path: root/redist/sba/sba_lapack.c

/////////////////////////////////////////////////////////////////////////////////
////
////  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 <float.h>
#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.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;
}