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/////////////////////////////////////////////////////////////////////////////////
//
// Levenberg - Marquardt non-linear minimization algorithm
// Copyright (C) 2004-05 Manolis Lourakis (lourakis@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.
//
/////////////////////////////////////////////////////////////////////////////////
#ifndef LM_REAL // not included by misc.c
#error This file should not be compiled directly!
#endif
/* precision-specific definitions */
#define LEVMAR_CHKJAC LM_ADD_PREFIX(levmar_chkjac)
#define FDIF_FORW_JAC_APPROX LM_ADD_PREFIX(fdif_forw_jac_approx)
#define FDIF_CENT_JAC_APPROX LM_ADD_PREFIX(fdif_cent_jac_approx)
#define TRANS_MAT_MAT_MULT LM_ADD_PREFIX(trans_mat_mat_mult)
#define LEVMAR_COVAR LM_ADD_PREFIX(levmar_covar)
#ifdef HAVE_LAPACK
#define LEVMAR_PSEUDOINVERSE LM_ADD_PREFIX(levmar_pseudoinverse)
static int LEVMAR_PSEUDOINVERSE(LM_REAL *A, LM_REAL *B, int m);
/* BLAS matrix multiplication & LAPACK SVD routines */
#define GEMM LM_CAT_(LM_BLAS_PREFIX, LM_ADD_PREFIX(gemm_))
/* C := alpha*op( A )*op( B ) + beta*C */
extern void GEMM(char *transa, char *transb, int *m, int *n, int *k,
LM_REAL *alpha, LM_REAL *a, int *lda, LM_REAL *b, int *ldb, LM_REAL *beta, LM_REAL *c, int *ldc);
#define GESVD LM_ADD_PREFIX(gesvd_)
#define GESDD LM_ADD_PREFIX(gesdd_)
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() */
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);
#else
#define LEVMAR_LUINVERSE LM_ADD_PREFIX(levmar_LUinverse_noLapack)
static int LEVMAR_LUINVERSE(LM_REAL *A, LM_REAL *B, int m);
#endif /* HAVE_LAPACK */
/* blocked multiplication of the transpose of the nxm matrix a with itself (i.e. a^T a)
* using a block size of bsize. The product is returned in b.
* Since a^T a is symmetric, its computation can be speeded up by computing only its
* upper triangular part and copying it to the lower part.
*
* More details on blocking can be found at
* http://www-2.cs.cmu.edu/afs/cs/academic/class/15213-f02/www/R07/section_a/Recitation07-SectionA.pdf
*/
void TRANS_MAT_MAT_MULT(LM_REAL *a, LM_REAL *b, int n, int m)
{
#ifdef HAVE_LAPACK /* use BLAS matrix multiply */
LM_REAL alpha=CNST(1.0), beta=CNST(0.0);
/* Fool BLAS to compute a^T*a avoiding transposing a: a is equivalent to a^T in column major,
* therefore BLAS computes a*a^T with a and a*a^T in column major, which is equivalent to
* computing a^T*a in row major!
*/
GEMM("N", "T", &m, &m, &n, &alpha, a, &m, a, &m, &beta, b, &m);
#else /* no LAPACK, use blocking-based multiply */
register int i, j, k, jj, kk;
register LM_REAL sum, *bim, *akm;
const int bsize=__BLOCKSZ__;
#define __MIN__(x, y) (((x)<=(y))? (x) : (y))
#define __MAX__(x, y) (((x)>=(y))? (x) : (y))
/* compute upper triangular part using blocking */
for(jj=0; jj<m; jj+=bsize){
for(i=0; i<m; ++i){
bim=b+i*m;
for(j=__MAX__(jj, i); j<__MIN__(jj+bsize, m); ++j)
bim[j]=0.0; //b[i*m+j]=0.0;
}
for(kk=0; kk<n; kk+=bsize){
for(i=0; i<m; ++i){
bim=b+i*m;
for(j=__MAX__(jj, i); j<__MIN__(jj+bsize, m); ++j){
sum=0.0;
for(k=kk; k<__MIN__(kk+bsize, n); ++k){
akm=a+k*m;
sum+=akm[i]*akm[j]; //a[k*m+i]*a[k*m+j];
}
bim[j]+=sum; //b[i*m+j]+=sum;
}
}
}
}
/* copy upper triangular part to the lower one */
for(i=0; i<m; ++i)
for(j=0; j<i; ++j)
b[i*m+j]=b[j*m+i];
#undef __MIN__
#undef __MAX__
#endif /* HAVE_LAPACK */
}
/* forward finite difference approximation to the jacobian of func */
void FDIF_FORW_JAC_APPROX(
void (*func)(LM_REAL *p, LM_REAL *hx, int m, int n, void *adata),
