From oleg@nrcbsa.bio.nrc.ca Thu May 23 08:52:24 1991
Return-Path: <oleg@nrcbsa.bio.nrc.ca>
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	id AA26046; Thu, 23 May 91 08:52:12 -0400
Message-Id: <9105231251.AA28692@ nrcbsa.bio.nrc.ca>
Date: Thu, 23 May 91 08:51:36 EDT
From: Dr. Oleg Keselyov <oleg@nrcbsa.bio.nrc.ca>
To: dongarra@cs.utk.edu
Subject: HL_vector.shar
Status: RO

#--------------------------------CUT HERE-------------------------------------
#! /bin/sh
#
# This is a shell archive. Save this into a file, edit it
# and delete all lines above this comment.  Then give this
# file to sh by executing the command 'sh file'. The files
# will be extracted into the current directory owned by
# you with default permissions.

# The archive contains a package of high level vector operations. Involves the 
# Aitken-Lagrange interpolation over the table of uniform or
# arbitrary mesh, Hook-Jeevse local multidimensional minimizer.

#			The contents of the archive
#
# Source code for the programs being submitted
# 	ali.c		Aitken-Lagrange interpolation over the table of 
#			uniform or arbitrary mesh
# 	hjmin.c		Hook-Jeevse local minimum finder
#
# Validation routines
# 	valin.c		Verify alin() function (Arbitrary mesh interpolation)
#	valis.c		Verify alis() function (Uniform mesh interpolation)
#	vhjmin.c	Verify Hook-Jeevse minimizer 
#
# Verification protocols
#	valin.dat	On Silicon Graphics IRIS 4D/280S
#	valis.dat
#	vhjmin.dat
#	valin.dat.vax	On VAX 6000 station 
#	valis.dat.vax
#	vhjmin.dat.vax

echo 'x - ali.c'
sed 's/^X//' << '________This_Is_The_END________' > ali.c
X/*
X ************************************************************************
X *	    		    C math library
X *		     Aitken-Lagrange interpolation
X *
X * function ALIS - interpolate function value for a given argument value
X *	  using function values tabulated over the uniform grid
X *
X * Input
X *	double alis(q,x0,s,y)
X *	double q 			Argument value specified
X *	double x0 			Argument for the 1. grid
X *	double s 			Grid mesh, >0
X *	VECTOR y			Vector of function values
X *					tabulated at points
X *					x0 + s*(i-V_lwb(v))
X *                                      Vector has to contain at 
X *					least 2 elements
X *
X * function ALIN - interpolate function value for a given argument value
X *	  using function values tabulated over the non-uniform grid
X *
X * Input
X *	double alin(q,x,y)
X *	const double q			Argument value specified
X *	const VECTOR x			Vector of argument values
X *	const VECTOR y			Vector of function values
X *					tabulated at points x[i]
X *                                      Vector has to contain at 
X *					least 2 elements
X * Output
X *	Both functions return the interpolated function value at point q
X *	Interpolation is terminated either
X *		- if the difference between two successive interpolated
X *		  values is absolutely less than EPSILON
X *		- if the absolute value of this difference stops
X *		  diminishing
X *		- after (N-1) steps, N being the no. of elements in vector y
X *
X * Algorithm
X *	Aitken scheme of Lagrange interpolation
X *	Algorithm is described in a book
X *		"Mathematical software for computers", Institute of
X *		 Mathematics of the Belorussian Academy of Sciences,
X *		 Minsk, 1974, issue #4
X *		 p. 146 (description of ALI, DALI subroutines)
X *		 p. 180 (description of ATSE, DATSE subroutines)
X *	
X ************************************************************************
X */
X
X
X#include "assert.h"
X#include "vector.h"
X#include "math.h"
X#include <malloc.h>
X
X
X/*
X *	Package private data
X */
X
Xstatic VECTOR Arg;		/* [1:n] Arranged table of arguments	*/
Xstatic VECTOR Val;		/* [1:n] Arranged table of function values*/
Xstatic double Q;           	/* Argument the function is to be 	*/
X				/* interpolated at			*/
X
X
Xstatic void allocate(n)
Xconst int n;			/* No. of points arrays must be allocated at*/
X{
X  if( Arg != (VECTOR)0 )	/* Vectors were allocated at previous calls*/
X    if( V_no_elems(Arg) == n )
X      return;			/* Vectors were allocated for a proper no.  */
X    else
X    {
X      V_free(Arg);
X      V_free(Val);
X    }
X  Arg = V_new(1,n);
X  Val = V_new(1,n);
X}
X
X
X/*
X *	Aitken - Lagrange process
X *
X *  Arg and Val tables are assumed to be arranged in the proper way
X *  (see root module)
X *
X */
X
Xstatic double aitken()
X{
X  register double valp = *V_elem(Val,1);	/* Best approximation found*/
X  register double diffp = HUGE_VAL;		/* abs(valp-prev. to valp)*/
X  register int i,j;
X
X/*  V_print(Arg,"Arg - interpolation nodes");  V_print(Val,"Val"); */
X			/* Compute j-th row of the Aitken scheme and	*/
X			/* place it in the Val array			*/
X  for(j=2; j<=V_upb(Val); j++)
X  {
X     register double argj = *V_elem(Arg,j);
X     register V$elem_type *valj = V_elem(Val,j);
X     register double diff;
X     for(i=1; i<=j-1; i++)
X        *valj = ( *V_elem(Val,i)*(Q-argj) - *valj * (Q-*V_elem(Arg,i)) )/
X		(*V_elem(Arg,i) - argj);
X
X/*     printf("\nValj = %g, valp = %g, argj=%g",*valj,valp,argj);  */
X     diff = fabs( *valj - valp );
X
X     if( j>2 && diff == 0 )  		/* Exact result has been achieved*/
X       break;
X
X     if( j>4 && diff > diffp )		/* Difference stops diminishing	*/
X       break;                           /* after the 4. step		*/
X
X     valp = *valj;  diffp = diff;
X  }
X
X  return valp;
X}
X
X
X
X/* 
X *=======================================================================
X *		          Function ALIS root module
X *
X * Arranges input data in the form suitable for aitken and call the latter
X *
X * Algorithm 
X * Fill in vectors arg and val[i] = y(arg[i]) so that distance abs(x-arg[i]) 
X * would increase as i increases. 
