland
3. Beebe, Nelson H.F., 1980, DOCUMENT, A Portable Text Formatting
Program, Department of Physics, University of Utah, Salt Lake City,
Utah 84112
4. Beebe, Nelson H.F., 1979, A User's Guide to <PLOT79>, Department of
Physics, University of Utah, Salt Lake City, Utah 84112
5. Barnhill, R.E., October 1983, A Survey of the Representation and
Design of Surfaces, IEEE Computer Graphics and Apllications, v. 3,
Number 7, pp. 9-16
6. de Boor, C., 1978, A Practical Guide to Splines, Springer Verlag
7. Courant, R., and Hilbert, D., 1962, Methods of Mathematical Physics,
v. 2, Wiley & Sons
8. Ryder, B.G., The PFORT Verifier, in Software Practice and Experience
4, pp. 369-377, Wiley & Sons
9. Schumaker, L., 1981, Spline Functions: Basic Theory, Wiley & Sons, 1
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 117
Acknowledgements
MICROSCOPE was developed over several years and has evolved
through several preliminary versions. This process was shaped by the work
of the Computer Aided Geometric Design Group at the Mathematics Department
of the University of Utah and has benefited from the interactions with Paul
Arner, R.E. Barnhill, Nelson Beebe, Gerald Farin, Tom Jensen, Chip
Petersen, Bruce Piper, Tracy Whelan, Andrew Worsey, and others.
This work was supported partly by the Department of Energy under
Contract No. DE-AC02-82ER12046 A000 to the University of Utah, by the
United States Army under Contract No. DAAG29-80C-0041, by two grants from
the University of Utah Research Committee, and by a sabbatical leave of the
first author from the University of Utah. A large part of the program
development, and this documentation, were carried out at the Mathematics
Research Center at the University of Wisconsin. Fred Sauer of MRC has been
most helpful with all computer related problems.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 118
INDEX
Abramowitz, 13, 116 CWINDOW, 51, 60-62 76-78, 83-84,
ACCENT, 51, 54-55, D, 7, 14, 22-32, 93-98, 105-7,
108 38-43, 46-49, 109, 111-12,
Alfeld, 1, 7, 30, 51-53, 56-57, 115; see also
92, 116 59, 64, 67, 83, function
alpha, 83-85, 87, 85-86, 88, DIVIDE, 51, 60-62
89-90 111-14, 116 DMNSN, 38, 51, 54,
Barnhill, 8, 116-17 d, 10-11, 16, 18, 58, 65
Beebe, 6, 8, 69, 78, 51, 54-55, DOCUMENT, 6, 116
116-17 59-60, 99-100 DOUBLE, 34, 51,
bivariate, 9, 16, 30 D1CHDIR, 51, 59 60-62, 73, 104,
BLSCRN, 77 D2CHDIR, 51, 59 111
de Boor, 21, 116 D3CHDIR, 51, 59 DSCALE, 39, 51,
C1CROSS, 51, 65-66 DCENTER, 51, 54-55, 54-55, 72
C2CROSS, 51, 65-66 72 DXAXIS, 51, 54-55,
C3CROSS, 51, 65-66 Delta, 10, 40, 45, 72
CBD, 104 49-50, 83-85, e, 8-11, 15-19,
CBDSW, 104 89-91 21-22, 27-28,
CBDU, 104 DEN, 94 34, 37-39, 52,
CCHANNL, 51, 63, derivative, 3, 5, 54-55, 58-61,
