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\begin{document}
\title{\mbox{ITPACK 2C: A FORTRAN Package for Solving}
\mbox{Large Sparse Linear Systems by Adaptive}
\mbox{Accelerated Iterative Methods}
\thanks{Work on this project was supported in part by National
Science Foundation Grant MCS 79-19829 at The University
of Texas at Austin.} }
\author{David R. Kincaid, John R. Respess, and David M. Young \\
University of Texas at Austin
\thanks{Center for Numerical Analysis, RLM Bldg. 13.150,
University of Texas, Austin, TX \ \ 78712}
\and
Roger G. Grimes \\
Boeing Computer Services Company
\thanks{Boeing Computer Services Company,
565 Andover Park West, Tukwila, WA \ \ 98067}}
\maketitle
\begin{abstract}
ITPACK 2C is a collection of seven FORTRAN subroutines for solving
large sparse linear systems by adaptive accelerated iterative
algorithms. Basic iterative procedures, such as the Jacobi method, the
Successive Overrelaxation method, the Symmetric Successive
Overrelaxation method, and the RS method for the reduced system are
combined, where possible, with acceleration procedures such as Chebyshev
(Semi-Iteration) and Conjugate Gradient for rapid convergence.
Automatic selection of the acceleration parameters and the use of
accurate stopping criteria are major features of this software package.
While the ITPACK routines can be called with any linear system
containing positive diagonal elements, they are the most successful in
solving systems with symmetric positive definite or mildly nonsymmetric
coefficient matrices.
\end{abstract}
\section{Introduction}
\label{intro}
For several years, we have been involved with the development and use of
research-oriented programs using iterative algorithms for solving large
sparse linear systems
\[ Au = b \]
with positive diagonal elements. One solves for the {\bf N} component
unknown vector $u$ given the ${\bf N \times N}$ nonsingular
coefficient matrix $A$ and the {\bf N} component right-hand side vector
$b$. The current ITPACK software package of subroutines, version 2C,
provides for the use of seven alternative iterative procedures. While
these subroutines are not designed as production software, they should
successfully handle industrial problems of moderate size, that is, ones
that fit in high-speed memory. This package is written in standard
FORTRAN-66 code. It has been tested over a wide variety of computer
systems using various FORTRAN compilers, including one which is
FORTRAN-77 compatible (see Acknowledgements).
The seven iterative solution modules are based on several basic
iterative procedures, such as the Jacobi method, the Successive
Overrelaxation (SOR) method, the Symmetric SOR (SSOR) method, and the RS
method for the reduced system. With the exception of SOR, the
convergence of these basic methods are accelerated by Chebyshev
(Semi-Iteration, SI) or Conjugate Gradient (CG) acceleration. All
methods are available with adaptive parameter estimation and automatic
stopping tests. When using the RS method it is required that the linear
system be reordered into a ``red-black"\footnote{In this ordering,
the components of the unknown vector $u$ are considered as either
``red" or ``black". A ``red-black ordering" is any ordering such
that every black unknown follows all of the red unknowns. This ordering
of unknowns leads to a $2\times 2$ ``red-black partitioning" of the
coefficient matrix, that is, a matrix of the form
\[ \lb \begin{array}{cc}
D_R & H \\
K & D_B \end{array} \rb \]
with diagonal submatrices $D_R$ and $D_B$. The original linear
system may require rearrangement in order to arrive at this form.}
system \cite{6,12}. A switch to compute, if possible, the red-black
indexing, permute the linear system, and permute associated vectors is
provided.
The successful convergence of iterative methods may be dependent on
conditions that are difficult to determine in advance. For example,
determining whether the coefficient matrix is positive definite can be
as costly to check as solving the system. On the other hand, some
conditions affecting convergence, such as positive diagonal elements,
diagonal dominance, and symmetry are relatively easy to verify. For
some applications, the theory may not exist to guarantee the convergence
of an iterative method. The algorithms in ITPACK have been tested most
extensively for linear systems arising from elliptic partial
differential equations. The routines can be applied, formally, to any
linear system which fits in high-speed memory. However, rapid
convergence, and indeed convergence itself cannot be guaranteed unless
the matrix of the system is symmetric and positive definite. Success
can be expected, though not guaranteed, for mildly nonsymmetric systems.
In other words, iterative methods may not converge when applied to
systems with coefficient matrices which are completely general with no
special properties.
This article discusses the usage of ITPACK and gives a few test
results. The description of the iterative methods is given in \cite{4}.
The underlying theory on which the iterative algorithms are based is
described in \cite{6}. A survey of the iterative methods in ITPACK
is presented in \cite{11}.
Throughout this paper, we adopt notation such as {\bf SOR()} when
referring to a subroutine and {\bf A(*)} for a single-dimensioned array.
The residual vector is $b-Au^{(n)}$ for the linear system $Au=b$ and the
pseudo-residual vector is $Gu^{(n)}+k-u^{(n)}$ for a basic iterative
method of the form $u^{(n+1)}=Gu^{(n)}+k$. The smallest and
largest eigenvalues of the iteration matrix $G$ are denoted $m(G)$ and
$M(G)$, respectively.
\section{Sparse Matrix Storage}
\label{storage}
The sparse storage scheme used in ITPACK is a common one. It is a
row-wise representation of the nonzero entries in the coefficient matrix
of the linear system. For a nonsymmetric coefficient matrix, all of the
nonzero values in each row are stored in a contiguous block of data in a
real-valued array {\bf A(*)}. If the matrix is symmetric, computer
memory can be saved by storing only the nonzero entries in each row on
and above the main diagonal. For either nonsymmetric or symmetric sparse
storage, associated column numbers are stored in an integer-valued
array {\bf JA(*)} such that ${\bf JA(K)}$ is the column number for the
value ${\bf A(K)}$. A mapping vector {\bf IA(*)} is used to denote the
starting locations of each of the contiguous blocks. The beginning of
the linear block for row $I$ is given by ${\bf IA(I)}$, the end by
${\bf IA(I+1)-1}$, and its length by ${\bf IA(I+1)-IA(I)}$. Thus,
{\bf IA(*)} will contain ${\bf N+1}$ elements to accommodate a linear
system of order {\bf N}. The entries for each row may be stored in any
order in the contiguous block for that row.
For example, the coefficient matrix
\[ \lb \begin{array}{ccccc}
11. & 0. & 0. & 14. & 15. \\
0. & 22. & 0. & 0. & 0. \\
0. & 0. & 33. & 0. & 0. \\
14. & 0. & 0. & 44. & 45. \\
15. & 0. & 0. & 45. & 55.