X *
X */
X
Xdouble alis(q,x0,s,y)
Xdouble q; 				/* Argument value specified	*/
Xdouble x0; 				/* Argument for the 1. grid   	*/
Xdouble s; 				/* Grid mesh			*/
XVECTOR y;				/* Vector of function values	*/
X{
X  int n = V_no_elems(y);		/* No. of grids			*/
X
X  assure( n > 1, "too few function values, DIM(y) must be > 1");
X  assure( s > 0, "grid mesh has to be a positive number");
X  allocate(n);
X  Q = q;
X
X			/* Preparation step				*/
X	                /* Selection is done by computing the subscript */
X			/* js of that argument, which is next to x.      */
X	                /* Afterwards neighbouring argument values are  */
X			/* tested and selected in the above sense.	*/
X   
X  {
X    int js = (int)( (q-x0)/s + 1.5 );	/* Index for a point 		*/
X  					/* x0+s*i closest to q 		*/
X    register int dir;			/* Direction to the next grid   */
X					/* close to q      		*/   
X    register int jcurr, jleft, jright;
X    register int i;
X
X    if( js<1 )                          /* Test for the case of extrapo-*/
X      js = 1;		                /* lation to the left end	*/
X    else if( js > n )
X      js = n;                           /* or to the right end		*/
X
X    dir = ( q > x0 + (js-1)*s ? 1 : -1 );
X
X    jcurr = js; jleft = js; jright = js;
X    for(i=1; i<=n; ++i)			/* Pick up elements x0+s*i	*/
X    {                                   /* in the neighbourhood of x 	*/
X       *V_elem(Arg,i) = x0 + (jcurr-1)*s;
X       *V_elem(Val,i) = *V_elem(y,jcurr-1+V_lwb(y));
X       if( jright >= n )
X	 dir = -1;
X       if( jleft <= 1 )
X	 dir = 1;
X       if( dir > 0 )
X       {
X         jcurr = ++jright;
X	 dir = -1;
X       }
X       else
X       {
X         jcurr = --jleft;
X	 dir = 1;
X       }
X    }
X  }
X
X  return aitken();
X}
X
X
X
X
X/*
X *	Function that defines the order while sorting 'indices' array
X *				with qsort
X *	Function gets pointers to two elements in indices array, i.e.
X *	to the two indices for the Arg[i] and Arg[j] elements.
X *	Function returns -1, 0, or +1 depending on the fact if 
X *	abs(Arg[i]-Q) is less, equal, or greater than abs(Arg[j]-Q)
X */
X
Xstatic int compare_indices(ip,jp)
Xconst int * ip;
Xconst int * jp;
X{
X  register double delta = fabs(*V_elem(Arg,*ip) - Q) -  
X	 		  fabs(*V_elem(Arg,*jp) - Q);
X  if( delta < 0 )
X    return -1;
X  else if(delta == 0)
X    return 0;
X  else
X    return 1;
X}
X
X
X/* 
X *=======================================================================
X *		          Function ALIN root module
X *
X * Arranges input data in the form suitable for aitken and call the latter
X *
X * Algorithm 
X * Fill in vectors arg and val[i] = y(arg[i]) so that distance abs(q-arg[i]) 
X * would increase as i increases. 
X *
X */
X
Xdouble alin(q,x,y)
Xconst double q;				/* Argument value specified	*/
Xconst VECTOR x; 			/* Vector of argument values   	*/
Xconst VECTOR y;				/* Vector of function values	*/
X{
X  int n = V_no_elems(x);		/* No. of grids			*/
X
X  V_are_compatible(x,y);
X  assure( n > 1, "too few function values, DIM(x) must be > 1");
X  allocate(n);
X  Q = q;
X			/* Preparation step				*/
X	                /* Selection is done by sorting x,y arrays	*/
X			/* in the way mentioned above. Fisrt an array	*/
X	                /* of indices is created and sorted, then arg,  */
X			/* val arrays are filled in with indices sorted	*/
X   
X  {     
X    int * indices = calloc(V_no_elems(Arg)+V_lwb(Arg),sizeof(int));
X    register int i;
X
X    for(i=V_lwb(Arg); i<=V_upb(Arg); i++)/* 1..n correspond to the	*/
X    {  		                 	 /* unsorted x,y		*/
X      indices[i] = i;	
X      *V_elem(Arg,i) = *V_elem(x,V_lwb(x)+i-V_lwb(Arg));
X    }
X                                                
X    					/* Sort indices in the way that	*/
X					/* |q-x[ind[i]]| < |q-x[ind[j]]|*/
X					/* for all j>i			*/
X    qsort(&indices[V_lwb(Arg)],n,sizeof(int),compare_indices);
X
X    for(i=V_lwb(Arg); i<=V_upb(Arg); i++)
X    {
X      register int ind = indices[i];
X      *V_elem(Arg,i) = *V_elem(x,V_lwb(x)+ind-V_lwb(Arg));
X      *V_elem(Val,i) = *V_elem(y,V_lwb(y)+ind-V_lwb(Arg));
X    }
X
X    free(indices);
X  }
X
X  return aitken();
X}
________This_Is_The_END________
if test `wc -l < ali.c` -ne 281; then
echo 'shar: ali.c was damaged during transit (should have had 281 lines)'
fi


echo 'x - hjmin.c'
sed 's/^X//' << '________This_Is_The_END________' > hjmin.c
X/*
X ************************************************************************
X *	    		    C math library
X *      function HJMIN - find local minimum of a given function
X *	   	        by the Hook-Jeevse method
X *
X * Input
X *	double hjmin(b,h0,funct)
X *	VECTOR b 			On input contains the initial
X *					guess to the minimum location
X *					On return has the vector
X *					specifying for the minimum location
X *	const VECTOR h0			Initial values for the steps along
X *					each directions
X *	double f(const VECTOR x)	Procedure to compute a function
X *					value at the specified point
X *
X * Output
X *	Hjmin returns the function value at the point of minimum
X *
X * Algorithm
X *	Hook-Jeevse method of direct search for a function minimum
X *	The method is of the 0. order (i.e. requiring no gradient computation)
X *	See
X *	B.Bondi. Methods of optimization. An Introduction - M.,
X *	"Radio i sviaz", 1988 - 127 p. (in Russian)
X *	
X *
X * Notes
X *	static VECTORs h and bas are used as work arrays. They are not
X *	destroyed (freed) on exit so that next call to Hjmin could use
X *	them if they are still appropriate for that call.
