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diff --git a/libmoped/libs/sba-1.6/sba_lapack.c b/libmoped/libs/sba-1.6/sba_lapack.c
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+/////////////////////////////////////////////////////////////////////////////////
+////
+//// Linear algebra operations for the sba package
+//// Copyright (C) 2004-2009 Manolis Lourakis (lourakis at ics forth gr)
+//// Institute of Computer Science, Foundation for Research & Technology - Hellas
+//// Heraklion, Crete, Greece.
+////
+//// This program is free software; you can redistribute it and/or modify
+//// it under the terms of the GNU General Public License as published by
+//// the Free Software Foundation; either version 2 of the License, or
+//// (at your option) any later version.
+////
+//// This program is distributed in the hope that it will be useful,
+//// but WITHOUT ANY WARRANTY; without even the implied warranty of
+//// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+//// GNU General Public License for more details.
+////
+///////////////////////////////////////////////////////////////////////////////////
+
+/* A note on memory alignment:
+ *
+ * Several of the functions below use a piece of dynamically allocated memory
+ * to store variables of different size (i.e., ints and doubles). To avoid
+ * alignment problems, care must be taken so that elements that are larger
+ * (doubles) are stored before smaller ones (ints). This ensures proper
+ * alignment under different alignment choices made by different CPUs:
+ * For instance, a double variable is aligned on x86 to 4 bytes but
+ * aligned to 8 bytes on AMD64 despite having the same size of 8 bytes.
+ */
+
+#include <stdio.h>
+#include <stdlib.h>
+#include <string.h>
+#include <math.h>
+#include <float.h>
+
+#include "compiler.h"
+#include "sba.h"
+
+#ifdef SBA_APPEND_UNDERSCORE_SUFFIX
+#define F77_FUNC(func) func ## _
+#else
+#define F77_FUNC(func) func
+#endif /* SBA_APPEND_UNDERSCORE_SUFFIX */
+
+
+/* declarations of LAPACK routines employed */
+
+/* QR decomposition */
+extern int F77_FUNC(dgeqrf)(int *m, int *n, double *a, int *lda, double *tau, double *work, int *lwork, int *info);
+extern int F77_FUNC(dorgqr)(int *m, int *n, int *k, double *a, int *lda, double *tau, double *work, int *lwork, int *info);
+
+/* solution of triangular system */
+extern int F77_FUNC(dtrtrs)(char *uplo, char *trans, char *diag, int *n, int *nrhs, double *a, int *lda, double *b, int *ldb, int *info);
+
+/* Cholesky decomposition, linear system solution and matrix inversion */
+extern int F77_FUNC(dpotf2)(char *uplo, int *n, double *a, int *lda, int *info); /* unblocked Cholesky */
+extern int F77_FUNC(dpotrf)(char *uplo, int *n, double *a, int *lda, int *info); /* block version of dpotf2 */
+extern int F77_FUNC(dpotrs)(char *uplo, int *n, int *nrhs, double *a, int *lda, double *b, int *ldb, int *info);
+extern int F77_FUNC(dpotri)(char *uplo, int *n, double *a, int *lda, int *info);
+
+/* LU decomposition, linear system solution and matrix inversion */
+extern int F77_FUNC(dgetrf)(int *m, int *n, double *a, int *lda, int *ipiv, int *info); /* blocked LU */
+extern int F77_FUNC(dgetf2)(int *m, int *n, double *a, int *lda, int *ipiv, int *info); /* unblocked LU */
+extern int F77_FUNC(dgetrs)(char *trans, int *n, int *nrhs, double *a, int *lda, int *ipiv, double *b, int *ldb, int *info);
+extern int F77_FUNC(dgetri)(int *n, double *a, int *lda, int *ipiv, double *work, int *lwork, int *info);
+
+/* SVD */
+extern int F77_FUNC(dgesvd)(char *jobu, char *jobvt, int *m, int *n,
+ double *a, int *lda, double *s, double *u, int *ldu,
+ double *vt, int *ldvt, double *work, int *lwork,
+ int *info);
+
+/* lapack 3.0 routine, faster than dgesvd() */
+extern int F77_FUNC(dgesdd)(char *jobz, int *m, int *n, double *a, int *lda,
+ double *s, double *u, int *ldu, double *vt, int *ldvt,
+ double *work, int *lwork, int *iwork, int *info);
+
+
+/* Bunch-Kaufman factorization of a real symmetric matrix A, solution of linear systems and matrix inverse */
+extern int F77_FUNC(dsytrf)(char *uplo, int *n, double *a, int *lda, int *ipiv, double *work, int *lwork, int *info); /* blocked ver. */
+extern int F77_FUNC(dsytrs)(char *uplo, int *n, int *nrhs, double *a, int *lda, int *ipiv, double *b, int *ldb, int *info);
+extern int F77_FUNC(dsytri)(char *uplo, int *n, double *a, int *lda, int *ipiv, double *work, int *info);
+
+
+/*
+ * This function returns the solution of Ax = b
+ *
+ * The function is based on QR decomposition with explicit computation of Q:
+ * If A=Q R with Q orthogonal and R upper triangular, the linear system becomes
+ * Q R x = b or R x = Q^T b.
+ *
+ * A is mxm, b is mx1. Argument iscolmaj specifies whether A is
+ * stored in column or row major order. Note that if iscolmaj==1
+ * this function modifies A!
+ *
+ * The function returns 0 in case of error, 1 if successfull
+ *
+ * This function is often called repetitively to solve problems of identical
+ * dimensions. To avoid repetitive malloc's and free's, allocated memory is
+ * retained between calls and free'd-malloc'ed when not of the appropriate size.
+ * A call with NULL as the first argument forces this memory to be released.
+ */
+int sba_Axb_QR(double *A, double *B, double *x, int m, int iscolmaj)
+{
+static double *buf=NULL;
+static int buf_sz=0, nb=0;
+
+double *a, *r, *tau, *work;
+int a_sz, r_sz, tau_sz, tot_sz;
+register int i, j;
+int info, worksz, nrhs=1;
+register double sum;
+
+ if(A==NULL){
+ if(buf) free(buf);
+ buf=NULL;
+ buf_sz=0;
+
+ return 1;
+ }
+
+ /* calculate required memory size */
+ a_sz=(iscolmaj)? 0 : m*m;
+ r_sz=m*m; /* only the upper triangular part really needed */
+ tau_sz=m;
+ if(!nb){
+#ifndef SBA_LS_SCARCE_MEMORY
+ double tmp;
+
+ worksz=-1; // workspace query; optimal size is returned in tmp
+ F77_FUNC(dgeqrf)((int *)&m, (int *)&m, NULL, (int *)&m, NULL, (double *)&tmp, (int *)&worksz, (int *)&info);
+ nb=((int)tmp)/m; // optimal worksize is m*nb
+#else
+ nb=1; // min worksize is m
+#endif /* SBA_LS_SCARCE_MEMORY */
+ }
+ worksz=nb*m;
+ tot_sz=a_sz + r_sz + tau_sz + worksz;
+
+ if(tot_sz>buf_sz){ /* insufficient memory, allocate a "big" memory chunk at once */
+ if(buf) free(buf); /* free previously allocated memory */
+
+ buf_sz=tot_sz;
+ buf=(double *)malloc(buf_sz*sizeof(double));
+ if(!buf){
+ fprintf(stderr, "memory allocation in sba_Axb_QR() failed!\n");
+ exit(1);
+ }
+ }
+
+ if(!iscolmaj){
+ a=buf;
+ /* store A (column major!) into a */
+ for(i=0; i<m; ++i)
+ for(j=0; j<m; ++j)
+ a[i+j*m]=A[i*m+j];
+ }
+ else a=A; /* no copying required */
+
+ r=buf+a_sz;
+ tau=r+r_sz;
+ work=tau+tau_sz;
+
+ /* QR decomposition of A */
+ F77_FUNC(dgeqrf)((int *)&m, (int *)&m, a, (int *)&m, tau, work, (int *)&worksz, (int *)&info);
+ /* error treatment */
+ if(info!=0){
+ if(info<0){
+ fprintf(stderr, "LAPACK error: illegal value for argument %d of dgeqrf in sba_Axb_QR()\n", -info);
+ exit(1);
+ }
+ else{
+ fprintf(stderr, "Unknown LAPACK error %d for dgeqrf in sba_Axb_QR()\n", info);
+ return 0;
+ }
+ }
+
+ /* R is now stored in the upper triangular part of a; copy it in r so that dorgqr() below won't destroy it */
+ for(i=0; i<r_sz; ++i)
+ r[i]=a[i];
+
+ /* compute Q using the elementary reflectors computed by the above decomposition */
+ F77_FUNC(dorgqr)((int *)&m, (int *)&m, (int *)&m, a, (int *)&m, tau, work, (int *)&worksz, (int *)&info);
+ if(info!=0){
+ if(info<0){
+ fprintf(stderr, "LAPACK error: illegal value for argument %d of dorgqr in sba_Axb_QR()\n", -info);
+ exit(1);
+ }
+ else{
+ fprintf(stderr, "Unknown LAPACK error (%d) in sba_Axb_QR()\n", info);
+ return 0;
+ }
+ }
+
+ /* Q is now in a; compute Q^T b in x */
+ for(i=0; i<m; ++i){
+ for(j=0, sum=0.0; j<m; ++j)
+ sum+=a[i*m+j]*B[j];
+ x[i]=sum;
+ }
+
+ /* solve the linear system R x = Q^t b */
+ F77_FUNC(dtrtrs)("U", "N", "N", (int *)&m, (int *)&nrhs, r, (int *)&m, x, (int *)&m, &info);
+ /* error treatment */
+ if(info!=0){
+ if(info<0){
+ fprintf(stderr, "LAPACK error: illegal value for argument %d of dtrtrs in sba_Axb_QR()\n", -info);
+ exit(1);
+ }
+ else{
+ fprintf(stderr, "LAPACK error: the %d-th diagonal element of A is zero (singular matrix) in sba_Axb_QR()\n", info);
+ return 0;
+ }
+ }
+
+ return 1;
+}
+
+/*
+ * This function returns the solution of Ax = b
+ *
+ * The function is based on QR decomposition without computation of Q:
+ * If A=Q R with Q orthogonal and R upper triangular, the linear system becomes
+ * (A^T A) x = A^T b or (R^T Q^T Q R) x = A^T b or (R^T R) x = A^T b.