\end{array} \rb \]
would be represented in nonsymmetric sparse storage as
\begin{eqnarray*}
{\bf A(*)} & = & [11.,14.,15.,22.,33.,14.,44.,45.,15.,45.,55.] \\
{\bf JA(*)} & = & [1,4,5,2,3,1,4,5,1,4,5] \\
{\bf IA(*)} & = & [1,4,5,6,9,12]
\end{eqnarray*}
and in symmetric sparse storage as
\begin{eqnarray*}
{\bf A(*)} & = & [11.,14.,15.,22.,33.,44.,45.,55.] \\
{\bf JA(*)} & = & [1, 4, 5, 2, 3, 4, 5, 5] \\
{\bf IA(*)} & = & [1, 4, 5, 6, 8, 9]
\end{eqnarray*}
\section{Usage}
\label{usage}
The user is expected to provide the coefficient matrix and the
right-hand side of the linear system to be solved. The data structure
for the matrix of the system is either the symmetric or nonsymmetric
sparse storage format described in Section~\ref{storage}. An initial
guess for the solution should be provided, if one is known; otherwise,
it can be set to all zero values. A series of approximations for the
solution are generated iteratively until the convergence criteria is
satisfied. The algorithms are performed in two work space arrays and
some control over the algorithmic procedure can be obtained from
switches in two parameter arrays.
There are seven main subroutines in ITPACK, each corresponding to an
iterative method. They are:
\bigskip
\begin{tabular}{ll} \hline
Subroutine & Method \\ \hline
{\bf JCG()} & Jacobi Conjugate Gradient \\
{\bf JSI()} & Jacobi Semi-Iteration \\
{\bf SOR()} & Successive Overrelaxation \\
{\bf SSORCG()} & Symmetric SOR Conjugate Gradient \\
{\bf SSORSI()} & Symmetric SOR Semi-Iteration \\
{\bf RSCG()} & Reduced System Conjugate Gradient \\
{\bf RSSI()} & Reduced System Semi-Iteration \\ \hline
\end{tabular}
\bigskip
\noindent
and the calling sequence is:
\bigskip
\centerline{\bf CALL $\la$ {\em method} $\ra$ (N, IA, JA, A, RHS, U, IWKSP,
NW, WKSP, IPARM,}
\centerline{\bf RPARM, IER)}
\bigskip
\noindent
where the parameters are defined in the following. Here ``input" means
that the subroutine expects the user to provide the necessary input data
and ``output" means that the routine passes back information in the
variable or array indicated. All parameters are linear arrays except
variables {\bf N}, {\bf NW}, and {\bf IER}. Moreover, all parameters may
be altered by the subroutine call except variables {\bf N} and
{\bf NW}. (See Section~\ref{notes} for additional details.)
\bigskip
\begin{description}
\item[N] is the order of the linear system. [integer; input]
\item[IA(*)] is a vector of length ${\bf N+1}$ used in the sparse
matrix storage format. It contains the row pointers
into {\bf JA(*)} and {\bf A(*)}. [integer array; input]
\item[JA(*)] is a vector of length {\bf NZ} (defined in {\bf A(*)} below)
used in the sparse matrix storage format. It contains
the column numbers for the corresponding entries in
{\bf A(*)}. [integer array; input]
\item[A(*)] is a vector of length {\bf NZ} used in the sparse matrix
storage format. It contains the nonzero entries of the
coefficient matrix with positive diagonal elements.
({\bf NZ} is the number of nonzero entries in the upper
triangular part of the coefficient matrix when symmetric
storage is used and is the total number of nonzeros when
nonsymmetric storage is used.) [real array; input]
\item[RHS(*)] is a vector of length {\bf N} containing the right-hand side
of the linear system. [real array; input]
\item[U(*)] is a vector of length {\bf N} containing the initial guess
to the solution of the linear system on input and the
latest approximate solution on output. [real array;
input/output]
\item[IWKSP(*)] is a vector of length ${\bf 3*N}$ used for integer workspace.
When reindexing for red-black ordering, the first
{\bf N} locations contain on output the permutation
vector for the red-black indexing, the next {\bf N}
locations contain its inverse, and the last {\bf N}
are used for integer workspace.\footnote{For the
red-black ordering, the {\bf I}th entry of a permutation
array {\bf P(*)} indicates the position {\bf J} into
which the {\bf I}th unknown of the original system is
being mapped, that is, if ${\bf P(I)=J}$ then unknown
{\bf I} is mapped into position {\bf J}. The {\bf J}th
entry of an inverse permutation array {\bf IP(*)}
indicates the position {\bf I} into which the {\bf J}th
unknown of the permuted system must be mapped to regain
the original ordering, that is, ${\bf IP(J)=I}$.}
[integer array; output]
\item[NW] is a scalar. On input, {\bf NW} is the available length for
{\bf WKSP(*)}. On output, {\bf IPARM(8)} is the actual
amount used (or needed). [integer; input]
\item[WKSP(*)] is a vector used for real working space whose length depends
on the iterative method being used. It must be at least
{\bf NW} entries long. (See the table near the end of this
section for the required amount of workspace for each
method.) [real array]
\item[IPARM(*)] is a vector of length 12 used to initialize various
integer and logical parameters. Default values may be
set by calling subroutine {\bf DFAULT()} described below.
On output, {\bf IPARM(*)} contains the values of the
parameters that were changed. (Further details are
given later in this section.) [integer array;
input/output]
\item[RPARM(*)] is a vector of length 12 used to initialize various real
parameters on input. Default values may be set by
calling subroutine {\bf DFAULT()} described below. On
output, {\bf RPARM(*)} contains the final values of
the parameters that were changed. (Further details
are given later in this section.) [real array;
input/output]
\item[IER] is the error flag which is set to zero for normal convergence
and to a nonzero integer when an error condition is
present. (See the table at the end of this section
for the meaning of nonzero values.) [integer; output]
\end{description}
\bigskip
The user may supply nondefault values for selected quantities in
{\bf IPARM(*)} and \newline {\bf RPARM(*)} by first executing
\bigskip
\centerline{\bf CALL DFAULT (IPARM, RPARM)}
\bigskip
\noindent
and then assigning the appropriate nondefault values before calling a
solution module of ITPACK.
The iterative algorithms used in ITPACK are quite complicated and some
knowledge of iterative methods is necessary to completely understand
them. The interested reader should consult the technical report \cite{4}
and the book \cite{6} for details. Important variables in this package
which may change adaptively are {\bf CME} (estimate of $M(B)$, the largest
eigenvalue of the Jacobi matrix), {\bf SME} (estimate of $m(B)$, the
smallest eigenvalue of the Jacobi matrix), {\bf OMEGA} (overrelaxation
parameter $\om$ for the SOR and SSOR methods), {\bf SPECR} (estimated spectral
radius of the SSOR matrix), {\bf BETAB} (estimate for the spectral radius
of the matrix $LU$ where $L$ and $U$ are strictly lower and upper
triangular matrices, respectively, such that the Jacobi matrix $B=L+U$).
The integer array {\bf IPARM(*)} and real array {\bf RPARM(*)} allow
the user to control certain parameters which affect the performance
of the iterative algorithms. Furthermore, these arrays allow the updated
parameters from the automatic adaptive procedures to be communicated
back to the user. The entries in {\bf IPARM(*)} and {\bf RPARM(*)} are:
\begin{description}
\item[IPARM(1)] {\bf ITMAX} is the maximum number of iterations
allowed. It is reset on output to the number of
iterations performed. Default: $100$
\item[IPARM(2)] {\bf LEVEL} is used to control the level of output.