X *
X ************************************************************************
X */
X
X
X#include "assert.h"
X#include "math.h"
X#include "vector.h"
X
X/*
X *	Static data
X */
X
Xstatic VECTOR h;			/* Steps along axes		*/
Xstatic VECTOR bas;			/* Base point			*/
Xstatic double f_bas;			/* Function value at it		*/
Xstatic double (*f)(const VECTOR x);	/* Function being minimized	*/
X
X
X/*
X * Examine the function f in the vicinity of the bas point
X * by making tentative steps fro/back along the every coordinate.
X * When the function is found to decrease, the point bas is correspondingly
X * updated. Examination() returns the minimal function value found in
X * the investigated region.
X *
X */
Xstatic double examination()
X{
X  register V_pointer(ph,h);
X  register V_pointer(pbas,bas); 
X  register int i;
X  
X			/* Perform step along a coordinate		*/
X  for(i=V_lwb(bas); i<=V_upb(bas); i++,ph++,pbas++)
X  {
X     register double basi_old = *pbas;  /* Old coordinate value		*/
X     register double f_new;         
X
X     if( *pbas = basi_old + *ph, f_new = (*f)(bas), f_new < f_bas )
X       f_bas = f_new;                   /* Step caused f to decrease, OK*/
X     else if( *pbas = basi_old - *ph, f_new = (*f)(bas), f_new < f_bas )
X       f_bas = f_new;
X     else
X       *pbas = basi_old;                /* No fall was found along this coord*/
X  }
X  return f_bas;
X}                                                 
X
X
X
X/*
X *	Root module
X */
X
Xdouble hjmin(b,h0,ff)
XVECTOR b;
Xconst VECTOR h0;
Xdouble (*ff)(const VECTOR x);
X{
X  double f_min;              		/* Min function value found	*/
X                                	/* beforehand			*/
X  
X  const double tau = 10*EPSILON;        /* Termination criterium	*/
X  const double threshold = 1e-8;        /* Threshold for the function	*/
X                                        /* decay to be treated as 	*/
X                                        /* significant    		*/
X  const double step_reduce_factor = 10;   
X  register int i;
X
X  f = ff;
X  V_are_compatible(b,h0);
X
X  			/* Allocate h and bas if needed			*/
X  if( h == (VECTOR)0 || V_lwb(h) != V_lwb(h0) || V_upb(h) != V_upb(h0) )
X  {     				/* h, bas nust be (re)allocated	*/
X     if( h != (VECTOR)0 )
X     {
X        V_free(h);
X        V_free(bas);
X     }
X     h = V_new(V_lwb(b),V_upb(b));
X     bas = V_new(V_lwb(b),V_upb(b));
X  }
X  V_assign(h,h0);
X  V_assign(bas,b); f_bas = f_min = (*f)(b);
X
X
X  for(;;)		/* Main iteration loop				*/
X  {                     /* b is a next approximation to min		*/
X     if( examination() < f_min - threshold )
X     {   			        /* Function falled markedly 	*/
X        do                              /* Perform search on pattern	*/
X        {                               /* bas - pattern, b - new min appr*/
X	  register V_pointer(pbas,bas); 
X	  register V_pointer(pmin,b); 
X          for(i=V_lwb(bas); i<=V_upb(bas); i++)
X          {
X             register double t = (*pbas - *pmin) + *pbas;
X	     *pmin++ = *pbas;
X	     *pbas++ = t;
X          }
X          f_min = f_bas;  f_bas = (*f)(bas);
X        }                               /* Continue search until f doesn't*/
X        while( examination() < f_min - threshold ); /* decrease		*/
X	V_assign(bas,b); f_bas = f_min;	/* Save the best approx found	*/
X     }
X     else                               /* Function didn't fall markedly*/
X     {                                  /* upon examination near bas	*/
X        register int success = 1;       /* Try to reduce the step	*/
X	register V_pointer(ph,h); 
X	register V_pointer(pbas,bas); 
X        for(i=V_lwb(h); i<=V_upb(h); i++)
X        {
X           *ph /= step_reduce_factor;
X	   success &= ( *ph++ /(1 + fabs(*pbas++)) < tau );
X        }
X        if( success )
X	  return f_min;
X     }
X
X  }
X}
________This_Is_The_END________
if test `wc -l < hjmin.c` -ne 155; then
echo 'shar: hjmin.c was damaged during transit (should have had 155 lines)'
fi


echo 'x - valin.c'
sed 's/^X//' << '________This_Is_The_END________' > valin.c
X/*
X *	Verify ALIN functions
X */
X
X#include "vector.h"
X#include "math.h"
X
Xstatic double f(x)  		/* Function involved in the test	*/
Xdouble x;
X{
X  return sin(x) * exp(-x/10);
X}
X
Xstatic VECTOR Nodes;
X
Xstatic void test(msg,q,ia,ib)	/* Test interpolation to q with		*/
Xconst char * msg;
Xdouble q;			/* x[ia,ib], y[ia:ib] 			*/
Xint ia,ib;
X{
X  VECTOR x = V_new(ia,ib);
X  VECTOR y = V_new(ia,ib);
X  register int i;
X
X  for(i=V_lwb(y); i<=V_upb(y); i++)
X     *V_elem(y,i) = f( (*V_elem(x,i) = *V_elem(Nodes,i)) );
X  
X  printf("\n\n%s\nNodes within the range [%d:%d] are used in interpolation\n",
X	  msg,ia,ib);
X  
X  printf("\nInterpolated value to x= %g is\t%g",q,alin(q,x,y));
X  printf("\nExact function value at this point\t%g\n",f(q));
X
X  V_free(x);
X  V_free(y);
X}
X
Xmain()
X{
X  printf("\n\n\t\tVerify ALIN procedure for a function sin(x) * exp(-x/10)\n");
X
X  {
X    register int i;
X    Nodes = V_new(2,51);
X    for(i=V_lwb(Nodes); i<=11; i++)
X       *V_elem(Nodes,i) = (i-1.)/10;
X    for(i=12; i<=V_upb(Nodes); i++)
X       *V_elem(Nodes,i) = (i-6.)/5;
X
X    printf("\t\tGrids 0.1 .. 1 have the mesh 0.1, and 1..9 have the mesh 0.2");
X  }
X
X  test("Extrapolation to the left",0.0,2,11);
X  test("Extrapolation to the right",8.0,2,41);
X  test("Interpolation just to the 16. node",2.0,2,31);
