axb_core.c

A sparse variant of the Levenberg-Marquardt algorithm implemented by levmar has been applied to bund

/////////////////////////////////////////////////////////////////////////////////
// 
//  Solution of linear systems involved in the Levenberg - Marquardt
//  minimization algorithm
//  Copyright (C) 2004  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.
//
/////////////////////////////////////////////////////////////////////////////////


/* Solvers for the linear systems Ax=b. Solvers should NOT modify their A & B arguments! */


#ifndef LM_REAL // not included by Axb.c
#error This file should not be compiled directly!
#endif


#ifdef LINSOLVERS_RETAIN_MEMORY
#define __STATIC__ static
#else
#define __STATIC__ // empty
#endif /* LINSOLVERS_RETAIN_MEMORY */

#ifdef HAVE_LAPACK

/* prototypes of LAPACK routines */

#define GEQRF LM_MK_LAPACK_NAME(geqrf)
#define ORGQR LM_MK_LAPACK_NAME(orgqr)
#define TRTRS LM_MK_LAPACK_NAME(trtrs)
#define POTF2 LM_MK_LAPACK_NAME(potf2)
#define POTRF LM_MK_LAPACK_NAME(potrf)
#define POTRS LM_MK_LAPACK_NAME(potrs)
#define GETRF LM_MK_LAPACK_NAME(getrf)
#define GETRS LM_MK_LAPACK_NAME(getrs)
#define GESVD LM_MK_LAPACK_NAME(gesvd)
#define GESDD LM_MK_LAPACK_NAME(gesdd)

/* QR decomposition */
extern int GEQRF(int *m, int *n, LM_REAL *a, int *lda, LM_REAL *tau, LM_REAL *work, int *lwork, int *info);
extern int ORGQR(int *m, int *n, int *k, LM_REAL *a, int *lda, LM_REAL *tau, LM_REAL *work, int *lwork, int *info);

/* solution of triangular systems */
extern int TRTRS(char *uplo, char *trans, char *diag, int *n, int *nrhs, LM_REAL *a, int *lda, LM_REAL *b, int *ldb, int *info);

/* Cholesky decomposition and systems solution */
extern int POTF2(char *uplo, int *n, LM_REAL *a, int *lda, int *info);
extern int POTRF(char *uplo, int *n, LM_REAL *a, int *lda, int *info); /* block version of dpotf2 */
extern int POTRS(char *uplo, int *n, int *nrhs, LM_REAL *a, int *lda, LM_REAL *b, int *ldb, int *info);

/* LU decomposition and systems solution */
extern int GETRF(int *m, int *n, LM_REAL *a, int *lda, int *ipiv, int *info);
extern int GETRS(char *trans, int *n, int *nrhs, LM_REAL *a, int *lda, int *ipiv, LM_REAL *b, int *ldb, int *info);

/* Singular Value Decomposition (SVD) */
extern int GESVD(char *jobu, char *jobvt, int *m, int *n, LM_REAL *a, int *lda, LM_REAL *s, LM_REAL *u, int *ldu,
                   LM_REAL *vt, int *ldvt, LM_REAL *work, int *lwork, int *info);

/* lapack 3.0 new SVD routine, faster than xgesvd().
 * In case that your version of LAPACK does not include them, use the above two older routines
 */
extern int GESDD(char *jobz, int *m, int *n, LM_REAL *a, int *lda, LM_REAL *s, LM_REAL *u, int *ldu, LM_REAL *vt, int *ldvt,
                   LM_REAL *work, int *lwork, int *iwork, int *info);

/* precision-specific definitions */
#define AX_EQ_B_QR LM_ADD_PREFIX(Ax_eq_b_QR)
#define AX_EQ_B_QRLS LM_ADD_PREFIX(Ax_eq_b_QRLS)
#define AX_EQ_B_CHOL LM_ADD_PREFIX(Ax_eq_b_Chol)
#define AX_EQ_B_LU LM_ADD_PREFIX(Ax_eq_b_LU)
#define AX_EQ_B_SVD LM_ADD_PREFIX(Ax_eq_b_SVD)