+ * This amounts to solving R^T y = A^T b for y and then R x = y for x
+ * Note that Q does not need to be explicitly computed
+ *
+ * A is mxm, b is mx1. Argument iscolmaj specifies whether A is
+ * stored in column or row major order. Note that if iscolmaj==1
+ * this function modifies A!
+ *
+ * The function returns 0 in case of error, 1 if successfull
+ *
+ * This function is often called repetitively to solve problems of identical
+ * dimensions. To avoid repetitive malloc's and free's, allocated memory is
+ * retained between calls and free'd-malloc'ed when not of the appropriate size.
+ * A call with NULL as the first argument forces this memory to be released.
+ */
+int sba_Axb_QRnoQ(double *A, double *B, double *x, int m, int iscolmaj)
+{
+static double *buf=NULL;
+static int buf_sz=0, nb=0;
+
+double *a, *tau, *work;
+int a_sz, tau_sz, tot_sz;
+register int i, j;
+int info, worksz, nrhs=1;
+register double sum;
+
+ if(A==NULL){
+ if(buf) free(buf);
+ buf=NULL;
+ buf_sz=0;
+
+ return 1;
+ }
+
+ /* calculate required memory size */
+ a_sz=(iscolmaj)? 0 : m*m;
+ tau_sz=m;
+ if(!nb){
+#ifndef SBA_LS_SCARCE_MEMORY
+ double tmp;
+
+ worksz=-1; // workspace query; optimal size is returned in tmp
+ F77_FUNC(dgeqrf)((int *)&m, (int *)&m, NULL, (int *)&m, NULL, (double *)&tmp, (int *)&worksz, (int *)&info);
+ nb=((int)tmp)/m; // optimal worksize is m*nb
+#else
+ nb=1; // min worksize is m
+#endif /* SBA_LS_SCARCE_MEMORY */
+ }
+ worksz=nb*m;
+ tot_sz=a_sz + tau_sz + worksz;
+
+ if(tot_sz>buf_sz){ /* insufficient memory, allocate a "big" memory chunk at once */
+ if(buf) free(buf); /* free previously allocated memory */
+
+ buf_sz=tot_sz;
+ buf=(double *)malloc(buf_sz*sizeof(double));
+ if(!buf){
+ fprintf(stderr, "memory allocation in sba_Axb_QRnoQ() failed!\n");
+ exit(1);
+ }
+ }
+
+ if(!iscolmaj){
+ a=buf;
+ /* store A (column major!) into a */
+ for(i=0; i<m; ++i)
+ for(j=0; j<m; ++j)
+ a[i+j*m]=A[i*m+j];
+ }
+ else a=A; /* no copying required */
+
+ tau=buf+a_sz;
+ work=tau+tau_sz;
+
+ /* compute A^T b in x */
+ for(i=0; i<m; ++i){
+ for(j=0, sum=0.0; j<m; ++j)
+ sum+=a[i*m+j]*B[j];
+ x[i]=sum;
+ }
+
+ /* QR decomposition of A */
+ F77_FUNC(dgeqrf)((int *)&m, (int *)&m, a, (int *)&m, tau, work, (int *)&worksz, (int *)&info);
+ /* error treatment */
+ if(info!=0){
+ if(info<0){
+ fprintf(stderr, "LAPACK error: illegal value for argument %d of dgeqrf in sba_Axb_QRnoQ()\n", -info);
+ exit(1);
+ }
+ else{
+ fprintf(stderr, "Unknown LAPACK error %d for dgeqrf in sba_Axb_QRnoQ()\n", info);
+ return 0;
+ }
+ }
+
+ /* R is stored in the upper triangular part of a */
+
+ /* solve the linear system R^T y = A^t b */
+ F77_FUNC(dtrtrs)("U", "T", "N", (int *)&m, (int *)&nrhs, a, (int *)&m, x, (int *)&m, &info);
+ /* error treatment */
+ if(info!=0){
+ if(info<0){
+ fprintf(stderr, "LAPACK error: illegal value for argument %d of dtrtrs in sba_Axb_QRnoQ()\n", -info);
+ exit(1);
+ }
+ else{
+ fprintf(stderr, "LAPACK error: the %d-th diagonal element of A is zero (singular matrix) in sba_Axb_QRnoQ()\n", info);
+ return 0;
+ }
+ }
+
+ /* solve the linear system R x = y */
+ F77_FUNC(dtrtrs)("U", "N", "N", (int *)&m, (int *)&nrhs, a, (int *)&m, x, (int *)&m, &info);
+ /* error treatment */
+ if(info!=0){
+ if(info<0){
+ fprintf(stderr, "LAPACK error: illegal value for argument %d of dtrtrs in sba_Axb_QRnoQ()\n", -info);
+ exit(1);
+ }
+ else{
+ fprintf(stderr, "LAPACK error: the %d-th diagonal element of A is zero (singular matrix) in sba_Axb_QRnoQ()\n", info);
+ return 0;
+ }
+ }
+
+ return 1;
+}
+
+/*
+ * This function returns the solution of Ax=b
+ *
+ * The function assumes that A is symmetric & positive definite and employs
+ * the Cholesky decomposition:
+ * If A=U^T U with U upper triangular, the system to be solved becomes
+ * (U^T U) x = b
+ * This amounts to solving U^T y = b for y and then U x = y for x
+ *
+ * A is mxm, b is mx1. Argument iscolmaj specifies whether A is
+ * stored in column or row major order. Note that if iscolmaj==1
+ * this function modifies A!
+ *
+ * The function returns 0 in case of error, 1 if successfull
+ *
+ * This function is often called repetitively to solve problems of identical
+ * dimensions. To avoid repetitive malloc's and free's, allocated memory is
+ * retained between calls and free'd-malloc'ed when not of the appropriate size.
+ * A call with NULL as the first argument forces this memory to be released.