Each higher value provides additional information.
Default: $0$
\begin{tabular}{rl}
[$<0$: & no output on unit {\bf IPARM(4)}; \\
$0$: & fatal error messages only; \\
$1$: & warning messages and minimum output; \\
$2$: & reasonable summary (progress of algorithm); \\
$3$: & parameter values and informative comments; \\
$4$: & approximate solution after each iteration
(primarily useful for debugging); \\
$5$: & original system]
\end{tabular}
\item[IPARM(3)] {\bf IRESET} is the communication switch. Default: $0$
\begin{tabular}{rl}
[$0$: & implies certain values of {\bf IPARM(*)} and
{\bf RPARM(*)} will be overwritten \\
& to communicate parameters back to the user; \\
$\neq 0$: & only {\bf IPARM(1)} and {\bf IPARM(8)}
will be reset.]
\end{tabular}
\item[IPARM(4)] {\bf NOUT} is the output unit number. Default: $6$
\item[IPARM(5)] {\bf ISYM} is the sparse storage format switch.
Default: $0$
\begin{tabular}{rl}
[$0$: & symmetric sparse storage; \\
$1$: & nonsymmetric sparse storage]
\end{tabular}
\item[IPARM(6)] {\bf IADAPT} is the adaptive switch. It determines
whether certain parameters have been set by the user or
should be computed automatically in either a fully
or partially adaptive sense. Default: $1$
\begin{tabular}{rl}
[$0$: & fixed iterative parameters used for {\bf SME},
{\bf CME}, {\bf OMEGA}, {\bf SPECR}, and \\
& {\bf BETAB} (nonadaptive); \\
$1$: & fully adaptive procedures used for all parameters; \\
$2$: & (SSOR methods only) {\bf SPECR} determined
adaptively and {\bf CME}, {\bf BETAB},\\
& and {\bf OMEGA} fixed; \\
$3$: & (SSOR methods only) {\bf BETAB} fixed and all other
parameters determined \\
& adaptively]
\end{tabular}
\noindent
(See \cite{4,6} for details and ${\bf RPARM(I), I=2,3,5,6,7}$
for {\bf CME}, {\bf SME}, {\bf OMEGA}, {\bf SPECR},
{\bf BETAB}, respectively. These parameters are set
by subroutine {\bf DFAULT()} or by the user.)
\item[IPARM(7)] {\bf ICASE} is the adaptive procedure case switch for the
JSI and SSOR methods. There are two strategies, called
Case I and Case II, for doing the adaptive procedure.
The choice of which case to select corresponds to knowledge
of the eigenvalues of the Jacobi matrix $B$ and their
estimates. Default: $1$
\begin{tabular}{rl}
[$\neq 2$ & Case I: Fixed ${\bf SME} \leq m(B)$
(general case); \\
$=2$ & Case II: Use when it is known that
$|m(B)| \leq M(B)]$
\end{tabular}
\noindent
The case switch determines how the estimates for
{\bf SME} and {\bf CME} are recomputed adaptively. In
Case I, {\bf SME} is fixed throughout and should be less
than or equal to $m(B)$. In Case II, {\bf SME} is set
to ${\bf -CME}$ which may adaptively change. As far as
the adaptive procedure is concerned, Case I is the most
general case and should be specified in the absence of
specific knowledge of the relationship between the
eigenvalues and their estimates. An example when Case II
is appropriate occurs when the Jacobi matrix has
Property A, since $m(B) = -M(B)$.\footnote{A matrix has
Property A if and only if it is a diagonal matrix or
else there exists a rearrangement of the rows and
corresponding columns of the matrix which corresponds to
a red-black partitioning.} Also, if $A$ is an $L$-matrix,
then for the Jacobi matrix, we have $|m(B)| \leq M(B)$
and {\bf SME} is always ${\bf -CME}$ (Case II).
\footnote{An $L$-matrix has positive diagonal elements
and nonpositive off-diagonal elements.} Selecting the
correct case may increase the rate of convergence of
the iterative method. (See \cite{6} for additional
discussion on Case I and II. Also, see
${\bf RPARM(I), I=2,3}$ for {\bf CME}, {\bf SME},
respectively.)
\item[IPARM(8)] {\bf NWKSP} is the amount of workspace used. It is used
for output only. If {\bf ITMAX} is set to a value just
over the actual number of iterations necessary for
convergence, the amount of memory for {\bf WKSP(*)} can
be reduced to just over the value returned here. This
may be done when rerunning a problem, for example.
Default: $0$
\item[IPARM(9)] {\bf NB} is the red-black ordering switch. On output,
if reindexing is done, {\bf NB} is set to the order of
the black subsystem. Default: $-1$
\noindent
[For RS methods,
\begin{tabular}{rl}
$<0$: & compute red-black indexing and permute
system; \\
$\geq 0$: & skip indexing---system already in red-black
form;
\end{tabular}
\noindent
For other methods,
\begin{tabular}{rl}
$<0$: & skip indexing---system already in desired
form; \\
$\geq 0$: & compute red-black indexing and permute
system]
\end{tabular}
\noindent
A negative integer value for {\bf IPARM(9)} causes the
equations to be handled in the most general way
appropriate for the solution method being used. For
methods other than RS methods this is the ``natural
order" while for RS methods it is the ``red-black order."
A non-negative value produces a red-black permutation
for all methods except for the RS methods which are
assumed to be in red-black order with the order of the
black subsystem {\bf NB} given. If reindexing is performed,
{\bf IPARM(9)} will contain the order of the black subsystem
on output.
\item[IPARM(10)] {\bf IREMOVE} is the switch for effectively removing
rows and columns when the diagonal entry is extremely
large compared to the nonzero off-diagonal entries in
that row. (See {\bf RPARM(8)} for additional details.)
Default: $0$
\noindent
[$0$: not done; $\neq 0$: test done]
\item[IPARM(11)] {\bf ITIME} is the timing switch. Default: $0$
\noindent
[$0$: time method; $\neq 0$: not done]
\item[IPARM(12)] {\bf IDGTS} is the error analysis switch. It determines
if an analysis is done to determine the accuracy of
the final computed solution. Default: $0$
\begin{tabular}{rl}
[$<0$: & skip error analysis \\
$0$: & compute {\bf DIGIT1} and {\bf DIGIT2} and
store in ${\bf RPARM(I), I=11,12}$, \\
& respectively; \\
$1$: & print {\bf DIGIT1} and {\bf DIGIT2}; \\
$2$: & print final approximate solution vector; \\
$3$: & print final approximate residual vector; \\
$4$: & print both solution and residual vectors; \\
& otherwise: no printing]
\end{tabular}
\noindent
(If ${\bf LEVEL} \leq 0$, no printing is done. See
${\bf RPARM(I), I=11,12}$ for details on {\bf DIGIT1}
and {\bf DIGIT2}.)