/*
 * 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.
 * The last equation can be solved directly.
 *
 * A is mxm, b is mx1
 *
 * The function returns 0 in case of error, 1 if successful
 *
 * 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 AX_EQ_B_QR(LM_REAL *A, LM_REAL *B, LM_REAL *x, int m)
{
__STATIC__ LM_REAL *buf=NULL;
__STATIC__ int buf_sz=0;

static int nb=0; /* no __STATIC__ decl. here! */

LM_REAL *a, *qtb, *tau, *r, *work;
int a_sz, qtb_sz, tau_sz, r_sz, tot_sz;
register int i, j;
int info, worksz, nrhs=1;
register LM_REAL sum;

    if(!A)
#ifdef LINSOLVERS_RETAIN_MEMORY
    {
      if(buf) free(buf);
      buf=NULL;
      buf_sz=0;

      return 1;
    }
#else
      return 1; /* NOP */
#endif /* LINSOLVERS_RETAIN_MEMORY */
   
    /* calculate required memory size */
    a_sz=m*m;
    qtb_sz=m;
    tau_sz=m;
    r_sz=m*m; /* only the upper triangular part really needed */
    if(!nb){
      LM_REAL tmp;

      worksz=-1; // workspace query; optimal size is returned in tmp
      GEQRF((int *)&m, (int *)&m, NULL, (int *)&m, NULL, (LM_REAL *)&tmp, (int *)&worksz, (int *)&info);
      nb=((int)tmp)/m; // optimal worksize is m*nb
    }
    worksz=nb*m;
    tot_sz=a_sz + qtb_sz + tau_sz + r_sz + worksz;

#ifdef LINSOLVERS_RETAIN_MEMORY
    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=(LM_REAL *)malloc(buf_sz*sizeof(LM_REAL));
      if(!buf){
        fprintf(stderr, RCAT("memory allocation in ", AX_EQ_B_QR) "() failed!\n");
        exit(1);
      }
    }
#else
      buf_sz=tot_sz;
      buf=(LM_REAL *)malloc(buf_sz*sizeof(LM_REAL));
      if(!buf){
        fprintf(stderr, RCAT("memory allocation in ", AX_EQ_B_QR) "() failed!\n");
        exit(1);
      }
#endif /* LINSOLVERS_RETAIN_MEMORY */

    a=buf;
    qtb=a+a_sz;
    tau=qtb+qtb_sz;
    r=tau+tau_sz;
    work=r+r_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];

  /* QR decomposition of A */
  GEQRF((int *)&m, (int *)&m, a, (int *)&m, tau, work, (int *)&worksz, (int *)&info);
  /* error treatment */
  if(info!=0){
    if(info<0){
      fprintf(stderr, RCAT(RCAT("LAPACK error: illegal value for argument %d of ", GEQRF) " in ", AX_EQ_B_QR) "()\n", -info);
      exit(1);
    }
    else{
      fprintf(stderr, RCAT(RCAT("Unknown LAPACK error %d for ", GEQRF) " in ", AX_EQ_B_QR) "()\n", info);
#ifndef LINSOLVERS_RETAIN_MEMORY
      free(buf);
#endif

      return 0;
    }
  }

  /* R is stored in the upper triangular part of a; copy it in r so that ORGQR() 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 */
  ORGQR((int *)&m, (int *)&m, (int *)&m, a, (int *)&m, tau, work, (int *)&worksz, (int *)&info);
  if(info!=0){
    if(info<0){
      fprintf(stderr, RCAT(RCAT("LAPACK error: illegal value for argument %d of ", ORGQR) " in ", AX_EQ_B_QR) "()\n", -info);
      exit(1);
    }
    else{
      fprintf(stderr, RCAT("Unknown LAPACK error (%d) in ", AX_EQ_B_QR) "()\n", info);
#ifndef LINSOLVERS_RETAIN_MEMORY
      free(buf);
#endif

      return 0;
    }
  }

  /* Q is now in a; compute Q^T b in qtb */
  for(i=0; i<m; i++){
    for(j=0, sum=0.0; j<m; j++)
      sum+=a[i*m+j]*B[j];
    qtb[i]=sum;
  }

  /* solve the linear system R x = Q^t b */
  TRTRS("U", "N", "N", (int *)&m, (int *)&nrhs, r, (int *)&m, qtb, (int *)&m, &info);
  /* error treatment */
  if(info!=0){
    if(info<0){
      fprintf(stderr, RCAT(RCAT("LAPACK error: illegal value for argument %d of ", TRTRS) " in ", AX_EQ_B_QR) "()\n", -info);
      exit(1);
    }
    else{
      fprintf(stderr, RCAT("LAPACK error: the %d-th diagonal element of A is zero (singular matrix) in ", AX_EQ_B_QR) "()\n", info);
#ifndef LINSOLVERS_RETAIN_MEMORY
      free(buf);
#endif

      return 0;
    }
  }

	/* copy the result in x */
	for(i=0; i<m; i++)
    x[i]=qtb[i];