+ */
+int sba_Axb_Chol(double *A, double *B, double *x, int m, int iscolmaj)
+{
+static double *buf=NULL;
+static int buf_sz=0;
+
+double *a;
+int a_sz, tot_sz;
+register int i, j;
+int info, nrhs=1;
+
+ if(A==NULL){
+ if(buf) free(buf);
+ buf=NULL;
+ buf_sz=0;
+
+ return 1;
+ }
+
+ /* calculate required memory size */
+ a_sz=(iscolmaj)? 0 : m*m;
+ tot_sz=a_sz;
+
+ if(tot_sz>buf_sz){ /* insufficient memory, allocate a "big" memory chunk at once */
+ if(buf) free(buf); /* free previously allocated memory */
+
+ buf_sz=tot_sz;
+ buf=(double *)malloc(buf_sz*sizeof(double));
+ if(!buf){
+ fprintf(stderr, "memory allocation in sba_Axb_Chol() failed!\n");
+ exit(1);
+ }
+ }
+
+ if(!iscolmaj){
+ a=buf;
+
+ /* store A into a and B into x; A is assumed to be symmetric, hence
+ * the column and row major order representations are the same
+ */
+ for(i=0; i<m; ++i){
+ a[i]=A[i];
+ x[i]=B[i];
+ }
+ for(j=m*m; i<j; ++i) // copy remaining rows; note that i is not re-initialized
+ a[i]=A[i];
+ }
+ else{ /* no copying is necessary for A */
+ a=A;
+ for(i=0; i<m; ++i)
+ x[i]=B[i];
+ }
+
+ /* Cholesky decomposition of A */
+ //F77_FUNC(dpotf2)("U", (int *)&m, a, (int *)&m, (int *)&info);
+ F77_FUNC(dpotrf)("U", (int *)&m, a, (int *)&m, (int *)&info);
+ /* error treatment */
+ if(info!=0){
+ if(info<0){
+ fprintf(stderr, "LAPACK error: illegal value for argument %d of dpotf2/dpotrf in sba_Axb_Chol()\n", -info);
+ exit(1);
+ }
+ else{
+ fprintf(stderr, "LAPACK error: the leading minor of order %d is not positive definite,\nthe factorization could not be completed for dpotf2/dpotrf in sba_Axb_Chol()\n", info);
+ return 0;
+ }
+ }
+
+ /* below are two alternative ways for solving the linear system: */
+#if 1
+ /* use the computed Cholesky in one lapack call */
+ F77_FUNC(dpotrs)("U", (int *)&m, (int *)&nrhs, a, (int *)&m, x, (int *)&m, &info);
+ if(info<0){
+ fprintf(stderr, "LAPACK error: illegal value for argument %d of dpotrs in sba_Axb_Chol()\n", -info);
+ exit(1);
+ }
+#else
+ /* solve the linear systems U^T y = b, U x = y */
+ F77_FUNC(dtrtrs)("U", "T", "N", (int *)&m, (int *)&nrhs, a, (int *)&m, x, (int *)&m, &info);
+ /* error treatment */
+ if(info!=0){
+ if(info<0){
+ fprintf(stderr, "LAPACK error: illegal value for argument %d of dtrtrs in sba_Axb_Chol()\n", -info);
+ exit(1);
+ }
+ else{
+ fprintf(stderr, "LAPACK error: the %d-th diagonal element of A is zero (singular matrix) in sba_Axb_Chol()\n", info);
+ return 0;
+ }
+ }
+
+ /* solve U x = y */
+ F77_FUNC(dtrtrs)("U", "N", "N", (int *)&m, (int *)&nrhs, a, (int *)&m, x, (int *)&m, &info);
+ /* error treatment */
+ if(info!=0){
+ if(info<0){
+ fprintf(stderr, "LAPACK error: illegal value for argument %d of dtrtrs in sba_Axb_Chol()\n", -info);
+ exit(1);
+ }
+ else{
+ fprintf(stderr, "LAPACK error: the %d-th diagonal element of A is zero (singular matrix) in sba_Axb_Chol()\n", info);
+ return 0;
+ }
+ }
+#endif /* 1 */
+
+ return 1;
+}
+
+/*
+ * This function returns the solution of Ax = b
+ *
+ * The function employs LU decomposition:
+ * If A=L U with L lower and U upper triangular, then the original system
+ * amounts to solving
+ * L y = b, U x = y
+ *
+ * A is mxm, b is mx1. Argument iscolmaj specifies whether A is
+ * stored in column or row major order. Note that if iscolmaj==1
+ * this function modifies A!
+ *
+ * The function returns 0 in case of error,
+ * 1 if successfull
+ *
+ * This function is often called repetitively to solve problems of identical
+ * dimensions. To avoid repetitive malloc's and free's, allocated memory is
+ * retained between calls and free'd-malloc'ed when not of the appropriate size.
+ * A call with NULL as the first argument forces this memory to be released.
+ */
+int sba_Axb_LU(double *A, double *B, double *x, int m, int iscolmaj)
+{
+static double *buf=NULL;
+static int buf_sz=0;
+
+int a_sz, ipiv_sz, tot_sz;
+register int i, j;
+int info, *ipiv, nrhs=1;
+double *a;
+
+ if(A==NULL){
+ if(buf) free(buf);
+ buf=NULL;
+ buf_sz=0;
+
+ return 1;
+ }
+
+ /* calculate required memory size */
+ ipiv_sz=m;
+ a_sz=(iscolmaj)? 0 : m*m;
+ tot_sz=a_sz*sizeof(double) + ipiv_sz*sizeof(int); /* should be arranged in that order for proper doubles alignment */
+
+ if(tot_sz>buf_sz){ /* insufficient memory, allocate a "big" memory chunk at once */
+ if(buf) free(buf); /* free previously allocated memory */
+
+ buf_sz=tot_sz;
+ buf=(double *)malloc(buf_sz);
+ if(!buf){
+ fprintf(stderr, "memory allocation in sba_Axb_LU() failed!\n");
+ exit(1);
+ }
+ }
+
+ if(!iscolmaj){
+ a=buf;
+ ipiv=(int *)(a+a_sz);
+
+ /* store A (column major!) into a and B into x */
+ for(i=0; i<m; ++i){
+ for(j=0; j<m; ++j)
+ a[i+j*m]=A[i*m+j];
+
+ x[i]=B[i];
+ }
+ }
+ else{ /* no copying is necessary for A */
+ a=A;
+ for(i=0; i<m; ++i)
+ x[i]=B[i];
+ ipiv=(int *)buf;
+ }
+
+ /* LU decomposition for A */
+ F77_FUNC(dgetrf)((int *)&m, (int *)&m, a, (int *)&m, ipiv, (int *)&info);
+ if(info!=0){
+ if(info<0){
+ fprintf(stderr, "argument %d of dgetrf illegal in sba_Axb_LU()\n", -info);
+ exit(1);
+ }
+ else{
+ fprintf(stderr, "singular matrix A for dgetrf in sba_Axb_LU()\n");
+ return 0;
+ }
+ }
+
+ /* solve the system with the computed LU */
+ F77_FUNC(dgetrs)("N", (int *)&m, (int *)&nrhs, a, (int *)&m, ipiv, x, (int *)&m, (int *)&info);
+ if(info!=0){
+ if(info<0){
+ fprintf(stderr, "argument %d of dgetrs illegal in sba_Axb_LU()\n", -info);
+ exit(1);
+ }
+ else{
+ fprintf(stderr, "unknown error for dgetrs in sba_Axb_LU()\n");
+ return 0;
+ }
+ }
+
+ return 1;
+}
+
+/*
+ * This function returns the solution of Ax = b
+ *
+ * The function is based on SVD decomposition:
+ * If A=U D V^T with U, V orthogonal and D diagonal, the linear system becomes
+ * (U D V^T) x = b or x=V D^{-1} U^T b
+ * Note that V D^{-1} U^T is the pseudoinverse A^+
+ *
+ * A is mxm, b is mx1. Argument iscolmaj specifies whether A is
+ * stored in column or row major order. Note that if iscolmaj==1
+ * this function modifies A!
+ *
+ * The function returns 0 in case of error, 1 if successfull
+ *
+ * This function is often called repetitively to solve problems of identical
+ * dimensions. To avoid repetitive malloc's and free's, allocated memory is
+ * retained between calls and free'd-malloc'ed when not of the appropriate size.
+ * A call with NULL as the first argument forces this memory to be released.