/* function to differentiate */
LM_REAL *p, /* I: current parameter estimate, mx1 */
LM_REAL *hx, /* I: func evaluated at p, i.e. hx=func(p), nx1 */
LM_REAL *hxx, /* W/O: work array for evaluating func(p+delta), nx1 */
LM_REAL delta, /* increment for computing the jacobian */
LM_REAL *jac, /* O: array for storing approximated jacobian, nxm */
int m,
int n,
void *adata)
{
register int i, j;
LM_REAL tmp;
register LM_REAL d;
for(j=0; j<m; ++j){
/* determine d=max(1E-04*|p[j]|, delta), see HZ */
d=CNST(1E-04)*p[j]; // force evaluation
d=FABS(d);
if(d<delta)
d=delta;
tmp=p[j];
p[j]+=d;
(*func)(p, hxx, m, n, adata);
p[j]=tmp; /* restore */
d=CNST(1.0)/d; /* invert so that divisions can be carried out faster as multiplications */
for(i=0; i<n; ++i){
jac[i*m+j]=(hxx[i]-hx[i])*d;
}
}
}
/* central finite difference approximation to the jacobian of func */
void FDIF_CENT_JAC_APPROX(
void (*func)(LM_REAL *p, LM_REAL *hx, int m, int n, void *adata),
/* function to differentiate */
LM_REAL *p, /* I: current parameter estimate, mx1 */
LM_REAL *hxm, /* W/O: work array for evaluating func(p-delta), nx1 */
LM_REAL *hxp, /* W/O: work array for evaluating func(p+delta), nx1 */
LM_REAL delta, /* increment for computing the jacobian */
LM_REAL *jac, /* O: array for storing approximated jacobian, nxm */
int m,
int n,
void *adata)
{
register int i, j;
LM_REAL tmp;
register LM_REAL d;
for(j=0; j<m; ++j){
/* determine d=max(1E-04*|p[j]|, delta), see HZ */
d=CNST(1E-04)*p[j]; // force evaluation
d=FABS(d);
if(d<delta)
d=delta;
tmp=p[j];
p[j]-=d;
(*func)(p, hxm, m, n, adata);
p[j]=tmp+d;
(*func)(p, hxp, m, n, adata);
p[j]=tmp; /* restore */
d=CNST(0.5)/d; /* invert so that divisions can be carried out faster as multiplications */
for(i=0; i<n; ++i){
jac[i*m+j]=(hxp[i]-hxm[i])*d;
}
}
}
/*
* Check the jacobian of a n-valued nonlinear function in m variables
* evaluated at a point p, for consistency with the function itself.
*
* Based on fortran77 subroutine CHKDER by
* Burton S. Garbow, Kenneth E. Hillstrom, Jorge J. More
* Argonne National Laboratory. MINPACK project. March 1980.
*
*
* func points to a function from R^m --> R^n: Given a p in R^m it yields hx in R^n
* jacf points to a function implementing the jacobian of func, whose correctness
* is to be tested. Given a p in R^m, jacf computes into the nxm matrix j the
* jacobian of func at p. Note that row i of j corresponds to the gradient of
* the i-th component of func, evaluated at p.
* p is an input array of length m containing the point of evaluation.
* m is the number of variables
* n is the number of functions
* adata points to possible additional data and is passed uninterpreted
* to func, jacf.
* err is an array of length n. On output, err contains measures
* of correctness of the respective gradients. if there is
* no severe loss of significance, then if err[i] is 1.0 the
* i-th gradient is correct, while if err[i] is 0.0 the i-th
* gradient is incorrect. For values of err between 0.0 and 1.0,
* the categorization is less certain. In general, a value of
* err[i] greater than 0.5 indicates that the i-th gradient is
* probably correct, while a value of err[i] less than 0.5
* indicates that the i-th gradient is probably incorrect.
*
*
* The function does not perform reliably if cancellation or
* rounding errors cause a severe loss of significance in the
* evaluation of a function. therefore, none of the components
* of p should be unusually small (in particular, zero) or any
* other value which may cause loss of significance.