#ifndef LINSOLVERS_RETAIN_MEMORY
  free(buf);
#endif

	return 1;
}

/*
 * This function returns the solution of min_x ||Ax - b||
 *
 * || . || is the second order (i.e. L2) norm. This is a least squares technique that
 * is based on QR decomposition:
 * If A=Q R with Q orthogonal and R upper triangular, the normal equations become
 * (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 mxn, b is mx1
 *
 * The function returns 0 in case of error, 1 if successful
 *
 * 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 AX_EQ_B_QRLS(LM_REAL *A, LM_REAL *B, LM_REAL *x, int m, int n)
{
__STATIC__ LM_REAL *buf=NULL;
__STATIC__ int buf_sz=0;

static int nb=0; /* no __STATIC__ decl. here! */

LM_REAL *a, *atb, *tau, *r, *work;
int a_sz, atb_sz, tau_sz, r_sz, tot_sz;
register int i, j;
int info, worksz, nrhs=1;
register LM_REAL sum;
   
    if(!A)
#ifdef LINSOLVERS_RETAIN_MEMORY
    {
      if(buf) free(buf);
      buf=NULL;
      buf_sz=0;

      return 1;
    }
#else
      return 1; /* NOP */
#endif /* LINSOLVERS_RETAIN_MEMORY */
   
    if(m<n){
		  fprintf(stderr, RCAT("Normal equations require that the number of rows is greater than number of columns in ", AX_EQ_B_QRLS) "() [%d x %d]! -- try transposing\n", m, n);
		  exit(1);
	  }
      
    /* calculate required memory size */
    a_sz=m*n;
    atb_sz=n;
    tau_sz=n;
    r_sz=n*n;
    if(!nb){
      LM_REAL tmp;

      worksz=-1; // workspace query; optimal size is returned in tmp
      GEQRF((int *)&m, (int *)&m, NULL, (int *)&m, NULL, (LM_REAL *)&tmp, (int *)&worksz, (int *)&info);
      nb=((int)tmp)/m; // optimal worksize is m*nb
    }
    worksz=nb*m;
    tot_sz=a_sz + atb_sz + tau_sz + r_sz + worksz;

#ifdef LINSOLVERS_RETAIN_MEMORY
    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=(LM_REAL *)malloc(buf_sz*sizeof(LM_REAL));
      if(!buf){
        fprintf(stderr, RCAT("memory allocation in ", AX_EQ_B_QRLS) "() failed!\n");
        exit(1);
      }
    }
#else
      buf_sz=tot_sz;
      buf=(LM_REAL *)malloc(buf_sz*sizeof(LM_REAL));
      if(!buf){
        fprintf(stderr, RCAT("memory allocation in ", AX_EQ_B_QRLS) "() failed!\n");
        exit(1);
      }
#endif /* LINSOLVERS_RETAIN_MEMORY */

    a=buf;
    atb=a+a_sz;
    tau=atb+atb_sz;
    r=tau+tau_sz;
    work=r+r_sz;

  /* store A (column major!) into a */
	for(i=0; i<m; i++)
		for(j=0; j<n; j++)
			a[i+j*m]=A[i*n+j];

  /* compute A^T b in atb */
  for(i=0; i<n; i++){
    for(j=0, sum=0.0; j<m; j++)
      sum+=A[j*n+i]*B[j];
    atb[i]=sum;
  }

  /* QR decomposition of A */
  GEQRF((int *)&m, (int *)&n, a, (int *)&m, tau, work, (int *)&worksz, (int *)&info);
  /* error treatment */
  if(info!=0){
    if(info<0){
      fprintf(stderr, RCAT(RCAT("LAPACK error: illegal value for argument %d of ", GEQRF) " in ", AX_EQ_B_QRLS) "()\n", -info);
      exit(1);
    }
    else{
      fprintf(stderr, RCAT(RCAT("Unknown LAPACK error %d for ", GEQRF) " in ", AX_EQ_B_QRLS) "()\n", info);
#ifndef LINSOLVERS_RETAIN_MEMORY
      free(buf);
#endif

      return 0;
    }
  }

  /* R is stored in the upper triangular part of a. Note that a is mxn while r nxn */
  for(j=0; j<n; j++){
    for(i=0; i<=j; i++)
      r[i+j*n]=a[i+j*m];