+ */
+int sba_Axb_SVD(double *A, double *B, double *x, int m, int iscolmaj)
+{
+static double *buf=NULL;
+static int buf_sz=0;
+static double eps=-1.0;
+
+register int i, j;
+double *a, *u, *s, *vt, *work;
+int a_sz, u_sz, s_sz, vt_sz, tot_sz;
+double thresh, one_over_denom;
+register double sum;
+int info, rank, worksz, *iwork, iworksz;
+
+ if(A==NULL){
+ if(buf) free(buf);
+ buf=NULL;
+ buf_sz=0;
+
+ return 1;
+ }
+
+ /* calculate required memory size */
+#ifndef SBA_LS_SCARCE_MEMORY
+ worksz=-1; // workspace query. Keep in mind that dgesdd requires more memory than dgesvd
+ /* note that optimal work size is returned in thresh */
+ F77_FUNC(dgesdd)("A", (int *)&m, (int *)&m, NULL, (int *)&m, NULL, NULL, (int *)&m, NULL, (int *)&m,
+ (double *)&thresh, (int *)&worksz, NULL, &info);
+ /* F77_FUNC(dgesvd)("A", "A", (int *)&m, (int *)&m, NULL, (int *)&m, NULL, NULL, (int *)&m, NULL, (int *)&m,
+ (double *)&thresh, (int *)&worksz, &info); */
+ worksz=(int)thresh;
+#else
+ worksz=m*(7*m+4); // min worksize for dgesdd
+ //worksz=5*m; // min worksize for dgesvd
+#endif /* SBA_LS_SCARCE_MEMORY */
+ iworksz=8*m;
+ a_sz=(!iscolmaj)? m*m : 0;
+ u_sz=m*m; s_sz=m; vt_sz=m*m;
+
+ tot_sz=(a_sz + u_sz + s_sz + vt_sz + worksz)*sizeof(double) + iworksz*sizeof(int); /* should be arranged in that order for proper doubles alignment */
+
+ if(tot_sz>buf_sz){ /* insufficient memory, allocate a "big" memory chunk at once */
+ if(buf) free(buf); /* free previously allocated memory */
+
+ buf_sz=tot_sz;
+ buf=(double *)malloc(buf_sz);
+ if(!buf){
+ fprintf(stderr, "memory allocation in sba_Axb_SVD() failed!\n");
+ exit(1);
+ }
+ }
+
+ if(!iscolmaj){
+ a=buf;
+ u=a+a_sz;
+ /* store A (column major!) into a */
+ for(i=0; i<m; ++i)
+ for(j=0; j<m; ++j)
+ a[i+j*m]=A[i*m+j];
+ }
+ else{
+ a=A; /* no copying required */
+ u=buf;
+ }
+
+ s=u+u_sz;
+ vt=s+s_sz;
+ work=vt+vt_sz;
+ iwork=(int *)(work+worksz);
+
+ /* SVD decomposition of A */
+ F77_FUNC(dgesdd)("A", (int *)&m, (int *)&m, a, (int *)&m, s, u, (int *)&m, vt, (int *)&m, work, (int *)&worksz, iwork, &info);
+ //F77_FUNC(dgesvd)("A", "A", (int *)&m, (int *)&m, a, (int *)&m, s, u, (int *)&m, vt, (int *)&m, work, (int *)&worksz, &info);
+
+ /* error treatment */
+ if(info!=0){
+ if(info<0){
+ fprintf(stderr, "LAPACK error: illegal value for argument %d of dgesdd/dgesvd in sba_Axb_SVD()\n", -info);
+ exit(1);
+ }
+ else{
+ fprintf(stderr, "LAPACK error: dgesdd (dbdsdc)/dgesvd (dbdsqr) failed to converge in sba_Axb_SVD() [info=%d]\n", info);
+
+ return 0;
+ }
+ }
+
+ if(eps<0.0){
+ double aux;
+
+ /* compute machine epsilon. DBL_EPSILON should do also */
+ for(eps=1.0; aux=eps+1.0, aux-1.0>0.0; eps*=0.5)
+ ;
+ eps*=2.0;
+ }
+
+ /* compute the pseudoinverse in a */
+ memset(a, 0, m*m*sizeof(double)); /* initialize to zero */
+ for(rank=0, thresh=eps*s[0]; rank<m && s[rank]>thresh; ++rank){
+ one_over_denom=1.0/s[rank];
+
+ for(j=0; j<m; ++j)
+ for(i=0; i<m; ++i)
+ a[i*m+j]+=vt[rank+i*m]*u[j+rank*m]*one_over_denom;
+ }
+
+ /* compute A^+ b in x */
+ for(i=0; i<m; ++i){
+ for(j=0, sum=0.0; j<m; ++j)
+ sum+=a[i*m+j]*B[j];
+ x[i]=sum;
+ }
+
+ return 1;
+}
+
+/*
+ * This function returns the solution of Ax = b for a real symmetric matrix A
+ *
+ * The function is based on UDUT factorization with the pivoting
+ * strategy of Bunch and Kaufman:
+ * A is factored as U*D*U^T where U is upper triangular and
+ * D symmetric and block diagonal (aka spectral decomposition,
+ * Banachiewicz factorization, modified Cholesky factorization)
+ *
+ * A is mxm, b is mx1. Argument iscolmaj specifies whether A is
+ * stored in column or row major order. Note that if iscolmaj==1
+ * this function modifies A!
+ *
+ * The function returns 0 in case of error,
+ * 1 if successfull
+ *
+ * This function is often called repetitively to solve problems of identical
+ * dimensions. To avoid repetitive malloc's and free's, allocated memory is
+ * retained between calls and free'd-malloc'ed when not of the appropriate size.
+ * A call with NULL as the first argument forces this memory to be released.
+ */
+int sba_Axb_BK(double *A, double *B, double *x, int m, int iscolmaj)
+{
+static double *buf=NULL;
+static int buf_sz=0, nb=0;
+
+int a_sz, ipiv_sz, work_sz, tot_sz;
+register int i, j;
+int info, *ipiv, nrhs=1;
+double *a, *work;
+
+ if(A==NULL){
+ if(buf) free(buf);
+ buf=NULL;
+ buf_sz=0;
+
+ return 1;
+ }
+
+ /* calculate required memory size */
+ ipiv_sz=m;
+ a_sz=(iscolmaj)? 0 : m*m;
+ if(!nb){
+#ifndef SBA_LS_SCARCE_MEMORY
+ double tmp;
+
+ work_sz=-1; // workspace query; optimal size is returned in tmp
+ F77_FUNC(dsytrf)("U", (int *)&m, NULL, (int *)&m, NULL, (double *)&tmp, (int *)&work_sz, (int *)&info);
+ nb=((int)tmp)/m; // optimal worksize is m*nb
+#else
+ nb=-1; // min worksize is 1
+#endif /* SBA_LS_SCARCE_MEMORY */
+ }
+ work_sz=(nb!=-1)? nb*m : 1;
+ tot_sz=(a_sz + work_sz)*sizeof(double) + ipiv_sz*sizeof(int); /* should be arranged in that order for proper doubles alignment */
+
+ if(tot_sz>buf_sz){ /* insufficient memory, allocate a "big" memory chunk at once */
+ if(buf) free(buf); /* free previously allocated memory */
+
+ buf_sz=tot_sz;
+ buf=(double *)malloc(buf_sz);
+ if(!buf){
+ fprintf(stderr, "memory allocation in sba_Axb_BK() failed!\n");
+ exit(1);
+ }
+ }
+
+ if(!iscolmaj){
+ a=buf;
+ work=a+a_sz;
+
+ /* store A into a and B into x; A is assumed to be symmetric, hence
+ * the column and row major order representations are the same
+ */
+ for(i=0; i<m; ++i){
+ a[i]=A[i];
+ x[i]=B[i];
+ }
+ for(j=m*m; i<j; ++i) // copy remaining rows; note that i is not re-initialized
+ a[i]=A[i];
+ }
+ else{ /* no copying is necessary for A */
+ a=A;
+ for(i=0; i<m; ++i)
+ x[i]=B[i];
+ work=buf;
+ }
+ ipiv=(int *)(work+work_sz);
+
+ /* factorization for A */
+ F77_FUNC(dsytrf)("U", (int *)&m, a, (int *)&m, ipiv, work, (int *)&work_sz, (int *)&info);
+ if(info!=0){
+ if(info<0){
+ fprintf(stderr, "argument %d of dsytrf illegal in sba_Axb_BK()\n", -info);
+ exit(1);
+ }
+ else{
+ fprintf(stderr, "singular block diagonal matrix D for dsytrf in sba_Axb_BK() [D(%d, %d) is zero]\n", info, info);
+ return 0;
+ }
+ }
+
+ /* solve the system with the computed factorization */
+ F77_FUNC(dsytrs)("U", (int *)&m, (int *)&nrhs, a, (int *)&m, ipiv, x, (int *)&m, (int *)&info);
+ if(info!=0){
+ if(info<0){
+ fprintf(stderr, "argument %d of dsytrs illegal in sba_Axb_BK()\n", -info);
+ exit(1);
+ }
+ else{
+ fprintf(stderr, "unknown error for dsytrs in sba_Axb_BK()\n");
+ return 0;
+ }
+ }
+
+ return 1;
+}
+
+/*
+ * This function computes the inverse of a square matrix whose upper triangle
+ * is stored in A into its lower triangle using LU decomposition
+ *
+ * The function returns 0 in case of error (e.g. A is singular),
+ * 1 if successfull
+ *
+ * This function is often called repetitively to solve problems of identical
+ * dimensions. To avoid repetitive malloc's and free's, allocated memory is
+ * retained between calls and free'd-malloc'ed when not of the appropriate size.