X  test("Interpolation to middle of table",1.55,2,50);
X
X}
________This_Is_The_END________
if test `wc -l < valin.c` -ne 58; then
echo 'shar: valin.c was damaged during transit (should have had 58 lines)'
fi


echo 'x - valis.c'
sed 's/^X//' << '________This_Is_The_END________' > valis.c
X/*
X *	Verify ALIS functions
X */
X
X#include "vector.h"
X#include "math.h"
X
Xstatic double s = 0.1;
X
Xstatic double f(x)  		/* Function involved in the test	*/
Xdouble x;
X{
X  return exp(-x);
X}
X
Xstatic void test(msg,x,ia,ib)	/* Test interpolation to x with		*/
Xconst char * msg;
Xdouble x;			/* y[ia:ib]				*/
Xint ia,ib;
X{
X  VECTOR y = V_new(ia,ib);
X  register int i;
X
X  for(i=V_lwb(y); i<=V_upb(y); i++)
X     *V_elem(y,i) = f(s*i);
X
X  printf("\n\n%s\nThe function being tabulated over [%g:%g]\n",
X	  msg,ia*s,ib*s);
X  
X  printf("\nInterpolated value to x= %g is\t%g",x,alis(x,ia*s,s,y));
X  printf("\nExact function value at this point\t%g\n",f(x));
X}
X
Xmain()
X{
X  printf("\n\n\t\tVerify ALIS procedure for a function exp(-x)\n");
X  printf("\t\t\tTabulated unifomly with the grid mesh of %g\n\n",s);
X
X  test("Extrapolation to the left",0.0,1,10);
X  test("Extrapolation to the right",30.5*s,0,28);
X  test("Interpolation just to the 10. node",10*s,5,15);
X  test("Interpolation to middle of table",5.6*s,1,10);
X
X}
________This_Is_The_END________
if test `wc -l < valis.c` -ne 44; then
echo 'shar: valis.c was damaged during transit (should have had 44 lines)'
fi


echo 'x - valin.dat'
sed 's/^X//' << '________This_Is_The_END________' > valin.dat
X
X
X		Verify ALIN procedure for a function sin(x) * exp(-x/10)
X		Grids 0.1 .. 1 have the mesh 0.1, and 1..9 have the mesh 0.2
X
XExtrapolation to the left
XNodes within the range [2:11] are used in interpolation
X
XInterpolated value to x= 0 is	8.51708e-07
XExact function value at this point	0
X
X
XExtrapolation to the right
XNodes within the range [2:41] are used in interpolation
X
XInterpolated value to x= 8 is	0.443167
XExact function value at this point	0.444547
X
X
XInterpolation just to the 16. node
XNodes within the range [2:31] are used in interpolation
X
XInterpolated value to x= 2 is	0.74447
XExact function value at this point	0.74447
X
X
XInterpolation to middle of table
XNodes within the range [2:50] are used in interpolation
X
XInterpolated value to x= 1.55 is	0.85623
XExact function value at this point	0.85623
________This_Is_The_END________
if test `wc -l < valin.dat` -ne 31; then
echo 'shar: valin.dat was damaged during transit (should have had 31 lines)'
fi


echo 'x - valis.dat'
sed 's/^X//' << '________This_Is_The_END________' > valis.dat
X
X
X		Verify ALIS procedure for a function exp(-x)
X			Tabulated unifomly with the grid mesh of 0.1
X
X
X
XExtrapolation to the left
XThe function being tabulated over [0.1:1]
X
XInterpolated value to x= 0 is	0.999998
XExact function value at this point	1
X
X
XExtrapolation to the right
XThe function being tabulated over [0:2.8]
X
XInterpolated value to x= 3.05 is	0.0473609
XExact function value at this point	0.0473589
X
X
XInterpolation just to the 10. node
XThe function being tabulated over [0.5:1.5]
X
XInterpolated value to x= 1 is	0.367879
XExact function value at this point	0.367879
X
X
XInterpolation to middle of table
XThe function being tabulated over [0.1:1]
X
XInterpolated value to x= 0.56 is	0.571209
XExact function value at this point	0.571209
________This_Is_The_END________
if test `wc -l < valis.dat` -ne 33; then
echo 'shar: valis.dat was damaged during transit (should have had 33 lines)'
fi


echo 'x - valin.dat.vax'
sed 's/^X//' << '________This_Is_The_END________' > valin.dat.vax
X
X
X		Verify ALIN procedure for a function sin(x) * exp(-x/10)
X		Grids 0.1 .. 1 have the mesh 0.1, and 1..9 have the mesh 0.2
X
XExtrapolation to the left
XNodes within the range [2:11] are used in interpolation
X
X
Xvector [1:10] - Arg - interpolation nodes
X
X     1	     2	     3	     4	     5	     6	     7	     8	     9	    10	
X-------------------------------------------------------------------------------
X   0.1	   0.2	   0.3	   0.4	   0.5	   0.6	   0.7	   0.8	   0.9	     1	
X
X
X
Xvector [1:10] - Val
X
X     1	     2	     3	     4	     5	     6	     7	     8	     9	    10	
X-------------------------------------------------------------------------------
X9.884006e-02	0.194735	0.286786	0.374149	0.456044	0.53176	0.600665	0.662203	0.715907	0.761394	
X
X
XValj = 2.944693e-03, valp = 9.884006e-02, argj=0.2
XValj = -8.998138e-04, valp = 2.944693e-03, argj=0.3
XValj = -5.620640e-05, valp = -8.998138e-04, argj=0.4
XValj = 7.443349e-06, valp = -5.620640e-05, argj=0.5
XValj = 1.169593e-06, valp = 7.443349e-06, argj=0.6
XValj = 8.517077e-07, valp = 1.169593e-06, argj=0.7
XValj = 1.696237e-06, valp = 8.517077e-07, argj=0.8
XInterpolated value to x= 0 is	8.517077e-07
XExact function value at this point	0
X
X
XExtrapolation to the right
XNodes within the range [2:41] are used in interpolation
X
X
Xvector [1:40] - Arg - interpolation nodes
X
X     1	     2	     3	     4	     5	     6	     7	     8	     9	    10	
X-------------------------------------------------------------------------------
X     7	   6.8	   6.6	   6.4	   6.2	     6	   5.8	   5.6	   5.4	   5.2	
X
X    11	    12	    13	    14	    15	    16	    17	    18	    19	    20	
X-------------------------------------------------------------------------------
X     5	   4.8	   4.6	   4.4	   4.2	     4	   3.8	   3.6	   3.4	   3.2	