\item[RPARM(1)] {\bf ZETA} is the stopping criterion or approximate
relative accuracy desired in the final computed solution.
If the method did not converge in {\bf IPARM(1)}
iterations, {\bf RPARM(1)} is reset to an estimate of
the relative accuracy achieved. The stopping criterion
is a test of whether {\bf ZETA} is greater than the
ratio of the two norm of the pseudo-residual vector and
the two norm of the current iteration vector times a
constant involving an eigenvalue estimate. (See \cite{4,6}
for details.) Default: $5 \times 10^{-6}$
\item[RPARM(2)] {\bf CME} is the estimate of the largest eigenvalue of
the Jacobi matrix. It changes to a new estimate if the
adaptive procedure is used. ${\bf CME} \leq M(B)$.
Default: $0.0$
\item[RPARM(3)] {\bf SME} is the estimate of the smallest eigenvalue of
the Jacobi matrix for the JSI method. In Case I,
{\bf SME} is fixed throughout at a value $\leq m(B)$.
In Case II, {\bf SME} is always set to ${\bf -CME}$
with {\bf CME} changing in the adaptive procedure. (See
{\bf IPARM(7)} for definitions of Case I and II.)
Default: $0.0$
\item[RPARM(4)] {\bf FF} is the adaptive procedure damping factor. Its
values are in the interval $(0.,1.]$ with $1.$ causing
the most frequent parameter changes when the fully
adaptive switch ${\bf IPARM(6)=1}$ is used.
Default: $0.75$
\item[RPARM(5)] {\bf OMEGA} is the overrelaxation parameter for the SOR
and the SSOR methods. If the method is fully adaptive,
{\bf OMEGA} changes. Default: $1.0$
\item[RPARM(6)] {\bf SPECR} is the estimated spectral radius for the
SSOR matrix. If the method is adaptive, {\bf SPECR}
changes. Default: $0.0$
\item[RPARM(7)] {\bf BETAB} is the estimate for the spectral radius of the
matrix $LU$ used in the SSOR methods. {\bf BETAB} may
change depending on the adaptive switch {\bf IPARM(6)}.
The matrix $L$ is the strictly lower triangular part of
the Jacobi matrix and $U$ is the strictly upper triangular
part. When the spectral radius of $LU$ is less than or
equal to $\frac{1}{4}$, the ``SSOR condition" is satisfied
for some problems provided one uses the natural ordering.
(See \cite{4,5,18} for additional details.) Default: $0.25$.
\item[RPARM(8)] {\bf TOL} is the tolerance factor near machine relative
precision, {\bf SRELPR}. In each row, if all nonzero
off-diagonal row entries are less than {\bf TOL} times
the value of the diagonal entry, then this row and
corresponding column are essentially removed from the
system. This is done by setting the nonzero off-diagonal
elements in the row and corresponding column to zero,
replacing the diagonal element with $1$, and adjusting
the elements on the right-hand side of the system so that
the new system is equivalent to to the original one.
\footnote{If the row and column corresponding to diagonal
entry $A_{i,i}$ are to be eliminated, then the
right-hand side is adjusted to $b_i \leftarrow b_i/A_{i,i}$
and $b_j \leftarrow b_j-b_i A_{i,j}$ for $j \neq i$.}
If the diagonal entry is the only nonzero
element in a row and is not greater than the reciprocal
of {\bf TOL}, then no elimination is done. This procedure
is useful for linear systems arising from finite element
discretizations of PDEs in which Dirichlet boundary
conditions are handled by giving the diagonal values in
the linear system extremely large values. (The installer
of this package should set the value of {\bf SRELPR}.
See comments in subroutine {\bf DFAULT()} for additional
details.) Default: $100.\times{\bf SRELPR}$
\item[RPARM(9)] {\bf TIME1} is the total time in seconds from the beginning
of the iterative algorithm until convergence. (A machine
dependent subprogram call for returning the time in
seconds is provided by the installer of this package.)
Default: $0.0$
\item[RPARM(10)] {\bf TIME2} is the total time in seconds for the entire
call. Default: $0.0$
\item[RPARM(11)] {\bf DIGIT1} is the approximate number of digits using
the estimated relative error with the final approximate
solution. It is computed as the negative of the logarithm
base ten of the final value of the stopping test. (See
details below or \cite{6}.) Default: $0.0$
\item[RPARM(12)] {\bf DIGIT2} is the approximate number of digits using
the estimated relative residual with the final approximate
solution. It is computed as the negative of the logarithm
base ten of the ratio of the two norm of the residual
vector and the two norm of the right-hand side vector.
This estimate is related to the condition number of the
original linear system and, therefore, it will not be
accurate if the system is ill-conditioned. (See details
below or \cite{6}.) Default: $0.0$
\end{description}
\bigskip
{\bf DIGIT1} is determined from the actual stopping test computed on
the final iteration, whereas {\bf DIGIT2} is based on the computed
residual vector using the final approximate solution after the algorithm
has converged. If these values differ greatly, then either the stopping
test has not worked successfully or the original system is ill-conditioned.
(See \cite{6} for additional details.)
For storage of certain intermediate results, the solution modules
require a real vector {\bf WKSP(*)} and a corresponding variable {\bf NW}
indicating the available space. The length of the workspace array varies
with each solution module and the maximum amount needed is given in
the following table.
\bigskip
\begin{tabular}{ll} \hline
Solution Module & Maximum Length of {\bf WKSP(*)} \\ \hline
{\bf JCG()} & ${\bf 4*N + NCG}$ \\
{\bf JSI()} & ${\bf 2*N}$ \\
{\bf SOR()} & ${\bf N}$ \\
{\bf SSORCG()} & ${\bf 6*N + NCG}$ \\
{\bf SSORSI()} & ${\bf 5*N}$ \\
{\bf RSCG()} & ${\bf N + 3*NB + NCG}$ \\
{\bf RSSI()} & ${\bf N + NB}$ \\ \hline
\end{tabular}
\bigskip
\noindent
The value of {\bf NCG} is ${\bf 2*IPARM(1)}$ for symmetric sparse storage
and ${\bf 4*IPARM(1)}$ for nonsymmetric sparse storage. It should be noted
that the actual amount of workspace used may be somewhat less than these
upper limits since some of the latter are dependent on the maximum number
of iterations allowed, {\bf ITMAX}, stored in {\bf IPARM(1)}. Clearly,
the array {\bf WKSP(*)} must be dimensioned to at least the value of
{\bf NW}.
Nonzero integer values of the error flag {\bf IER} indicate that an error
condition was detected. These values are listed below according to
their numerical value and to the name of the routine in which the flag
was set.