+ * A call with NULL as the first argument forces this memory to be released.
+ */
+int sba_symat_invert_LU(double *A, int m)
+{
+static double *buf=NULL;
+static int buf_sz=0, nb=0;
+
+int a_sz, ipiv_sz, work_sz, tot_sz;
+register int i, j;
+int info, *ipiv;
+double *a, *work;
+
+ if(A==NULL){
+ if(buf) free(buf);
+ buf=NULL;
+ buf_sz=0;
+
+ return 1;
+ }
+
+ /* calculate required memory size */
+ ipiv_sz=m;
+ a_sz=m*m;
+ if(!nb){
+#ifndef SBA_LS_SCARCE_MEMORY
+ double tmp;
+
+ work_sz=-1; // workspace query; optimal size is returned in tmp
+ F77_FUNC(dgetri)((int *)&m, NULL, (int *)&m, NULL, (double *)&tmp, (int *)&work_sz, (int *)&info);
+ nb=((int)tmp)/m; // optimal worksize is m*nb
+#else
+ nb=1; // min worksize is m
+#endif /* SBA_LS_SCARCE_MEMORY */
+ }
+ work_sz=nb*m;
+ tot_sz=(a_sz + work_sz)*sizeof(double) + ipiv_sz*sizeof(int); /* should be arranged in that order for proper doubles alignment */
+
+ if(tot_sz>buf_sz){ /* insufficient memory, allocate a "big" memory chunk at once */
+ if(buf) free(buf); /* free previously allocated memory */
+
+ buf_sz=tot_sz;
+ buf=(double *)malloc(buf_sz);
+ if(!buf){
+ fprintf(stderr, "memory allocation in sba_symat_invert_LU() failed!\n");
+ exit(1);
+ }
+ }
+
+ a=buf;
+ work=a+a_sz;
+ ipiv=(int *)(work+work_sz);
+
+ /* store A (column major!) into a */
+ for(i=0; i<m; ++i)
+ for(j=i; j<m; ++j)
+ a[i+j*m]=a[j+i*m]=A[i*m+j]; // copy A's upper part to a's upper & lower
+
+ /* LU decomposition for A */
+ F77_FUNC(dgetrf)((int *)&m, (int *)&m, a, (int *)&m, ipiv, (int *)&info);
+ if(info!=0){
+ if(info<0){
+ fprintf(stderr, "argument %d of dgetrf illegal in sba_symat_invert_LU()\n", -info);
+ exit(1);
+ }
+ else{
+ fprintf(stderr, "singular matrix A for dgetrf in sba_symat_invert_LU()\n");
+ return 0;
+ }
+ }
+
+ /* (A)^{-1} from LU */
+ F77_FUNC(dgetri)((int *)&m, a, (int *)&m, ipiv, work, (int *)&work_sz, (int *)&info);
+ if(info!=0){
+ if(info<0){
+ fprintf(stderr, "argument %d of dgetri illegal in sba_symat_invert_LU()\n", -info);
+ exit(1);
+ }
+ else{
+ fprintf(stderr, "singular matrix A for dgetri in sba_symat_invert_LU()\n");
+ return 0;
+ }
+ }
+
+ /* store (A)^{-1} in A's lower triangle */
+ for(i=0; i<m; ++i)
+ for(j=0; j<=i; ++j)
+ A[i*m+j]=a[i+j*m];
+
+ return 1;
+}
+
+/*
+ * This function computes the inverse of a square symmetric positive definite
+ * matrix whose upper triangle is stored in A into its lower triangle using
+ * Cholesky factorization
+ *
+ * The function returns 0 in case of error (e.g. A is not positive definite or singular),
+ * 1 if successfull
+ *
+ * This function is often called repetitively to solve problems of identical
+ * dimensions. To avoid repetitive malloc's and free's, allocated memory is
+ * retained between calls and free'd-malloc'ed when not of the appropriate size.
+ * A call with NULL as the first argument forces this memory to be released.
+ */
+int sba_symat_invert_Chol(double *A, int m)
+{
+static double *buf=NULL;
+static int buf_sz=0;
+
+int a_sz, tot_sz;
+register int i, j;
+int info;
+double *a;
+
+ if(A==NULL){
+ if(buf) free(buf);
+ buf=NULL;
+ buf_sz=0;
+
+ return 1;
+ }
+
+ /* calculate required memory size */
+ a_sz=m*m;
+ tot_sz=a_sz;
+
+ if(tot_sz>buf_sz){ /* insufficient memory, allocate a "big" memory chunk at once */
+ if(buf) free(buf); /* free previously allocated memory */
+
+ buf_sz=tot_sz;
+ buf=(double *)malloc(buf_sz*sizeof(double));
+ if(!buf){
+ fprintf(stderr, "memory allocation in sba_symat_invert_Chol() failed!\n");
+ exit(1);
+ }
+ }
+
+ a=(double *)buf;
+
+ /* store A into a; A is assumed symmetric, hence no transposition is needed */
+ for(i=0, j=a_sz; i<j; ++i)
+ a[i]=A[i];
+
+ /* Cholesky factorization for A; a's lower part corresponds to A's upper */
+ //F77_FUNC(dpotrf)("L", (int *)&m, a, (int *)&m, (int *)&info);
+ F77_FUNC(dpotf2)("L", (int *)&m, a, (int *)&m, (int *)&info);
+ /* error treatment */
+ if(info!=0){
+ if(info<0){
+ fprintf(stderr, "LAPACK error: illegal value for argument %d of dpotrf in sba_symat_invert_Chol()\n", -info);
+ exit(1);
+ }
+ else{
+ fprintf(stderr, "LAPACK error: the leading minor of order %d is not positive definite,\nthe factorization could not be completed for dpotrf in sba_symat_invert_Chol()\n", info);
+ return 0;
+ }
+ }
+
+ /* (A)^{-1} from Cholesky */
+ F77_FUNC(dpotri)("L", (int *)&m, a, (int *)&m, (int *)&info);
+ if(info!=0){
+ if(info<0){
+ fprintf(stderr, "argument %d of dpotri illegal in sba_symat_invert_Chol()\n", -info);
+ exit(1);
+ }
+ else{
+ fprintf(stderr, "the (%d, %d) element of the factor U or L is zero, singular matrix A for dpotri in sba_symat_invert_Chol()\n", info, info);
+ return 0;
+ }
+ }
+
+ /* store (A)^{-1} in A's lower triangle. The lower triangle of the symmetric A^{-1} is in the lower triangle of a */
+ for(i=0; i<m; ++i)
+ for(j=0; j<=i; ++j)
+ A[i*m+j]=a[i+j*m];
+
+ return 1;
+}
+
+/*
+ * This function computes the inverse of a symmetric indefinite
+ * matrix whose upper triangle is stored in A into its lower triangle
+ * using LDLT factorization with the pivoting strategy of Bunch and Kaufman
+ *
+ * The function returns 0 in case of error (e.g. A is singular),
+ * 1 if successfull
+ *
+ * This function is often called repetitively to solve problems of identical
+ * dimensions. To avoid repetitive malloc's and free's, allocated memory is
+ * retained between calls and free'd-malloc'ed when not of the appropriate size.