X
X    21	    22	    23	    24	    25	    26	    27	    28	    29	    30	
X-------------------------------------------------------------------------------
X     3	   2.8	   2.6	   2.4	   2.2	     2	   1.8	   1.6	   1.4	   1.2	
X
X    31	    32	    33	    34	    35	    36	    37	    38	    39	    40	
X-------------------------------------------------------------------------------
X     1	   0.9	   0.8	   0.7	   0.6	   0.5	   0.4	   0.3	   0.2	   0.1	
X
X
X
Xvector [1:40] - Val
X
X     1	     2	     3	     4	     5	     6	     7	     8	     9	    10	
X-------------------------------------------------------------------------------
X0.32625	0.250326	0.161021	6.145556e-02	-4.469759e-02	-0.153346	-0.26013	-0.360585	-0.450327	-0.525232	
X
X    11	    12	    13	    14	    15	    16	    17	    18	    19	    20	
X-------------------------------------------------------------------------------
X-0.581617	-0.61641	-0.627301	-0.612866	-0.572666	-0.5073	-0.418426	-0.308736	-0.181887	-4.238836e-02	
X
X    21	    22	    23	    24	    25	    26	    27	    28	    29	    30	
X-------------------------------------------------------------------------------
X0.104544	0.253179	0.397478	0.531338	0.648834	0.74447	0.813426	0.85178	0.856709	0.826645	
X
X    31	    32	    33	    34	    35	    36	    37	    38	    39	    40	
X-------------------------------------------------------------------------------
X0.761394	0.715907	0.662203	0.600665	0.53176	0.456044	0.374149	0.286786	0.194735	9.884006e-02	
X
X
XValj = 0.705868, valp = 0.32625, argj=6.8
XValj = 0.505138, valp = 0.705868, argj=6.6
XValj = 0.395847, valp = 0.505138, argj=6.4
XValj = 0.434259, valp = 0.395847, argj=6.2
XValj = 0.450398, valp = 0.434259, argj=6
XValj = 0.445535, valp = 0.450398, argj=5.8
XValj = 0.443449, valp = 0.445535, argj=5.6
XValj = 0.443167, valp = 0.443449, argj=5.4
XValj = 0.441998, valp = 0.443167, argj=5.2
XInterpolated value to x= 8 is	0.443167
XExact function value at this point	0.444547
X
X
XInterpolation just to the 16. node
XNodes within the range [2:31] are used in interpolation
X
X
Xvector [1:30] - Arg - interpolation nodes
X
X     1	     2	     3	     4	     5	     6	     7	     8	     9	    10	
X-------------------------------------------------------------------------------
X     2	   2.2	   1.8	   1.6	   2.4	   2.6	   1.4	   2.8	   1.2	     1	
X
X    11	    12	    13	    14	    15	    16	    17	    18	    19	    20	
X-------------------------------------------------------------------------------
X     3	   0.9	   0.8	   3.2	   0.7	   0.6	   3.4	   0.5	   3.6	   0.4	
X
X    21	    22	    23	    24	    25	    26	    27	    28	    29	    30	
X-------------------------------------------------------------------------------
X   0.3	   3.8	   0.2	   0.1	     4	   4.2	   4.4	   4.6	   4.8	     5	
X
X
X
Xvector [1:30] - Val
X
X     1	     2	     3	     4	     5	     6	     7	     8	     9	    10	
X-------------------------------------------------------------------------------
X0.74447	0.648834	0.813426	0.85178	0.531338	0.397478	0.856709	0.253179	0.826645	0.761394	
X
X    11	    12	    13	    14	    15	    16	    17	    18	    19	    20	
X-------------------------------------------------------------------------------
X0.104544	0.715907	0.662203	-4.238836e-02	0.600665	0.53176	-0.181887	0.456044	-0.308736	0.374149	
X
X    21	    22	    23	    24	    25	    26	    27	    28	    29	    30	
X-------------------------------------------------------------------------------
X0.286786	-0.418426	0.194735	9.884006e-02	-0.5073	-0.572666	-0.612866	-0.627301	-0.61641	-0.581617	
X
X
XValj = 0.74447, valp = 0.74447, argj=2.2
XValj = 0.74447, valp = 0.74447, argj=1.8
XInterpolated value to x= 2 is	0.74447
XExact function value at this point	0.74447
X
X
XInterpolation to middle of table
XNodes within the range [2:50] are used in interpolation
X
X
Xvector [1:49] - Arg - interpolation nodes
X
X     1	     2	     3	     4	     5	     6	     7	     8	     9	    10	
X-------------------------------------------------------------------------------
X   1.6	   1.4	   1.8	   1.2	     2	     1	   0.9	   2.2	   0.8	   0.7	
X
X    11	    12	    13	    14	    15	    16	    17	    18	    19	    20	
X-------------------------------------------------------------------------------
X   2.4	   0.6	   2.6	   0.5	   0.4	   2.8	   0.3	   0.2	   0.1	     3	
X
X    21	    22	    23	    24	    25	    26	    27	    28	    29	    30	
X-------------------------------------------------------------------------------
X   3.2	   3.4	   3.6	   3.8	     4	   4.2	   4.4	   4.6	   4.8	     5	
X
X    31	    32	    33	    34	    35	    36	    37	    38	    39	    40	
X-------------------------------------------------------------------------------
X   5.2	   5.4	   5.6	   5.8	     6	   6.2	   6.4	   6.6	   6.8	     7	
X
X    41	    42	    43	    44	    45	    46	    47	    48	    49	
X-------------------------------------------------------------------------------
X   7.2	   7.4	   7.6	   7.8	     8	   8.2	   8.4	   8.6	   8.8	
X
X
X
Xvector [1:49] - Val
X
X     1	     2	     3	     4	     5	     6	     7	     8	     9	    10	
X-------------------------------------------------------------------------------
X0.85178	0.856709	0.813426	0.826645	0.74447	0.761394	0.715907	0.648834	0.662203	0.600665	
X
X    11	    12	    13	    14	    15	    16	    17	    18	    19	    20	