\bigskip
\begin{tabular}{rrl} \hline
\multicolumn{2}{l}{Error Flag} & Meaning \\ \hline
${\bf IER} =$ & $0$, & Normal convergence was obtained. \\
$ =$ & $1+{\rm Mth}$, & Invalid order of the system, {\bf N}. \\
$ =$ & $2+{\rm Mth}$, & Workspace array {\bf WKSP(*)} is not large
enough. {\bf IPARM(8)} \\
& & is set to the amount of required workspace,
{\bf NW}. \\
$ =$ & $3+{\rm Mth}$, & Failure to converge in {\bf IPARM(1)}
iterations. {\bf RPARM(1)} \\
& & is reset to the last stopping value computed. \\
$ =$ & $4+{\rm Mth}$, & Invalid order of the black subsystem, {\bf NB}. \\
$ =$ & $101$, & A diagonal element is not positive. \\
$ =$ & $102$, & No diagonal element in a row. \\
$ =$ & $201$, & Red-black indexing is not possible. \\
$ =$ & $301$, & No entry in a row of the original matrix. \\
$ =$ & $302$, & No entry in a row of the permuted matrix. \\
$ =$ & $303$, & Sorting error in a row of the permuted matrix. \\
$ =$ & $401$, & A diagonal element is not positive. \\
$ =$ & $402$, & No diagonal element in a row. \\
$ =$ & $501$, & Failure to converge in {\bf ITMAX}
function evaluations. \\
$ =$ & $502$, & Function does not change sign at the endpoints. \\
$ =$ & $601$, & Successive iterates are not monotone increasing. \\
\hline
\end{tabular}
\bigskip
\noindent
{\bf JCG()}, {\bf JSI()}, {\bf SOR()}, {\bf SSORCG()}, {\bf SSORSI()},
{\bf RSCG()}, {\bf RSSI()} assign values to Mth of 10,20,30,40,50,60,70,
respectively. {\bf SBELM()}, {\bf PRBNDX()}, {\bf PERMAT()},
{\bf SCAL()}, {\bf ZBRENT()}, {\bf EQRT1S()} are subroutines with
error flags in the 100's, 200's, 300's, 400's, 500's, 600's, respectively.
These routines perform the following functions: {\bf SBELM()} removes
rows and columns, {\bf PRBNDX()} determines the red-black indexing,
{\bf SCAL()} scales the system, {\bf ZBRENT()} is a modified IMSL routine
for computing a zero of a function which changes sign in a given
interval, {\bf EQRT1S()} is a modified IMSL routine for computing the
largest eigenvalue of a symmetric tridiagonal matrix.\footnote{IMSL
(International Mathematical and Statistical Libraries, Inc.),
Sixth Floor NBC Bldg., 7500 Bellaire Blvd., Houston, TX, 77036.}
\section{User-Oriented Modules}
\label{modules}
The array {\bf U(*)} should contain an initial approximation to the
solution of the linear system before any ITPACK module is called. If
the user has no information for making such a guess, then the zero vector
may be used as the starting vector. The subroutine {\bf VFILL()} can be
used to fill a vector with a constant:
\bigskip
\centerline{\bf CALL VFILL (N, U, VAL)}
\bigskip
\noindent
fills the array {\bf U(*)} of length {\bf N} with the value {\bf VAL} in
each entry.
To aid the user in using the iterative methods of ITPACK, four modules
for constructing the sparse matrix storage arrays are included. The
modules are:
\bigskip
\begin{tabular}{ll}
{\bf SBINI()} & is called at the beginning to initialize the arrays
{\bf IA(*)}, {\bf JA(*)}, \\
& {\bf A(*)}, and {\bf IWORK(*)}; \\
{\bf SBSIJ()} & is called repeatedly to set the individual entries in the
matrix and \\
& build a link list representation of the matrix
structure; \\
{\bf SBEND()} & is called at the end to restructure the link list into
final sparse \\
& storage form; \\
{\bf SBAGN()} & is called to return again to the link list representation
if {\bf SBEND()} \\
& has been called but additional elements are
to be added or modified.
\end{tabular}
\bigskip
\noindent
These modules are described below.
\bigskip
(a) Initialization:
\bigskip
\centerline{\bf CALL SBINI (N,NZ,IA,JA,A,IWORK)}
\bigskip
\noindent
Initializes {\bf IA(*)}, {\bf JA(*)}, {\bf A(*)}, and {\bf IWORK(*)} for
a system of order {\bf N}. {\bf IA(*)}, {\bf JA(*)}, and {\bf IWORK(*)}
are integer arrays of length at least ${\bf N+1}$, {\bf NZ}, and {\bf NZ},
respectively. {\bf A(*)} is a real array of length at least {\bf NZ}.
\bigskip
(b) Set individual entries:
\bigskip
\centerline{\bf CALL SBSIJ (N,NZ,IA,JA,A,IWORK,I,J,VAL,MODE,LEVEL,NOUT,IER)}
\bigskip
\noindent
Inserts the value, {\bf VAL}, of the {\bf (I,J)} entry of the user's
matrix into the link list representation for that matrix. When using
symmetric sparse storage, {\bf J} must be greater than or equal to
{\bf I}. If the {\bf (I,J)} entry has already been set then {\bf MODE}
specifies the way in which the entry is to be treated:
\bigskip
\begin{tabular}{rl}
${\bf MODE}<0$, & Current entry value is left as is; \\
$=0$, & Current entry value is reset to {\bf VAL}; \\
$>0$, & {\bf VAL} is added to the current entry value.
\end{tabular}
\bigskip
\noindent
If {\bf LEVEL} is less than $0$, {\bf SBSIJ()} causes no printing. If
{\bf LEVEL} is $0$, fatal errors messages are written to output unit
number {\bf NOUT}; and if {\bf LEVEL} is $1$ or greater, a message is
printed when {\bf SBSIJ()} encounters a value it has already set with the
value being reset according to the value of {\bf MODE}. {\bf IER} is
an error parameter and returns values of
\bigskip
\begin{tabular}{rrl} \hline
\multicolumn{2}{l}{Error Flag} & Meaning \\ \hline
${\bf IER} =$ & $0$, & New {\bf (I,J)} entry is established. \\
$ =$ & $700$, & {\bf (I,J)} entry is already set---reset according
to {\bf MODE}. \\
$ =$ & $701$, & Improper values for either {\bf I} or {\bf J}. \\
$ =$ & $702$, & {\bf NZ} is too small---no room for the new entry. \\ \hline
\end{tabular}
\bigskip
(c) Finalization:
\bigskip
\centerline{\bf CALL SBEND (N,NZ,IA,JA,A,IWORK)}
\bigskip
\noindent
Restructures the link list data structure built by {\bf SBINI()} and
{\bf SBSIJ()} into the final data structure required by ITPACK.