+ * A call with NULL as the first argument forces this memory to be released.
+ */
+int sba_symat_invert_BK(double *A, int m)
+{
+static double *buf=NULL;
+static int buf_sz=0, nb=0;
+
+int a_sz, ipiv_sz, work_sz, tot_sz;
+register int i, j;
+int info, *ipiv;
+double *a, *work;
+
+ if(A==NULL){
+ if(buf) free(buf);
+ buf=NULL;
+ buf_sz=0;
+
+ return 1;
+ }
+
+ /* calculate required memory size */
+ ipiv_sz=m;
+ a_sz=m*m;
+ if(!nb){
+#ifndef SBA_LS_SCARCE_MEMORY
+ double tmp;
+
+ work_sz=-1; // workspace query; optimal size is returned in tmp
+ F77_FUNC(dsytrf)("L", (int *)&m, NULL, (int *)&m, NULL, (double *)&tmp, (int *)&work_sz, (int *)&info);
+ nb=((int)tmp)/m; // optimal worksize is m*nb
+#else
+ nb=-1; // min worksize is 1
+#endif /* SBA_LS_SCARCE_MEMORY */
+ }
+ work_sz=(nb!=-1)? nb*m : 1;
+ work_sz=(work_sz>=m)? work_sz : m; /* ensure that work is at least m elements long, as required by dsytri */
+ tot_sz=(a_sz + work_sz)*sizeof(double) + ipiv_sz*sizeof(int); /* should be arranged in that order for proper doubles alignment */
+
+ if(tot_sz>buf_sz){ /* insufficient memory, allocate a "big" memory chunk at once */
+ if(buf) free(buf); /* free previously allocated memory */
+
+ buf_sz=tot_sz;
+ buf=(double *)malloc(buf_sz);
+ if(!buf){
+ fprintf(stderr, "memory allocation in sba_symat_invert_BK() failed!\n");
+ exit(1);
+ }
+ }
+
+ a=buf;
+ work=a+a_sz;
+ ipiv=(int *)(work+work_sz);
+
+ /* store A into a; A is assumed symmetric, hence no transposition is needed */
+ for(i=0, j=a_sz; i<j; ++i)
+ a[i]=A[i];
+
+ /* LDLT factorization for A; a's lower part corresponds to A's upper */
+ F77_FUNC(dsytrf)("L", (int *)&m, a, (int *)&m, ipiv, work, (int *)&work_sz, (int *)&info);
+ if(info!=0){
+ if(info<0){
+ fprintf(stderr, "argument %d of dsytrf illegal in sba_symat_invert_BK()\n", -info);
+ exit(1);
+ }
+ else{
+ fprintf(stderr, "singular block diagonal matrix D for dsytrf in sba_symat_invert_BK() [D(%d, %d) is zero]\n", info, info);
+ return 0;
+ }
+ }
+
+ /* (A)^{-1} from LDLT */
+ F77_FUNC(dsytri)("L", (int *)&m, a, (int *)&m, ipiv, work, (int *)&info);
+ if(info!=0){
+ if(info<0){
+ fprintf(stderr, "argument %d of dsytri illegal in sba_symat_invert_BK()\n", -info);
+ exit(1);
+ }
+ else{
+ fprintf(stderr, "D(%d, %d)=0, matrix is singular and its inverse could not be computed in sba_symat_invert_BK()\n", info, info);
+ return 0;
+ }
+ }
+
+ /* store (A)^{-1} in A's lower triangle. The lower triangle of the symmetric A^{-1} is in the lower triangle of a */
+ for(i=0; i<m; ++i)
+ for(j=0; j<=i; ++j)
+ A[i*m+j]=a[i+j*m];
+
+ return 1;
+}
+
+
+#define __CG_LINALG_BLOCKSIZE 8
+
+/* Dot product of two vectors x and y using loop unrolling and blocking.
+ * see http://www.abarnett.demon.co.uk/tutorial.html
+ */
+
+inline static double dprod(const int n, const double *const x, const double *const y)
+{
+register int i, j1, j2, j3, j4, j5, j6, j7;
+int blockn;
+register double sum0=0.0, sum1=0.0, sum2=0.0, sum3=0.0,
+ sum4=0.0, sum5=0.0, sum6=0.0, sum7=0.0;
+
+ /* n may not be divisible by __CG_LINALG_BLOCKSIZE,
+ * go as near as we can first, then tidy up.
+ */
+ blockn = (n / __CG_LINALG_BLOCKSIZE) * __CG_LINALG_BLOCKSIZE;
+
+ /* unroll the loop in blocks of `__CG_LINALG_BLOCKSIZE' */
+ for(i=0; i<blockn; i+=__CG_LINALG_BLOCKSIZE){
+ sum0+=x[i]*y[i];
+ j1=i+1; sum1+=x[j1]*y[j1];
+ j2=i+2; sum2+=x[j2]*y[j2];
+ j3=i+3; sum3+=x[j3]*y[j3];
+ j4=i+4; sum4+=x[j4]*y[j4];
+ j5=i+5; sum5+=x[j5]*y[j5];
+ j6=i+6; sum6+=x[j6]*y[j6];
+ j7=i+7; sum7+=x[j7]*y[j7];
+ }
+
+ /*
+ * There may be some left to do.
+ * This could be done as a simple for() loop,
+ * but a switch is faster (and more interesting)
+ */
+
+ if(i<n){
+ /* Jump into the case at the place that will allow
+ * us to finish off the appropriate number of items.
+ */
+
+ switch(n - i){
+ case 7 : sum0+=x[i]*y[i]; ++i;
+ case 6 : sum1+=x[i]*y[i]; ++i;
+ case 5 : sum2+=x[i]*y[i]; ++i;
+ case 4 : sum3+=x[i]*y[i]; ++i;
+ case 3 : sum4+=x[i]*y[i]; ++i;
+ case 2 : sum5+=x[i]*y[i]; ++i;
+ case 1 : sum6+=x[i]*y[i]; ++i;
+ }
+ }
+
+ return sum0+sum1+sum2+sum3+sum4+sum5+sum6+sum7;
+}
+
+
+/* Compute z=x+a*y for two vectors x and y and a scalar a; z can be one of x, y.
+ * Similarly to the dot product routine, this one uses loop unrolling and blocking
+ */
+
+inline static void daxpy(const int n, double *const z, const double *const x, const double a, const double *const y)
+{
+register int i, j1, j2, j3, j4, j5, j6, j7;
+int blockn;
+
+ /* n may not be divisible by __CG_LINALG_BLOCKSIZE,
+ * go as near as we can first, then tidy up.
+ */
+ blockn = (n / __CG_LINALG_BLOCKSIZE) * __CG_LINALG_BLOCKSIZE;
+
+ /* unroll the loop in blocks of `__CG_LINALG_BLOCKSIZE' */
+ for(i=0; i<blockn; i+=__CG_LINALG_BLOCKSIZE){
+ z[i]=x[i]+a*y[i];
+ j1=i+1; z[j1]=x[j1]+a*y[j1];
+ j2=i+2; z[j2]=x[j2]+a*y[j2];
+ j3=i+3; z[j3]=x[j3]+a*y[j3];
+ j4=i+4; z[j4]=x[j4]+a*y[j4];
+ j5=i+5; z[j5]=x[j5]+a*y[j5];
+ j6=i+6; z[j6]=x[j6]+a*y[j6];
+ j7=i+7; z[j7]=x[j7]+a*y[j7];
+ }
+
+ /*
+ * There may be some left to do.
+ * This could be done as a simple for() loop,
+ * but a switch is faster (and more interesting)
+ */
+
+ if(i<n){
+ /* Jump into the case at the place that will allow
+ * us to finish off the appropriate number of items.