67-68 9-11, 14-16, 63-64, 68-69,
CDIRCTN, 51, 65-66 19, 21, 24-27, 71, 78, 82-83,
CHVALUE, 51, 65-66 30-32, 37-60, 89, 92, 94, 98,
Clough-Tocher, 30, 64-66, 70, 104-5, 107,
116 73-74, 82, 110-11, 113
CLSCRN, 77 84-86, 88-91, EGRAPH, 40, 51,
command, 3, 5-6, 98, 105-6, 109, 54-55, 108
17-18, 22, 33, 113-14 eps, 7, 83
35-38, 41, DF, 105-6 eta, 38, 42-43,
44-74, 78-79, DGRAPH, 39, 51, 82-84, 89-90,
92-94, 101-2, 54-55, 108 110-13
104, 107-11 Dirac, 10, 45, 84 examination, 3, 10,
CORDER, 51, 65-66 direction, 3, 10-11, 14-16, 19, 23,
Courant, 45, 116 14-16, 18, 28, 31, 42,
critical, 9-10, 27, 22-32, 38-43, 50-53, 55,
85; see also 46-49, 51-53, 58-59, 63-64,
set 55-60, 63-68, 66, 68, 71-72,
cross, 3, 14-16, 72-74, 96, 99, 74, 82, 95-96,
18-19, 24, 30, 104, 106-8, 99, 110, 114
44, 51-53, 58, 111-14; see EXIT, 51, 62-63, 68
63-66, 68, also External, 10
72-74, 104, investigation F, 10-13, 34, 73,
106, 109; see display, 3-6, 8-11, 78, 82, 110-11,
also 13-20, 22-35, 116
derivative, 37-42, 44, 46, FLIP, 51, 54-56
direction 49-51, 53-61, Font, 77
CRT, 8, 10, 13, 15 63, 65-66, FORCE, 33, 51-52,
cusp, 9, 49, 114 69-72, 74, 54, 61-64, 73,
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 119
INDEX
92, 109 51-53, 55, 66, 108
front, 7, 10, 66, 58-60, 63-64, NUM, 94
112 66, 68, 72, 74, NUMRCL, 34-35, 109
FS, 96, 100, 105, 95-98, 107-8, nw, 96-97, 99,
108 112 101-4
function, 2-5, 8-10, IPOINT, 51, 58 OUTPUT, 34-35, 52,
13-17, 19-22, ITYPE, 92-94 65, 107
24, 26-31, k, 11, 13, 15-16, P, 10-13, 16, 45,
34-36, 38, 40, 45, 50-55, 57, 51-53, 58, 73,
42-50, 55-56, 65-66, 70, 99-100
61-64, 66, 71, 82-91 P1CHPNT, 52, 58
73-74, 82-84, LC, 75-76, 80-81 P2CHPNT, 52, 58
89-93, 95-98, LCBD, 104 P3CHPNT, 52, 58
100, 105, LCROSS, 51, 65-66 package, 3-4, 6-9,
110-14, 116 LDEF, 100, 105 13, 17, 20, 35,
Gateaux, 18 LDRCTN, 51, 59 38, 64, 68-69,
GO, 18, 33, 38, LGO, 92-94, 107 75-79, 92-94,
51-52, 54, 58, LINES, 34-35, 107 104, 110-14
61-64, 73, 80, LIST, 51, 64-65 PAUSE, 52, 67-68
92, 94 LOG, 51, 65, 105 PCURSR, 77
GRAPHC, 34 LPOINT, 51, 58-59 PEN07, 70, 75, 78
h, 5, 11-12, 14, LSCRN, 93, 107 PEN18, 70, 75, 78
16, 22-32, MANLC, 75, 80-81 PFORT, 8, 116; see
38-43, 46-52, MANUAL, 75, 79-81 also Verifier,
56-57, 60-61, MANUC, 75, 80-81 Portable,
63-64, 66-67, MCRSCP, 3, 33-35, Portability
83, 85, 87-91, 58, 62-63, 68, PLCHRS, 77
95-97, 102, 74, 76, 92-93, PLOT, 3, 34-35, 37,
104, 108, 101, 104 52, 69-72, 74,
113-14 MICROSCOPE, 1-4, 77-78, 109-10
HALVE, 41, 51, 60-62 6-8, 10-11, PLOT79, 3-4, 8, 35,
Harris, 1, 92 13-14, 17-23, 52, 54, 69-72,
HELP, 34, 51, 58, 25, 32-45, 49, 77-78, 80,
64-65, 79, 109 52-54, 57-79, 115-16
HELPLC, 75, 79-81 84, 87, 92-110, PLTDMY, 75, 78,