\bigskip
(d) Undo Finalization:
\bigskip
\centerline{\bf CALL SBAGN (N,NZ,IA,JA,A,IWORK,LEVEL,NOUT,IER)}
\bigskip
\noindent
Returns to link list representation for modification or addition of
elements to the system. Repeated calls to {\bf SBSIJ()} can then be made
followed by a single call to {\bf SBEND()} to close-out the sparse matrix
representation. If {\bf LEVEL} is less than $0$, no printing is done and
if {\bf LEVEL} is $0$ or greater, fatal error information is written to
the output unit number {\bf NOUT}. {\bf IER} is an error flag indicating:
\bigskip
\begin{tabular}{rrl} \hline
\multicolumn{2}{l}{Error Flag} & Meaning \\ \hline
${\bf IER} =$ & $0$, & Successful completion. \\
$ =$ & $703$, & {\bf NZ} is too small---no room for the new entry. \\ \hline
\end{tabular}
\bigskip
\noindent
Note that {\bf SBINI()} should not be called after {\bf SBAGN()} is
called since it would destroy the previous data.
\section{Examples}
\label{examples}
Given a linear system $Au=b$ with
\[ A = \lb \begin{array}{rrrr}
4 & -1 & -1 & 0 \\
-1 & 4 & 0 & -1 \\
-1 & 0 & 4 & -1 \\
0 & -1 & -1 & 4
\end{array} \rb, \hspace{0.3in} b = \lb \begin{array}{c}
6 \\ 0 \\ 0 \\ 6 \end{array} \rb, \]
a program to solve this problem with an initial guess of
$u^T = (0, 0, 0, 0)$ using {\bf JCG()} with symmetric sparse storage and
printing the final approximate solution vector follows.
\begin{verbatim}
INTEGER IA(5), JA(8), IPARM(12), IWKSP(12)
REAL A(8), RHS(4), u(4), WKSP(24), RPARM(12)
DATA A(1),A(2),A(3),A(4) / 4.0,-1.0,-1.0,4.0 /
DATA A(5),A(6),A(7),A(8) / -1.0,4.0,-1.0,4.0 /
DATA JA(1),JA(2),JA(3),JA(4) / 1,2,3,2 /
DATA JA(5),JA(6),JA(7),JA(8) / 4,3,4,4 /
DATA IA(1),IA(2),IA(3),IA(4),IA(5) / 1,4,6,8,9 /
DATA RHS(1),RHS(2),RHS(3),RHS(4) / 6.0,0.0,0.0,6.0 /
DATA N /4/, NW /24/, ITMAX /4/, LEVEL/1/, IDGTS/2/
C
CALL DFAULT (IPARM, RPARM)
IPARM(1) = ITMAX
IPARM(2) = LEVEL
IPARM(12) = IDGTS
CALL VFILL (N, U, 0.E0)
CALL JCG (N,IA,JA,A,RHS,U,IWKSP,NW,WKSP,IPARM,RPARM,IER)
STOP
END
\end{verbatim}
\noindent
The output for this run would be
\begin{verbatim}
BEGINNING OF ITPACK SOLUTION MODULE JCG
JCG HAS CONVERGED IN 2 ITERATIONS.
APPROX. NO. OF DIGITS (EST. REL. ERROR) = 14.6 (DIGIT1)
APPROX. NO. OF DIGITS (EST. REL. RESIDUAL) = 14.3 (DIGIT2)
SOLUTION VECTOR.
1 2 3 4
-------------------------------------------------------------
2.00000E+00 1.00000E+00 1.00000E+00 2.00000E+00
\end{verbatim}
Textbook methods such as the Jacobi (J), Gauss-Seidel (GS), Successive
Overrelaxation (SOR---fixed relaxation factor omega), Symmetric
Successive Overrelaxation (SSOR---fixed relaxation factor omega), and
the RS method can be obtained from this package by resetting appropriate
parameters after the subroutine {\bf DFAULT()} is called but before
ITPACK routines are called.
\bigskip
\begin{tabular}{lll} \hline
Method & Use & Parameters \\ \hline
J & {\bf JSI()} & ${\bf IPARM(6)=0, IPARM(7)=2}$ \\
GS & {\bf SOR()} & ${\bf IPARM(6)=0}$ \\
SOR---fixed omega & {\bf SOR()} & ${\bf IPARM(6)=0, RPARM(5)=OMEGA}$ \\
SSOR---fixed omega & {\bf SSORSI()} & ${\bf IPARM(6)=0, RPARM(5)=OMEGA}$ \\
RS & {\bf RSSI()} & ${\bf IPARM(6)=0}$ \\ \hline
\end{tabular}
\bigskip
\noindent
These methods were not included as separate routines because they are
usually slower than the accelerated methods included in this package.
On the black unknowns, the Cyclic Chebyshev Semi-Iterative (CCSI)
method of Golub and Varga \cite{2} gives the same result as the RSSI
method. The CCSI and RSSI methods converge at the same rate, and each
of them converges twice as fast as the JSI method. This is a theoretical
result \cite{6} and does not count the time involved in establishing
the red-black indexing and the red-black partitioned system. Similarly,
the Cyclic Conjugate Gradient (CCG) method with respect to the black
unknowns, considered by Reid \cite{16} (see also Hageman and Young
\cite{6}), gives the same results as the RSCG method. Also, the CCG and
the RSCG methods converge at the same rate, and each of them converges,
theoretically, exactly twice as fast as the JCG method. Hence, the
accelerated RS methods are preferable to the accelerated J methods
when using a red-black indexing.
\section{Numerical Results}
\label{results}
The iterative algorithms in ITPACK have been tested over a wide class
of matrix problems arising from elliptic partial differential equations
with Dirichlet, Neumann, and mixed boundary conditions on arbitrary
two-dimensional regions (including cracks and holes) and on rectangular
three-dimensional regions \cite{1}. Both finite-difference and
finite-element procedures have been employed to obtain the linear systems.
The two sample problems presented here, while simple to pose, are
representative of the behavior of the ITPACK routines for more complex
problems. The iterative algorithms make no use of the constant
coefficients in these two problems or of the particular structure of the
resulting linear system. Because the ITPACK code is not tailored to any
particular class of partial differential equations or discretization
procedure, but rather to sparse linear systems, it is felt that the
package can be used to solve a wider class of problems.
We now consider two simple partial differential equations which when
discretized by finite-difference methods give rise to large sparse
linear systems. We obtain the solution of each of these systems by the
seven algorithms in ITPACK 2C. These numerical results should aid the
user of ITPACK in determining the amount of time required when solving
more complicated sparse systems. However, one should not interpret
these execution times as conclusive by themselves. Variances introduced
by different compilers, computer systems, and timing functions can
sometimes be significant. Moreover, the number of iterations required
by an iterative method is dependent on the problem being solved, the
initial estimate for the solution, the parameter estimates used, and the
relative accuracy requested in the stopping criterion {\bf RPARM(1)}.
These tests were run on the CDC Cyber 170/750 at the University of Texas
with the FTN 4.8 compiler (OPT=2).