X-------------------------------------------------------------------------------
X0.531338	0.53176	0.397478	0.456044	0.374149	0.253179	0.286786	0.194735	9.884006e-02	0.104544	
X
X    21	    22	    23	    24	    25	    26	    27	    28	    29	    30	
X-------------------------------------------------------------------------------
X-4.238836e-02	-0.181887	-0.308736	-0.418426	-0.5073	-0.572666	-0.612866	-0.627301	-0.61641	-0.581617	
X
X    31	    32	    33	    34	    35	    36	    37	    38	    39	    40	
X-------------------------------------------------------------------------------
X-0.525232	-0.450327	-0.360585	-0.26013	-0.153346	-4.469759e-02	6.145556e-02	0.161021	0.250326	0.32625	
X
X    41	    42	    43	    44	    45	    46	    47	    48	    49	
X-------------------------------------------------------------------------------
X0.38632	0.428786	0.452664	0.457738	0.444547	0.414328	0.368939	0.310769	0.242614	
X
X
XValj = 0.853013, valp = 0.85178, argj=1.4
XValj = 0.856146, valp = 0.853013, argj=1.8
XValj = 0.856207, valp = 0.856146, argj=1.2
XValj = 0.856229, valp = 0.856207, argj=2
XValj = 0.85623, valp = 0.856229, argj=1
XValj = 0.85623, valp = 0.85623, argj=0.9
XInterpolated value to x= 1.55 is	0.85623
XExact function value at this point	0.85623
________This_Is_The_END________
if test `wc -l < valin.dat.vax` -ne 191; then
echo 'shar: valin.dat.vax was damaged during transit (should have had 191 lines)'
fi


echo 'x - valis.dat.vax'
sed 's/^X//' << '________This_Is_The_END________' > valis.dat.vax
X
X
X		Verify ALIS procedure for a function exp(-x)
X			Tabulated unifomly with the grid mesh of 0.1
X
X
X
XExtrapolation to the left
XThe function being tabulated over [0.1:1]
X
X
Xvector [1:10] - Arg - interpolation nodes
X
X     1	     2	     3	     4	     5	     6	     7	     8	     9	    10	
X-------------------------------------------------------------------------------
X   0.1	   0.2	   0.3	   0.4	   0.5	   0.6	   0.7	   0.8	   0.9	     1	
X
X
X
Xvector [1:10] - Val
X
X     1	     2	     3	     4	     5	     6	     7	     8	     9	    10	
X-------------------------------------------------------------------------------
X0.904837	0.818731	0.740818	0.67032	0.606531	0.548812	0.496585	0.449329	0.40657	0.367879	
X
X
XValj = 0.990944, valp = 0.904837, argj=0.2
XValj = 0.999138, valp = 0.990944, argj=0.3
XValj = 0.999918, valp = 0.999138, argj=0.4
XValj = 0.999992, valp = 0.999918, argj=0.5
XValj = 0.999998, valp = 0.999992, argj=0.6
XValj = 0.999998, valp = 0.999998, argj=0.7
XInterpolated value to x= 0 is	0.999998
XExact function value at this point	1
X
X
XExtrapolation to the right
XThe function being tabulated over [0:2.8]
X
X
Xvector [1:29] - Arg - interpolation nodes
X
X     1	     2	     3	     4	     5	     6	     7	     8	     9	    10	
X-------------------------------------------------------------------------------
X   2.8	   2.7	   2.6	   2.5	   2.4	   2.3	   2.2	   2.1	     2	   1.9	
X
X    11	    12	    13	    14	    15	    16	    17	    18	    19	    20	
X-------------------------------------------------------------------------------
X   1.8	   1.7	   1.6	   1.5	   1.4	   1.3	   1.2	   1.1	     1	   0.9	
X
X    21	    22	    23	    24	    25	    26	    27	    28	    29	
X-------------------------------------------------------------------------------
X   0.8	   0.7	   0.6	   0.5	   0.4	   0.3	   0.2	   0.1	     0	
X
X
X
Xvector [1:29] - Val
X
X     1	     2	     3	     4	     5	     6	     7	     8	     9	    10	
X-------------------------------------------------------------------------------
X6.081006e-02	6.720551e-02	7.427358e-02	8.208500e-02	9.071796e-02	0.100259	0.110803	0.122456	0.135335	0.149569	
X
X    11	    12	    13	    14	    15	    16	    17	    18	    19	    20	
X-------------------------------------------------------------------------------
X0.165299	0.182684	0.201897	0.22313	0.246597	0.272532	0.301194	0.332871	0.367879	0.40657	
X
X    21	    22	    23	    24	    25	    26	    27	    28	    29	
X-------------------------------------------------------------------------------
X0.449329	0.496585	0.548812	0.606531	0.67032	0.740818	0.818731	0.904837	     1	
X
X
XValj = 4.482143e-02, valp = 6.081006e-02, argj=2.7
XValj = 4.776407e-02, valp = 4.482143e-02, argj=2.6
XValj = 4.729968e-02, valp = 4.776407e-02, argj=2.5
XValj = 4.736654e-02, valp = 4.729968e-02, argj=2.4
XValj = 4.735743e-02, valp = 4.736654e-02, argj=2.3
XValj = 4.736087e-02, valp = 4.735743e-02, argj=2.2
XValj = 4.737027e-02, valp = 4.736087e-02, argj=2.1
XInterpolated value to x= 3.05 is	4.736087e-02
XExact function value at this point	4.735892e-02
X
X
XInterpolation just to the 10. node
XThe function being tabulated over [0.5:1.5]
X
X
Xvector [1:11] - Arg - interpolation nodes
X
X     1	     2	     3	     4	     5	     6	     7	     8	     9	    10	
X-------------------------------------------------------------------------------
X     1	   0.9	   1.1	   0.8	   1.2	   0.7	   1.3	   0.6	   1.4	   0.5	
X
X    11	
X-------------------------------------------------------------------------------
X   1.5	
X
X
X
Xvector [1:11] - Val
X
X     1	     2	     3	     4	     5	     6	     7	     8	     9	    10	
X-------------------------------------------------------------------------------
X0.367879	0.40657	0.332871	0.449329	0.301194	0.496585	0.272532	0.548812	0.246597	0.606531	
X
X    11	
X-------------------------------------------------------------------------------