HELPUC, 75, 79, 81 117 80-81
Hershey, 77; see Mode, 3, 35-37, PLTLC, 75, 77-78,
also Font 52-53, 57, 65, 80-81
Hilbert, 45, 116 68-69, 73, 94, PLTUC, 75, 77-78, 81
HSUMRY, 51, 64-65, 107, 110-11 point, 3, 6, 10-11,
79, 109 multivariate, 1-3, 13-16, 18-19,
ICBD, 104 8-9, 18-19, 44, 22-32, 35,
IDIRCTN, 51, 59 112 37-43, 45-49,
IHVALUE, 51, 60-62 MULTPLY, 52, 60-62 51-59, 62-64,
IINTVL, 38, 51, NEWS, 52, 64-65 66-68, 70-74,
58-59 nh, 96, 99, 101-4, 77-78, 82-85,
investigation, 3, 7, 107 95-100, 105-8,
10, 14-19, 35, NORMAL, 52, 59-60, 111-14; see
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> page 120
INDEX
also 51, 54-55, terminal, 8, 10, 15,
examination 63-64, 72, 74, 17, 19, 33-34,
pole, 9, 113-14 106, 108 54, 68, 77-78,
Portability, 6, 33, Schumaker, 21, 116 101, 105, 107
75 SCREEN, 75-77, test, 4-9, 38, 44,
Portable, 2, 8, 33, 80-81, 93, 107, 64, 78, 110-14;
75-76, 79, 116 109 see also
POS, 75-76, 80-81 screen, 6, 8, 10, package
precision, 3, 6, 13, 17-18, TNOTE, 52, 65
28-29, 36, 73, 33-39, 51-54, trial, 8-10, 13-16,
104 57-58, 61-65, 19, 34-36, 38,
primitive, 8, 10, 69, 72, 76-78, 40, 42, 44, 49,
21-22, 27, 44; 80-81, 92, 94, 60-63, 66,
see also 101, 105-8, 110 73-74, 82, 89,
function SCROLL, 75-77, 81 110-12, 114;
PROMPT, 34-35 scroll, 33, 94 see also
PS, 99-100 set, 7, 9-10, 14-15, function
psi, 85, 89-90 17-18, 27-28, trivariate, 9, 116
QUIT, 52, 62-63 34, 42, 59, TYPE, 37, 42, 52, 54,
random, 7, 10, 63-66, 75, 84, 57
23-24, 45, 83, 92, 94, 99, UC, 75-76, 81
112 102, 104, 107-8 UNDO, 52, 62, 64, 107
RCROSS, 52, 65-66, SETDF, 52, 62-63, USER, 37-38, 52,
68 111 62-63, 73-74,
RDIRCTN, 52, 59, 68 SETPEN, 70-71 78, 104, 109-11
RESTART, 34, 52, SETPN, 71 USRDMY, 75, 78
62-63, 111 SHIFT, 39, 52, 58-59 USRLC, 75, 78,
ROPPNT, 100, 107-8 spline, 21, 111, 116 80-81, 110
ROPUDI, 100, 108 ss, 95-99, 101-2, USRUC, 75, 78, 81,
ROTATE, 52, 59 104 110
round-off, 3-5, 10, Stegun, 13, 116 Verifier, 8, 116
13, 22-24, STORE, 52, 62-63, w, 16, 49-50, 60,
28-32, 35, 42, 111 63-64, 85
44-45, 60-61, STPEN, 71 WAIT, 52, 62-63
82-90, 94, 112 SUBUSR, 35, 73-74, WIDTH, 34-35, 107
RPOINT, 52, 58-59, 78, 110 width, 5, 15-16, 19,
68 SUPPRT, 75-76, 80-81 35, 41, 49-52,
RSCREEN, 33, 52, 54, surface, 9, 19 60-61, 63-64,
58, 94 sw, 95-98, 100-102, 74, 85, 95-98,
RWIND, 52, 67-68 108 101-4, 108, 113
Ryder, 8, 116 tangential, 11, window, 3, 5, 16,
s, 11, 14, 16, 14-16, 18-19, 19, 41-42, 46,
22-32, 38-43, 44-45, 54-55, 49-52, 60-61,
46-50, 56-57, 66, 70-71, 63-64, 66, 74,
60, 67, 84-85, 73-74, 82, 92, 85, 89, 98,
95, 101-2, 110, 97, 109; see 101, 108, 113
113-14 also derivative XS, 96, 99-100, 105
scale, 15, 39, 41, TCENTER, 52, 54, 57 ZOOM, 52, 60-62