To obtain representative sparse linear systems, we discretize the
following two self-adjoint elliptic partial differential equations in a
region with prescribed conditions on the boundary. Here $u_{xx}$,
$u_{yy}$, $u_{zz}$ are partial derivatives and $du/dn$ is the derivative
in the normal direction.
\begin{eqnarray}
u_{xx} + 2u_{yy} = 0, && \mbox{$(x,y)$ in $S=(0,1)\times (0,1)$} \\
u = 1 + xy, && \mbox{$(x,y)$ on the boundary of $S$}
\nonumber
\end{eqnarray}
\noindent
Using the standard 5-point symmetric finite-difference operator with
$h=\frac{1}{20}$, we obtain a sparse linear system with $1729$ nonzero
elements and $361$ unknowns.
\begin{eqnarray}
u_{xx}+2u_{yy}+3u_{zz}=0, && \mbox{$(x,y,z)$ in
$C=(0,1)\times (0,1)\times (0,1)$} \nonumber \\
\mbox{On the boundary of C:} && \\
u=1, && \mbox{$(0,y,z)$, $(x,0,z)$, or $(x,y,0)$} \nonumber \\
du/dn = yz(1 + yz), && (1,y,z) \nonumber \\
du/dn = xz(1 + xz), && (x,1,z) \nonumber \\
du/dn = xy(1 + xy), && (x,y,1) \nonumber
\end{eqnarray}
\noindent
Using the standard 7-point symmetric finite difference operator with
$h=\frac{1}{7}$, we obtain a sparse linear system with $1296$ nonzero
elements and $216$ unknowns.
Tables~\ref{prob1} and \ref{prob2} display the number of iterations and
execution times (in seconds) for the seven methods in ITPACK 2C for the
linear systems corresponding to problems (1) and (2), respectively, using
symmetric sparse storage. Both the time for the iteration algorithm and
the total time for the subroutine call are given. The stopping criterion
was set to $5\times 10^{-6}$. To illustrate how effective the adaptive
procedures are, we have included in these tables the number of iterations
and the time when the optimum iteration parameters were used with no
adaptive procedures.
\begin{table}
\begin{center}
\begin{tabular}{llrrr} \hline
Routine & Ordering & Iterations & Iteration time & Total time \\ \hline
{\bf JCG()} & Natural & 61 (61) & .250 (.247) & .281 (.271) \\
& Red-black & 61 (61) & .232 (.246) & .402 (.413) \\
& & & & \\
{\bf JSI()} & Natural & 108 (95) & .408 (.344) & .439 (.375) \\
& Red-black & 108 (95) & .393 (.332) & .569 (.498) \\
& & & & \\
{\bf SOR()} & Natural & 72 (54) & .356 (.280) & .368 (.307) \\
& Red-black & 65 (47) & .311 (.224) & .469 (.411) \\
& & & & \\
{\bf SSORCG()} & Natural & 17 (13) & .232 (.173) & .264 (.185) \\
& & & & \\
{\bf SSORSI()} & Natural & 23 (22) & .242 (.213) & .273 (.244) \\
& & & & \\
{\bf RSCG()} & Red-black & 31 (31) & .104 (.117) & .269 (.297) \\
& & & & \\
{\bf RSSI()} & Red-black & 60 (48) & .207 (.166) & .358 (.344) \\
\hline
\end{tabular}
\caption{Number of Iterations and Execution Times for
Problem (1) Using Adaptive and Nonadaptive
Procedures (Nonadaptive Data in Parentheses)}
\label{prob1}
\end{center}
\end{table}
\begin{table}
\begin{center}
\begin{tabular}{llrrr} \hline
Routine & Ordering & Iterations & Iteration time & Total time \\ \hline
{\bf JCG()} & Natural & 28 (28) & .092 (.090) & .107 (.090) \\
& Red-black & 28 (28) & .079 (.074) & .191 (.202) \\
& & & & \\
{\bf JSI()} & Natural & 64 (54) & .166 (.136) & .196 (.152) \\
& Red-black & 64 (54) & .160 (.130) & .268 (.266) \\
& & & & \\
{\bf SOR()} & Natural & 42 (29) & .139 (.095) & .150 (.110) \\
& Red-black & 38 (29) & .124 (.097) & .236 (.231) \\
& & & & \\
{\bf SSORCG()} & Natural & 15 (11) & .136 (.097) & .167 (.111) \\
& & & & \\
{\bf SSORSI()} & Natural & 19 (15) & .138 (.101) & .153 (.117) \\
& & & & \\
{\bf RSCG()} & Red-black & 15 (15) & .032 (.051) & .150 (.169) \\
& & & & \\
{\bf RSSI()} & Red-black & 31 (27) & .075 (.064) & .186 (.196) \\
\hline
\end{tabular}
\caption{Number of Iterations and Execution Times for
Problem (2) Using Adaptive and Nonadaptive
Procedures (Nonadaptive Data in Parentheses)}
\label{prob2}
\end{center}
\end{table}
Values corresponding to the red-black ordering with the SSOR methods
are omitted from the tables since it is known that these methods are
ineffective with this ordering. Since the RS methods are defined for
only the red-black ordering, the table entries for these methods with
the natural ordering are not included.
\section{Notes on Use}
\label{notes}
Before an iterative algorithm is called to solve a linear system, the
values in the array {\bf A(*)} are permuted and scaled. Afterwards, these
values are unpermuted and unscaled. Consequently, the values in arrays
{\bf A(*)} and {\bf RHS(*)} may change slightly due to roundoff errors
in the computer arithmetic. Moreover, since entries in each row of the
linear system may be stored in any order within a contiguous block of
data, the locations of elements of {\bf A(*)} and of corresponding ones
in {\bf JA(*)} may change from those given before the permuting and
unpermuting was done. The same linear system is defined by the arrays
{\bf A(*)}, {\bf JA(*)}, and {\bf IA(*)} whether or not corresponding
elements in {\bf A(*)} and {\bf JA(*)} have changed locations within
contiguous blocks.
Scaling of the linear system is done as follows to reduce the number
of arithmetic operations. The diagonal entries of the linear system are
checked for positivity and are moved to the first {\bf N} locations of the
array {\bf A(*)}. The nonzero off-diagonal entries of the linear system
$Au=b$ are scaled. The scaling involves the diagonal matrix $D^{\ha}$ of
square roots of the diagonal entries of the linear system, that is,
\[ (D^{-\ha}AD^{-\ha})(D^{\ha}u) = (D^{-\ha}b). \]
The algorithms iterate until convergence is reached based on the
relative accuracy requested via the stopping criterion set in {\bf RPARM(1)}
for the scaled solution vector $(D^{\ha}u)$. Unscaling solves for $u$ and
returns the linear system to its original form subject to roundoff
errors in the arithmetic and to possible movement of entries within
contiguous blocks of data.
When requested, a red-black permutation of the data will be done
before and after the iterative algorithm is called. Otherwise, the
linear system is used in the order it is given which we call the
``natural ordering."
The Successive Overrelaxation (SOR) method has been shown to be more
effective with the red-black ordering than with the natural ordering for
some problems \cite{18}. In the SOR algorithm, the first iteration uses
$\om = 1$ and the stopping criterion is set to a large value so
that at least one Gauss-Seidel iteration is performed before an
approximate value for the optimum relaxation parameter is computed.