X0.22313	
X
X
XValj = 0.367879, valp = 0.367879, argj=0.9
XValj = 0.367879, valp = 0.367879, argj=1.1
XInterpolated value to x= 1 is	0.367879
XExact function value at this point	0.367879
X
X
XInterpolation to middle of table
XThe function being tabulated over [0.1:1]
X
X
Xvector [1:10] - Arg - interpolation nodes
X
X     1	     2	     3	     4	     5	     6	     7	     8	     9	    10	
X-------------------------------------------------------------------------------
X   0.6	   0.5	   0.7	   0.4	   0.8	   0.3	   0.9	   0.2	     1	   0.1	
X
X
X
Xvector [1:10] - Val
X
X     1	     2	     3	     4	     5	     6	     7	     8	     9	    10	
X-------------------------------------------------------------------------------
X0.548812	0.606531	0.496585	0.67032	0.449329	0.740818	0.40657	0.818731	0.367879	0.904837	
X
X
XValj = 0.571899, valp = 0.548812, argj=0.5
XValj = 0.57124, valp = 0.571899, argj=0.7
XValj = 0.571208, valp = 0.57124, argj=0.4
XValj = 0.571209, valp = 0.571208, argj=0.8
XValj = 0.571209, valp = 0.571209, argj=0.3
XInterpolated value to x= 0.56 is	0.571209
XExact function value at this point	0.571209
________This_Is_The_END________
if test `wc -l < valis.dat.vax` -ne 141; then
echo 'shar: valis.dat.vax was damaged during transit (should have had 141 lines)'
fi


echo 'x - vhjmin.c'
sed 's/^X//' << '________This_Is_The_END________' > vhjmin.c
X/*	Verify HJMIN function			*/
X
X#include "math.h"
X#include "vector.h"
X
Xdouble hjmin(VECTOR b, const VECTOR h0, double (*f)(const VECTOR x));
X
Xstatic int iter_count;
X
X			/* Simplified vector print			*/
Xstatic void pr_vector(v)
XVECTOR v;
X{
X  register int i;
X  for(i=V_lwb(v); i<=V_upb(v); i++)
X     printf("%9.3g  ",*V_elem(v,i));
X}
X
X
X/*			Rosenbroke function				*/
X
Xdouble f1(v)
Xconst VECTOR v;
X{
X  register double x1 = *V_elem(v,1);
X  register double x2 = *V_elem(v,2);
X  iter_count++;
X  return 100*powi(x2 - x1*x1, 2) + powi(1 - x1, 2);
X}
X 
Xvoid test1()
X{
X  const int n = 2;
X  VECTOR b0 = V_build(1,n,-1.2,1.,"END");	/* Initial guess	*/
X  VECTOR b  = V_new(1,n);			/* Min location found	*/
X  VECTOR bm = V_build(1,n,1.0,1.0,"END");	/* Exact min location	*/
X  VECTOR h0  = V_build(1,n,10.,10.,"END");	/* Initial step		*/
X
X  V_assign(b,b0);
X  iter_count = 0;
X  printf("\n\n\tRosenbroke function\n");
X  printf("\n\n\tf = 100*(x2-x1^2)^2 + (1-x1)^2\n\n");
X  printf("\nInitial guess         b0 = "); pr_vector(b0);
X  printf("\nFunction value at it  f0 = %g",f1(b0));
X  printf("\nInitial steps         h0 = "); pr_vector(h0);
X  printf("\n");
X  printf("\nMinimum f value found f  = %g", hjmin(b,h0,f1));
X  printf("\n                  at  b  = "); pr_vector(b);
X  printf("\nExact min location    bm = "); pr_vector(bm);
X  printf("\nNo. of iterations     ni = %d\n",iter_count);
X
X}
X
X
X
X/*			     Bocks function 				*/
X
Xdouble f2(v)
Xconst VECTOR v;
X{
X  register double x1 = *V_elem(v,1);
X  register double x2 = *V_elem(v,2);
X  iter_count++;
X  return powi( exp(-1./10) - exp(-x1/10) + exp(-10./10) - exp(-x2/10), 2);
X}
X 
Xvoid test2()
X{
X  const int n = 2;
X  VECTOR b0 = V_build(1,n,0.,0.,"END");		/* Initial guess	*/
X  VECTOR b  = V_new(1,n);			/* Min location found	*/
X  VECTOR bm = V_build(1,n,10.0,1.0,"END");	/* Exact min location	*/
X  VECTOR h0  = V_build(1,n,10.,10.,"END");	/* Initial step		*/
X
X  V_assign(b,b0);
X  iter_count = 0;
X  printf("\n\n\tBocks function\n");
X  printf("\n\n\tf = [ exp(-1/10) - exp(-x1/10) + exp(-10/10) -exp(-x2/10) ]^2\n\n");
X  printf("\nInitial guess         b0 = "); pr_vector(b0);
X  printf("\nFunction value at it  f0 = %g",f2(b0));
X  printf("\nInitial steps         h0 = "); pr_vector(h0);
X  printf("\n");
X  printf("\nMinimum f value found f  = %g", hjmin(b,h0,f2));
X  printf("\n                  at  b  = "); pr_vector(b);
X  printf("\nExact min location    bm = "); pr_vector(bm);
X  printf("\nNo. of iterations     ni = %d\n",iter_count);
X
X}
X
X
X
X/*			Mile & Cuntrell function 			*/
X
Xdouble f3(v)
Xconst VECTOR v;
X{
X  register double x1 = *V_elem(v,1);
X  register double x2 = *V_elem(v,2);
X  register double x3 = *V_elem(v,3);
X  register double x4 = *V_elem(v,4);
X  iter_count++;
X  return powi( exp(x1)-x2, 4) + 100*powi(x2-x3,6) + powi(atan(x3-x4),4) +
X	 powi(x1,8);
X}
X 
Xvoid test3()
X{
X  const int n = 4;
X  VECTOR b0 = V_build(1,n,1.,2.,2.,2.,"END");	/* Initial guess	*/
X  VECTOR b  = V_new(1,n);			/* Min location found	*/
X  VECTOR bm = V_build(1,n,0.,1.,1.,1.,"END");	/* Exact min location	*/
X  VECTOR h0  = V_build(1,n,10.,10.,10.,10.,"END");	/* Initial step	*/
X
X  V_assign(b,b0);
X  iter_count = 0;
X  printf("\n\n\tMile & Cuntrell function\n");
X  printf("\n\n\tf = [ exp(x1)-x2 ]^4 +100(x2-x3)^6 + atan(x3-x4)^4 + x1^8\n\n");
X  printf("\nInitial guess         b0 = "); pr_vector(b0);
X  printf("\nFunction value at it  f0 = %g",f3(b0));
X  printf("\nInitial steps         h0 = "); pr_vector(h0);
X  printf("\n");
X  printf("\nMinimum f value found f  = %g", hjmin(b,h0,f3));
X  printf("\n                  at  b  = "); pr_vector(b);
X  printf("\nExact min location    bm = "); pr_vector(bm);
X  printf("\nNo. of iterations     ni = %d\n",iter_count);
X
X}
X
X/*			  Powell function 				*/
X
Xdouble f4(v)
Xconst VECTOR v;
X{
X  register double x1 = *V_elem(v,1);
X  register double x2 = *V_elem(v,2);
X  register double x3 = *V_elem(v,3);
X  register double x4 = *V_elem(v,4);