+ */
+
+ switch(n - i){
+ case 7 : z[i]=x[i]+a*y[i]; ++i;
+ case 6 : z[i]=x[i]+a*y[i]; ++i;
+ case 5 : z[i]=x[i]+a*y[i]; ++i;
+ case 4 : z[i]=x[i]+a*y[i]; ++i;
+ case 3 : z[i]=x[i]+a*y[i]; ++i;
+ case 2 : z[i]=x[i]+a*y[i]; ++i;
+ case 1 : z[i]=x[i]+a*y[i]; ++i;
+ }
+ }
+}
+
+/*
+ * This function returns the solution of Ax = b where A is posititive definite,
+ * based on the conjugate gradients method; see "An intro to the CG method" by J.R. Shewchuk, p. 50-51
+ *
+ * A is mxm, b, x are is mx1. Argument niter specifies the maximum number of
+ * iterations and eps is the desired solution accuracy. niter<0 signals that
+ * x contains a valid initial approximation to the solution; if niter>0 then
+ * the starting point is taken to be zero. Argument prec selects the desired
+ * preconditioning method as follows:
+ * 0: no preconditioning
+ * 1: jacobi (diagonal) preconditioning
+ * 2: SSOR preconditioning
+ * Argument iscolmaj specifies whether A is stored in column or row major order.
+ *
+ * The function returns 0 in case of error,
+ * the number of iterations performed if successfull
+ *
+ * This function is often called repetitively to solve problems of identical
+ * dimensions. To avoid repetitive malloc's and free's, allocated memory is
+ * retained between calls and free'd-malloc'ed when not of the appropriate size.
+ * A call with NULL as the first argument forces this memory to be released.
+ */
+int sba_Axb_CG(double *A, double *B, double *x, int m, int niter, double eps, int prec, int iscolmaj)
+{
+static double *buf=NULL;
+static int buf_sz=0;
+
+register int i, j;
+register double *aim;
+int iter, a_sz, res_sz, d_sz, q_sz, s_sz, wk_sz, z_sz, tot_sz;
+double *a, *res, *d, *q, *s, *wk, *z;
+double delta0, deltaold, deltanew, alpha, beta, eps_sq=eps*eps;
+register double sum;
+int rec_res;
+
+ if(A==NULL){
+ if(buf) free(buf);
+ buf=NULL;
+ buf_sz=0;
+
+ return 1;
+ }
+
+ /* calculate required memory size */
+ a_sz=(iscolmaj)? m*m : 0;
+ res_sz=m; d_sz=m; q_sz=m;
+ if(prec!=SBA_CG_NOPREC){
+ s_sz=m; wk_sz=m;
+ z_sz=(prec==SBA_CG_SSOR)? m : 0;
+ }
+ else
+ s_sz=wk_sz=z_sz=0;
+
+ tot_sz=a_sz+res_sz+d_sz+q_sz+s_sz+wk_sz+z_sz;
+
+ if(tot_sz>buf_sz){ /* insufficient memory, allocate a "big" memory chunk at once */
+ if(buf) free(buf); /* free previously allocated memory */
+
+ buf_sz=tot_sz;
+ buf=(double *)malloc(buf_sz*sizeof(double));
+ if(!buf){
+ fprintf(stderr, "memory allocation request failed in sba_Axb_CG()\n");
+ exit(1);
+ }
+ }
+
+ if(iscolmaj){
+ a=buf;
+ /* store A (row major!) into a */
+ for(i=0; i<m; ++i)
+ for(j=0, aim=a+i*m; j<m; ++j)
+ aim[j]=A[i+j*m];
+ }
+ else a=A; /* no copying required */
+
+ res=buf+a_sz;
+ d=res+res_sz;
+ q=d+d_sz;
+ if(prec!=SBA_CG_NOPREC){
+ s=q+q_sz;
+ wk=s+s_sz;
+ z=(prec==SBA_CG_SSOR)? wk+wk_sz : NULL;
+
+ for(i=0; i<m; ++i){ // compute jacobi (i.e. diagonal) preconditioners and save them in wk
+ sum=a[i*m+i];
+ if(sum>DBL_EPSILON || -sum<-DBL_EPSILON) // != 0.0
+ wk[i]=1.0/sum;
+ else
+ wk[i]=1.0/DBL_EPSILON;
+ }
+ }
+ else{
+ s=res;
+ wk=z=NULL;
+ }
+
+ if(niter>0){
+ for(i=0; i<m; ++i){ // clear solution and initialize residual vector: res <-- B
+ x[i]=0.0;
+ res[i]=B[i];
+ }
+ }
+ else{
+ niter=-niter;
+
+ for(i=0; i<m; ++i){ // initialize residual vector: res <-- B - A*x
+ for(j=0, aim=a+i*m, sum=0.0; j<m; ++j)
+ sum+=aim[j]*x[j];
+ res[i]=B[i]-sum;
+ }
+ }
+
+ switch(prec){
+ case SBA_CG_NOPREC:
+ for(i=0, deltanew=0.0; i<m; ++i){
+ d[i]=res[i];
+ deltanew+=res[i]*res[i];
+ }
+ break;
+ case SBA_CG_JACOBI: // jacobi preconditioning
+ for(i=0, deltanew=0.0; i<m; ++i){
+ d[i]=res[i]*wk[i];
+ deltanew+=res[i]*d[i];
+ }
+ break;
+ case SBA_CG_SSOR: // SSOR preconditioning; see the "templates" book, fig. 3.2, p. 44
+ for(i=0; i<m; ++i){
+ for(j=0, sum=0.0, aim=a+i*m; j<i; ++j)
+ sum+=aim[j]*z[j];
+ z[i]=wk[i]*(res[i]-sum);
+ }
+
+ for(i=m-1; i>=0; --i){
+ for(j=i+1, sum=0.0, aim=a+i*m; j<m; ++j)
+ sum+=aim[j]*d[j];
+ d[i]=z[i]-wk[i]*sum;
+ }
+ deltanew=dprod(m, res, d);
+ break;
+ default:
+ fprintf(stderr, "unknown preconditioning option %d in sba_Axb_CG\n", prec);
+ exit(1);
+ }
+
+ delta0=deltanew;
+
+ for(iter=1; deltanew>eps_sq*delta0 && iter<=niter; ++iter){
+ for(i=0; i<m; ++i){ // q <-- A d
+ aim=a+i*m;
+/***
+ for(j=0, sum=0.0; j<m; ++j)
+ sum+=aim[j]*d[j];
+***/
+ q[i]=dprod(m, aim, d); //sum;
+ }
+
+/***
+ for(i=0, sum=0.0; i<m; ++i)
+ sum+=d[i]*q[i];
+***/
+ alpha=deltanew/dprod(m, d, q); // deltanew/sum;
+
+/***
+ for(i=0; i<m; ++i)
+ x[i]+=alpha*d[i];
+***/
+ daxpy(m, x, x, alpha, d);
+
+ if(!(iter%50)){
+ for(i=0; i<m; ++i){ // accurate computation of the residual vector
+ aim=a+i*m;
+/***
+ for(j=0, sum=0.0; j<m; ++j)
+ sum+=aim[j]*x[j];
+***/
+ res[i]=B[i]-dprod(m, aim, x); //B[i]-sum;
+ }
+ rec_res=0;
+ }
+ else{
+/***
+ for(i=0; i<m; ++i) // approximate computation of the residual vector
+ res[i]-=alpha*q[i];
+***/
+ daxpy(m, res, res, -alpha, q);
+ rec_res=1;
+ }
+
+ if(prec){
+ switch(prec){
+ case SBA_CG_JACOBI: // jacobi
+ for(i=0; i<m; ++i)
+ s[i]=res[i]*wk[i];
+ break;
+ case SBA_CG_SSOR: // SSOR
+ for(i=0; i<m; ++i){
+ for(j=0, sum=0.0, aim=a+i*m; j<i; ++j)
+ sum+=aim[j]*z[j];
+ z[i]=wk[i]*(res[i]-sum);
+ }
+
+ for(i=m-1; i>=0; --i){
+ for(j=i+1, sum=0.0, aim=a+i*m; j<m; ++j)
+ sum+=aim[j]*s[j];
+ s[i]=z[i]-wk[i]*sum;
+ }
+ break;
+ }
+ }
+
+ deltaold=deltanew;
+/***
+ for(i=0, sum=0.0; i<m; ++i)
+ sum+=res[i]*s[i];
+***/
+ deltanew=dprod(m, res, s); //sum;
+
+ /* make sure that we get around small delta that are due to
+ * accumulated floating point roundoff errors
+ */
+ if(rec_res && deltanew<=eps_sq*delta0){
+ /* analytically recompute delta */
+ for(i=0; i<m; ++i){
+ for(j=0, aim=a+i*m, sum=0.0; j<m; ++j)
+ sum+=aim[j]*x[j];
+ res[i]=B[i]-sum;
+ }
+ deltanew=dprod(m, res, s);
+ }
+
+ beta=deltanew/deltaold;
+
+/***
+ for(i=0; i<m; ++i)
+ d[i]=s[i]+beta*d[i];
+***/
+ daxpy(m, d, s, beta, d);
+ }
+
+ return iter;
+}
+
+/*
+ * This function computes the Cholesky decomposition of the inverse of a symmetric
+ * (covariance) matrix A into B, i.e. B is s.t. A^-1=B^t*B and B upper triangular.