Optional features of this package are red-black ordering, effective
removal of rows and columns when the diagonal entry is extremely large,
and error analysis. In the event that one is not using some of these
options and needs additional memory space for a very large linear
system, the relevant subroutines which can be replaced with dummy
subroutines are as follows: red-black ordering [{\bf PRBNDX()},
{\bf PERMAT()}, {\bf PERVEC()}, {\bf QSORT()}], removal of rows
[{\bf SBELM()}], error analysis [{\bf PERROR()}].
The timing routine {\bf TIMER()} should call a routine which returns
the run time in seconds.
The value of the machine relative precision is contained in the
variable {\bf SRELPR} which is set in the subroutine {\bf DFAULT()}
and in the test program. This and other default values may be
permanently changed when the code is installed by changing their
values in the subroutine {\bf DFAULT()}. {\bf SRELPR} must be changed
when moving the code to another computer. If the installer of this
package does not know its value, an approximate value can be determined
from a simple FORTRAN program given in the comment statements of
subroutine {\bf DFAULT()}.
Since the amount of precision may change from computer to computer,
the relative accuracy requested in the stopping criterion {\bf ZETA}
must not be less than about $500$ times the machine relative precision
{\bf SRELPR}. If a value of {\bf ZETA} is requested that is too small
then the code resets it to this value. The current default value for
{\bf ZETA}, $5 \times 10^{-6}$, is set by the routine {\bf DFAULT()}
into {\bf RPARM(1)}.
The distribution tape contains the ITPACK 2C software package of $71$
subprograms and a testing program {\bf MAIN()} together with its $27$
subprograms. The routines {\bf DFAULT()} and {\bf TIMER()}
in ITPACK and the program {\bf MAIN()} are the only ones requiring
editing by the installer of the package. ITPACK can be made into
a compiled program library although not all of it would normally
be used in a particular application.
\section{ITPACK History}
\label{history}
The 2C version of the ITPACK codes described here is the result of
several years of research and development. The development of ITPACK
began in the early 1970's when Professor Garrett Birkhoff suggested that
general purpose software for solving linear systems should be developed
for iterative methods as well as for direct methods. Initially,
prototype programs were written based on preliminary iterative
algorithms involving adaptive selection of parameters and automatic
stopping procedures. These programs were tested on a large set of
elliptic partial differential equations over domains compatible with the
subroutine {\bf REGION()} \cite{8} which superimposed a square grid over the
domain. These routines were designed for solving self-adjoint elliptic
partial differential equations. Next a preliminary version of ITPACK
was coded in standard FORTRAN. The ITPACK routines used iterative
algorithms which were refined from the prototype programs. However,
these routines were designed to solve large sparse linear systems of
algebraic equations instead of partial differential equations. The use
of three interchangeable symmetric sparse storage modes in ITPACK 1.0
\cite{3} allowed for great flexibility and made it possible to solve a wider
class of problems than the prototype programs and to study different
storage modes for iterative methods. The next version, ITPACK 2.0
\cite{4}, was significantly faster than its predecessor since it was
restricted to allow only one sparse symmetric storage format. Most of
the iterative algorithms utilized in the 2.0 version of this package
assume that the coefficient matrix of the linear system is symmetric
positive definite. As with many packages, the need to handle a slightly
larger class of problems, namely, nearly symmetric systems, soon became
evident. This required adapting the routines to allow a switch for
either a symmetric or nonsymmetric storage mode in ITPACK 2A \cite{5}.
Moreover, a modification of the Conjugate Gradient algorithms was
developed to handle nearly symmetric systems \cite{12}. ITPACK has been
improved in the 2B version \cite{14} by (a) writing more efficient
versions of several key subroutines, (b) incorporating Basic Linear
Algebra Subprograms, BLAS \cite{15}, and (c) improving the user interface
with better printing and documentation. Some additional improvements
and corrections were made in the 2C version. The algorithms in ITPACK
are not guaranteed to converge for all linear systems but have been
shown to work successfully for a large number of symmetric and
nonsymmetric systems which arise from solving elliptic partial
differential equations \cite{1,13}.
The numerical algorithms in ITPACK 2C correspond to those described in
the appendix of technical report \cite{5} and outlined in the book
\cite{7}. In particular, the SOR code is based on an algorithm
suggested to us by L. Hageman. Various other algorithms exist for
iterative methods. For example, S. Eisenstat has an implementation
of the Symmetric Successive Overrelaxation preconditioned Conjugate
Gradient procedure.\footnote{Private communication.}
Modules based on the seven iterative routines in ITPACK have been
incorporated into the elliptic partial differential equation solving
package ELLPACK \cite{17} together with all the necessary translation
routines needed. The user-oriented modules described in
Section~\ref{modules} are
not in ELLPACK. Moreover, if the ELLPACK system is not being used to
generate the linear system for ITPACK, it is recommended that ITPACK
be used as a stand-alone package apart from ELLPACK.
\bigskip
\noindent
{\bf Acknowledgements}
\bigskip
The authors wish to thank the referee for carefully going through the
code and documentation for several different versions. Test runs were
made on a variety of computer systems and helpful suggestions were made
by R. Boisvert, W. Coughran, J. Dongarra, W. Dyksen, S. Eisenstat,
S. Fillebrown, P. Gaffney, W. Gordon, R. Hanson, R. Lynch, J. Rice,
B. Ward, and others. These suggestions and comments together with those
of the referee have resulted in an improved software package. ITPACK
has been tested on the following computing machines: CDC 6400, 6500,
6600, 7600, Cyber 170/750, 203, 205; Cray 1; DEC 10, 20, PDP 10, VAX
11/750, 11/780; IBM 195, 370/158, 3033; PRIME 400, 750; and others.
\newpage
\begin{thebibliography}{99}
\bibitem{1} S. Eisenstat, A. George, R. Grimes, D. Kincaid, and
A. Sherman. ``Some Comparisons of Software Packages for Large
Sparse Linear Systems," in {\em Advances in Computer Methods for
Partial Differential Equations III}, (R. Vichnevetsky and
R. Stepleman, eds.), Publ. IMACS, Department of Computer Science,
Rutgers University, New Brunswick, New Jersey, 08903, 1979,
pp. 98-106.
\bibitem{2} G. Golub and R. Varga. ``Chebyshev Semi-Iterative Methods,
Successive Overrelaxation Iterative Methods, and Second-Order
Richardson Iterative Methods," Parts I \& II, {\em Numerische
Mathematik}, Vol. 3, 1961, pp. 147-168.
\bibitem{3} R. Grimes, D. Kincaid, W. Macgregor, and D. Young.
``ITPACK Report: Adaptive Iterative Algorithms Using Symmetric
Sparse Storage," CNA-139, Center for Numerical Analysis, University of
Texas, Austin, Texas, 78712, August 1978.
\bibitem{4} R. Grimes, D. Kincaid, and D. Young. ``ITPACK 2.0 User's
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\end{thebibliography}
\end{document}