    /* lower part is zero */
    for(i=j+1; i<n; i++)
      r[i+j*n]=0.0;
  }

  /* solve the linear system R^T y = A^t b */
  TRTRS("U", "T", "N", (int *)&n, (int *)&nrhs, r, (int *)&n, atb, (int *)&n, &info);
  /* error treatment */
  if(info!=0){
    if(info<0){
      fprintf(stderr, RCAT(RCAT("LAPACK error: illegal value for argument %d of ", TRTRS) " in ", AX_EQ_B_QRLS) "()\n", -info);
      exit(1);
    }
    else{
      fprintf(stderr, RCAT("LAPACK error: the %d-th diagonal element of A is zero (singular matrix) in ", AX_EQ_B_QRLS) "()\n", info);
#ifndef LINSOLVERS_RETAIN_MEMORY
      free(buf);
#endif

      return 0;
    }
  }

  /* solve the linear system R x = y */
  TRTRS("U", "N", "N", (int *)&n, (int *)&nrhs, r, (int *)&n, atb, (int *)&n, &info);
  /* error treatment */
  if(info!=0){
    if(info<0){
      fprintf(stderr, RCAT(RCAT("LAPACK error: illegal value for argument %d of ", TRTRS) " in ", AX_EQ_B_QRLS) "()\n", -info);
      exit(1);
    }
    else{
      fprintf(stderr, RCAT("LAPACK error: the %d-th diagonal element of A is zero (singular matrix) in ", AX_EQ_B_QRLS) "()\n", info);
#ifndef LINSOLVERS_RETAIN_MEMORY
      free(buf);
#endif

      return 0;
    }
  }

	/* copy the result in x */
	for(i=0; i<n; i++)
    x[i]=atb[i];

#ifndef LINSOLVERS_RETAIN_MEMORY
  free(buf);
#endif

	return 1;
}

/*
 * This function returns the solution of Ax=b
 *
 * The function assumes that A is symmetric & postive 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 amount to solving U^T y = b for y and then U x = y for x
 *
 * A is mxm, b is mx1
 *
 * The function returns 0 in case of error, 1 if successful
 *
 * 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 AX_EQ_B_CHOL(LM_REAL *A, LM_REAL *B, LM_REAL *x, int m)
{
__STATIC__ LM_REAL *buf=NULL;
__STATIC__ int buf_sz=0;

LM_REAL *a, *b;
int a_sz, b_sz, tot_sz;
register int i;
int info, nrhs=1;
   
    if(!A)
#ifdef LINSOLVERS_RETAIN_MEMORY
    {
      if(buf) free(buf);
      buf=NULL;
      buf_sz=0;

      return 1;
    }
#else
      return 1; /* NOP */
#endif /* LINSOLVERS_RETAIN_MEMORY */
   
    /* calculate required memory size */
    a_sz=m*m;
    b_sz=m;
    tot_sz=a_sz + b_sz;

#ifdef LINSOLVERS_RETAIN_MEMORY
    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=(LM_REAL *)malloc(buf_sz*sizeof(LM_REAL));
      if(!buf){
        fprintf(stderr, RCAT("memory allocation in ", AX_EQ_B_CHOL) "() failed!\n");
        exit(1);
      }
    }
#else
      buf_sz=tot_sz;
      buf=(LM_REAL *)malloc(buf_sz*sizeof(LM_REAL));
      if(!buf){
        fprintf(stderr, RCAT("memory allocation in ", AX_EQ_B_CHOL) "() failed!\n");
        exit(1);
      }
#endif /* LINSOLVERS_RETAIN_MEMORY */

    a=buf;
    b=a+a_sz;

    /* store A into a anb B into b. A is assumed symmetric,
     * hence no transposition is needed
     */
    for(i=0; i<m; i++){
      a[i]=A[i];
      b[i]=B[i];
    }
    for(i=m; i<m*m; i++)
      a[i]=A[i];

  /* Cholesky decomposition of A */
  //POTF2("U", (int *)&m, a, (int *)&m, (int *)&info);
  POTRF("U", (int *)&m, a, (int *)&m, (int *)&info);
  /* error treatment */
  if(info!=0){
    if(info<0){
      fprintf(stderr, RCAT(RCAT(RCAT("LAPACK error: illegal value for argument %d of ", POTF2) "/", POTRF) " in ",
                      AX_EQ_B_CHOL) "()\n", -info);
      exit(1);
    }
    else{
      fprintf(stderr, RCAT(RCAT(RCAT("LAPACK error: the leading minor of order %d is not positive definite,\nthe factorization could not be completed for ", POTF2) "/", POTRF) " in ", AX_EQ_B_CHOL) "()\n", info);
#ifndef LINSOLVERS_RETAIN_MEMORY
      free(buf);
#endif

      return 0;