+ * A and B can coincide
+ *
+ * The function returns 0 in case of error (e.g. A is singular),
+ * 1 if successfull
+ *
+ * This function is often called repetitively to operate on matrices of identical
+ * dimensions. To avoid repetitive malloc's and free's, allocated memory is
+ * retained between calls and free'd-malloc'ed when not of the appropriate size.
+ * A call with NULL as the first argument forces this memory to be released.
+ *
+ */
+#if 0
+int sba_mat_cholinv(double *A, double *B, int m)
+{
+static double *buf=NULL;
+static int buf_sz=0, nb=0;
+
+int a_sz, ipiv_sz, work_sz, tot_sz;
+register int i, j;
+int info, *ipiv;
+double *a, *work;
+
+ if(A==NULL){
+ if(buf) free(buf);
+ buf=NULL;
+ buf_sz=0;
+
+ return 1;
+ }
+
+ /* calculate the required memory size */
+ ipiv_sz=m;
+ a_sz=m*m;
+ if(!nb){
+#ifndef SBA_LS_SCARCE_MEMORY
+ double tmp;
+
+ work_sz=-1; // workspace query; optimal size is returned in tmp
+ F77_FUNC(dgetri)((int *)&m, NULL, (int *)&m, NULL, (double *)&tmp, (int *)&work_sz, (int *)&info);
+ nb=((int)tmp)/m; // optimal worksize is m*nb
+#else
+ nb=1; // min worksize is m
+#endif /* SBA_LS_SCARCE_MEMORY */
+ }
+ work_sz=nb*m;
+ tot_sz=(a_sz + work_sz)*sizeof(double) + ipiv_sz*sizeof(int); /* should be arranged in that order for proper doubles alignment */
+
+ if(tot_sz>buf_sz){ /* insufficient memory, allocate a "big" memory chunk at once */
+ if(buf) free(buf); /* free previously allocated memory */
+
+ buf_sz=tot_sz;
+ buf=(double *)malloc(buf_sz);
+ if(!buf){
+ fprintf(stderr, "memory allocation in sba_mat_cholinv() failed!\n");
+ exit(1);
+ }
+ }
+
+ a=buf;
+ work=a+a_sz;
+ ipiv=(int *)(work+work_sz);
+
+ /* step 1: invert A */
+ /* store A into a; A is assumed symmetric, hence no transposition is needed */
+ for(i=0; i<m*m; ++i)
+ a[i]=A[i];
+
+ /* LU decomposition for A (Cholesky should also do) */
+ F77_FUNC(dgetf2)((int *)&m, (int *)&m, a, (int *)&m, ipiv, (int *)&info);
+ //F77_FUNC(dgetrf)((int *)&m, (int *)&m, a, (int *)&m, ipiv, (int *)&info);
+ if(info!=0){
+ if(info<0){
+ fprintf(stderr, "argument %d of dgetf2/dgetrf illegal in sba_mat_cholinv()\n", -info);
+ exit(1);
+ }
+ else{
+ fprintf(stderr, "singular matrix A for dgetf2/dgetrf in sba_mat_cholinv()\n");
+ return 0;
+ }
+ }
+
+ /* (A)^{-1} from LU */
+ F77_FUNC(dgetri)((int *)&m, a, (int *)&m, ipiv, work, (int *)&work_sz, (int *)&info);
+ if(info!=0){
+ if(info<0){
+ fprintf(stderr, "argument %d of dgetri illegal in sba_mat_cholinv()\n", -info);
+ exit(1);
+ }
+ else{
+ fprintf(stderr, "singular matrix A for dgetri in sba_mat_cholinv()\n");
+ return 0;
+ }
+ }
+
+ /* (A)^{-1} is now in a (in column major!) */
+
+ /* step 2: Cholesky decomposition of a: A^-1=B^t B, B upper triangular */
+ F77_FUNC(dpotf2)("U", (int *)&m, a, (int *)&m, (int *)&info);
+ /* error treatment */
+ if(info!=0){
+ if(info<0){
+ fprintf(stderr, "LAPACK error: illegal value for argument %d of dpotf2 in sba_mat_cholinv()\n", -info);
+ exit(1);
+ }
+ else{
+ fprintf(stderr, "LAPACK error: the leading minor of order %d is not positive definite,\n%s\n", info,
+ "and the Cholesky factorization could not be completed in sba_mat_cholinv()");
+ return 0;
+ }
+ }
+
+ /* the decomposition is in the upper part of a (in column-major order!).
+ * copying it to the lower part and zeroing the upper transposes
+ * a in row-major order
+ */
+ for(i=0; i<m; ++i)
+ for(j=0; j<i; ++j){
+ a[i+j*m]=a[j+i*m];
+ a[j+i*m]=0.0;
+ }
+ for(i=0; i<m*m; ++i)
+ B[i]=a[i];
+
+ return 1;
+}
+#endif
+
+int sba_mat_cholinv(double *A, double *B, int m)
+{
+int a_sz;
+register int i, j;
+int info;
+double *a;
+
+ if(A==NULL){
+ return 1;
+ }
+
+ a_sz=m*m;
+ a=B; /* use B as working memory, result is produced in it */
+
+ /* step 1: invert A */
+ /* store A into a; A is assumed symmetric, hence no transposition is needed */
+ for(i=0; i<a_sz; ++i)
+ a[i]=A[i];
+
+ /* Cholesky decomposition for A */
+ F77_FUNC(dpotf2)("U", (int *)&m, a, (int *)&m, (int *)&info);
+ if(info!=0){
+ if(info<0){
+ fprintf(stderr, "argument %d of dpotf2 illegal in sba_mat_cholinv()\n", -info);
+ exit(1);
+ }
+ else{
+ fprintf(stderr, "LAPACK error: the leading minor of order %d is not positive definite,\n%s\n", info,
+ "and the Cholesky factorization could not be completed in sba_mat_cholinv()");
+ return 0;
+ }
+ }
+
+ /* (A)^{-1} from Cholesky */
+ F77_FUNC(dpotri)("U", (int *)&m, a, (int *)&m, (int *)&info);
+ if(info!=0){
+ if(info<0){
+ fprintf(stderr, "argument %d of dpotri illegal in sba_mat_cholinv()\n", -info);
+ exit(1);
+ }
+ else{
+ fprintf(stderr, "the (%d, %d) element of the factor U or L is zero, singular matrix A for dpotri in sba_mat_cholinv()\n", info, info);
+ return 0;
+ }
+ }
+
+ /* (A)^{-1} is now in a (in column major!) */
+
+ /* step 2: Cholesky decomposition of a: A^-1=B^t B, B upper triangular */
+ F77_FUNC(dpotf2)("U", (int *)&m, a, (int *)&m, (int *)&info);
+ /* error treatment */
+ if(info!=0){
+ if(info<0){
+ fprintf(stderr, "LAPACK error: illegal value for argument %d of dpotf2 in sba_mat_cholinv()\n", -info);
+ exit(1);
+ }
+ else{
+ fprintf(stderr, "LAPACK error: the leading minor of order %d is not positive definite,\n%s\n", info,
+ "and the Cholesky factorization could not be completed in sba_mat_cholinv()");
+ return 0;
+ }
+ }
+
+ /* the decomposition is in the upper part of a (in column-major order!).
+ * copying it to the lower part and zeroing the upper transposes
+ * a in row-major order
+ */
+ for(i=0; i<m; ++i)
+ for(j=0; j<i; ++j){
+ a[i+j*m]=a[j+i*m];
+ a[j+i*m]=0.0;
+ }
+
+ return 1;
+}