X  iter_count++;
X  return powi( x1+10*x2, 2) + 5*powi(x3-x4,2) + powi(x2-2*x3,4) +
X	 10*powi(x1-x4,4);
X}
X 
Xvoid test4()
X{
X  const int n = 4;
X  VECTOR b0 = V_build(1,n,3.,-1.,0.,1.,"END");	/* Initial guess	*/
X  VECTOR b  = V_new(1,n);			/* Min location found	*/
X  VECTOR bm = V_build(1,n,0.,0.,0.,0.,"END");	/* Exact min location	*/
X  VECTOR h0  = V_build(1,n,10.,10.,10.,10.,"END");	/* Initial step	*/
X
X  V_assign(b,b0);
X  iter_count = 0;
X  printf("\n\n\tPowell function\n");
X  printf("\n\n\tf = (x1+10*x2)^2 + 5(x3-x4)^2 + (x2-2x3)^4 + 10(x1-x4)^4\n\n");
X  printf("\nInitial guess         b0 = "); pr_vector(b0);
X  printf("\nFunction value at it  f0 = %g",f4(b0));
X  printf("\nInitial steps         h0 = "); pr_vector(h0);
X  printf("\n");
X  printf("\nMinimum f value found f  = %g", hjmin(b,h0,f4));
X  printf("\n                  at  b  = "); pr_vector(b);
X  printf("\nExact min location    bm = "); pr_vector(bm);
X  printf("\nNo. of iterations     ni = %d\n",iter_count);
X
X}
X
X
Xmain()
X{
X  printf("\n\n\n\tVerify HJMIN program to minimize a function value\n");
X  test1();
X  test2();
X  test3();
X  test4();
X}
________This_Is_The_END________
if test `wc -l < vhjmin.c` -ne 174; then
echo 'shar: vhjmin.c was damaged during transit (should have had 174 lines)'
fi


echo 'x - vhjmin.dat'
sed 's/^X//' << '________This_Is_The_END________' > vhjmin.dat
X
X
X
X	Verify HJMIN program to minimize a function value
X
X
X	Rosenbroke function
X
X
X	f = 100*(x2-x1^2)^2 + (1-x1)^2
X
X
XInitial guess         b0 =      -1.2          1  
XFunction value at it  f0 = 24.2
XInitial steps         h0 =        10         10  
X
XMinimum f value found f  = 6.97611e-11
X                  at  b  =         1          1  
XExact min location    bm =         1          1  
XNo. of iterations     ni = 270
X
X
X	Bocks function
X
X
X	f = [ exp(-1/10) - exp(-x1/10) + exp(-10/10) -exp(-x2/10) ]^2
X
X
XInitial guess         b0 =         0          0  
XFunction value at it  f0 = 0.528941
XInitial steps         h0 =        10         10  
X
XMinimum f value found f  = 0
X                  at  b  =        10          1  
XExact min location    bm =        10          1  
XNo. of iterations     ni = 89
X
X
X	Mile & Cuntrell function
X
X
X	f = [ exp(x1)-x2 ]^4 +100(x2-x3)^6 + atan(x3-x4)^4 + x1^8
X
X
XInitial guess         b0 =         1          2          2          2  
XFunction value at it  f0 = 1.26618
XInitial steps         h0 =        10         10         10         10  
X
XMinimum f value found f  = 4.1179e-27
X                  at  b  =  1.49e-08          1          1          1  
XExact min location    bm =         0          1          1          1  
XNo. of iterations     ni = 227
X
X
X	Powell function
X
X
X	f = (x1+10*x2)^2 + 5(x3-x4)^2 + (x2-2x3)^4 + 10(x1-x4)^4
X
X
XInitial guess         b0 =         3         -1          0          1  
XFunction value at it  f0 = 215
XInitial steps         h0 =        10         10         10         10  
X
XMinimum f value found f  = 0
X                  at  b  =         0          0          0          0  
XExact min location    bm =         0          0          0          0  
XNo. of iterations     ni = 154
________This_Is_The_END________
if test `wc -l < vhjmin.dat` -ne 68; then
echo 'shar: vhjmin.dat was damaged during transit (should have had 68 lines)'
fi


echo 'x - vhjmin.dat.vax'
sed 's/^X//' << '________This_Is_The_END________' > vhjmin.dat.vax
X
X
X
X	Verify HJMIN program to minimize a function value
X
X
X	Rosenbroke function
X
X
X	f = 100*(x2-x1^2)^2 + (1-x1)^2
X
X
XInitial guess         b0 =      -1.2          1  
XFunction value at it  f0 = 24.2
XInitial steps         h0 =        10         10  
X
XMinimum f value found f  = 2.402913e-10
X                  at  b  =         1          1  
XExact min location    bm =         1          1  
XNo. of iterations     ni = 272
X
X
X	Bocks function
X
X
X	f = [ exp(-1/10) - exp(-x1/10) + exp(-10/10) -exp(-x2/10) ]^2
X
X
XInitial guess         b0 =         0          0  
XFunction value at it  f0 = 0.528941
XInitial steps         h0 =        10         10  
X
XMinimum f value found f  = 0
X                  at  b  =        10          1  
XExact min location    bm =        10          1  
XNo. of iterations     ni = 93
X
X
X	Mile & Cuntrell function
X
X
X	f = [ exp(x1)-x2 ]^4 +100(x2-x3)^6 + atan(x3-x4)^4 + x1^8
X
X
XInitial guess         b0 =         1          2          2          2  
XFunction value at it  f0 = 1.26618
XInitial steps         h0 =        10         10         10         10  
X
XMinimum f value found f  = 7.888609e-27
X                  at  b  = 5.960e-08          1          1          1  
XExact min location    bm =         0          1          1          1  
XNo. of iterations     ni = 235
X
X
X	Powell function
X
X
X	f = (x1+10*x2)^2 + 5(x3-x4)^2 + (x2-2x3)^4 + 10(x1-x4)^4
X
X
XInitial guess         b0 =         3         -1          0          1  
XFunction value at it  f0 = 215
XInitial steps         h0 =        10         10         10         10  
X
XMinimum f value found f  = 0
X                  at  b  =         0          0          0          0  
XExact min location    bm =         0          0          0          0  
XNo. of iterations     ni = 162
________This_Is_The_END________
if test `wc -l < vhjmin.dat.vax` -ne 68; then
echo 'shar: vhjmin.dat.vax was damaged during transit (should have had 68 lines)'
fi