*/
void LEVMAR_CHKJAC(
void (*func)(LM_REAL *p, LM_REAL *hx, int m, int n, void *adata),
void (*jacf)(LM_REAL *p, LM_REAL *j, int m, int n, void *adata),
LM_REAL *p, int m, int n, void *adata, LM_REAL *err)
{
LM_REAL factor=CNST(100.0);
LM_REAL one=CNST(1.0);
LM_REAL zero=CNST(0.0);
LM_REAL *fvec, *fjac, *pp, *fvecp, *buf;
register int i, j;
LM_REAL eps, epsf, temp, epsmch;
LM_REAL epslog;
int fvec_sz=n, fjac_sz=n*m, pp_sz=m, fvecp_sz=n;
epsmch=LM_REAL_EPSILON;
eps=(LM_REAL)sqrt(epsmch);
buf=(LM_REAL *)malloc((fvec_sz + fjac_sz + pp_sz + fvecp_sz)*sizeof(LM_REAL));
if(!buf){
fprintf(stderr, LCAT(LEVMAR_CHKJAC, "(): memory allocation request failed\n"));
exit(1);
}
fvec=buf;
fjac=fvec+fvec_sz;
pp=fjac+fjac_sz;
fvecp=pp+pp_sz;
/* compute fvec=func(p) */
(*func)(p, fvec, m, n, adata);
/* compute the jacobian at p */
(*jacf)(p, fjac, m, n, adata);
/* compute pp */
for(j=0; j<m; ++j){
temp=eps*FABS(p[j]);
if(temp==zero) temp=eps;
pp[j]=p[j]+temp;
}
/* compute fvecp=func(pp) */
(*func)(pp, fvecp, m, n, adata);
epsf=factor*epsmch;
epslog=(LM_REAL)log10(eps);
for(i=0; i<n; ++i)
err[i]=zero;
for(j=0; j<m; ++j){
temp=FABS(p[j]);
if(temp==zero) temp=one;
for(i=0; i<n; ++i)
err[i]+=temp*fjac[i*m+j];
}
for(i=0; i<n; ++i){
temp=one;
if(fvec[i]!=zero && fvecp[i]!=zero && FABS(fvecp[i]-fvec[i])>=epsf*FABS(fvec[i]))
temp=eps*FABS((fvecp[i]-fvec[i])/eps - err[i])/(FABS(fvec[i])+FABS(fvecp[i]));
err[i]=one;
if(temp>epsmch && temp<eps)
err[i]=((LM_REAL)log10(temp) - epslog)/epslog;
if(temp>=eps) err[i]=zero;
}
free(buf);
return;
}
#ifdef HAVE_LAPACK
/*
* This function computes the pseudoinverse of a square matrix A
* into B using SVD. A and B can coincide
*
* The function returns 0 in case of error (e.g. A is singular),
* the rank of A if successfull
*
* A, B are mxm
*
*/
static int LEVMAR_PSEUDOINVERSE(LM_REAL *A, LM_REAL *B, int m)
{
LM_REAL *buf=NULL;
int buf_sz=0;
static LM_REAL eps=CNST(-1.0);
register int i, j;
LM_REAL *a, *u, *s, *vt, *work;
int a_sz, u_sz, s_sz, vt_sz, tot_sz;
LM_REAL thresh, one_over_denom;
int info, rank, worksz, *iwork, iworksz;
/* calculate required memory size */
worksz=16*m; /* more than needed */
iworksz=8*m;
a_sz=m*m;
u_sz=m*m; s_sz=m; vt_sz=m*m;
tot_sz=iworksz*sizeof(int) + (a_sz + u_sz + s_sz + vt_sz + worksz)*sizeof(LM_REAL);
buf_sz=tot_sz;
buf=(LM_REAL *)malloc(buf_sz);
if(!buf){
fprintf(stderr, RCAT("memory allocation in ", LEVMAR_PSEUDOINVERSE) "() failed!\n");
exit(1);
}
iwork=(int *)buf;
a=(LM_REAL *)(iwork+iworksz);
/* 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];
u=a + a_sz;
s=u+u_sz;
vt=s+s_sz;
work=vt+vt_sz;
/* SVD decomposition of A */
GESVD("A", "A", (int *)&m, (int *)&m, a, (int *)&m, s, u, (int *)&m, vt, (int *)&m, work, (int *)&worksz, &info);
//GESDD("A", (int *)&m, (int *)&m, a, (int *)&m, s, u, (int *)&m, vt, (int *)&m, work, (int *)&worksz, iwork, &info);
/* error treatment */
if(info!=0){
if(info<0){
fprintf(stderr, RCAT(RCAT(RCAT("LAPACK error: illegal value for argument %d of ", GESVD), "/" GESDD) " in ", LEVMAR_PSEUDOINVERSE) "()\n", -info);
exit(1);
}
else{
fprintf(stderr, RCAT("LAPACK error: dgesdd (dbdsdc)/dgesvd (dbdsqr) failed to converge in ", LEVMAR_PSEUDOINVERSE) "() [info=%d]\n", info);
free(buf);
return 0;
}
}
if(eps<0.0){
LM_REAL aux;
/* compute machine epsilon */
for(eps=CNST(1.0); aux=eps+CNST(1.0), aux-CNST(1.0)>0.0; eps*=CNST(0.5))
;
eps*=CNST(2.0);
}
/* compute the pseudoinverse in B */
for(i=0; i<a_sz; i++) B[i]=0.0; /* initialize to zero */
for(rank=0, thresh=eps*s[0]; rank<m && s[rank]>thresh; rank++){
one_over_denom=CNST(1.0)/s[rank];
for(j=0; j<m; j++)
for(i=0; i<m; i++)
B[i*m+j]+=vt[rank+i*m]*u[j+rank*m]*one_over_denom;
}
free(buf);
return rank;
}
#else // no LAPACK
/*
* This function computes the inverse of A in B. A and B can coincide
*
* The function employs LAPACK-free LU decomposition of A to solve m linear
* systems A*B_i=I_i, where B_i and I_i are the i-th columns of B and I.
*
* A and B are mxm
*
* The function returns 0 in case of error,
* 1 if successfull
*
*/
static int LEVMAR_LUINVERSE(LM_REAL *A, LM_REAL *B, int m)
{
void *buf=NULL;
int buf_sz=0;
register int i, j, k, l;
int *idx, maxi=-1, idx_sz, a_sz, x_sz, work_sz, tot_sz;
LM_REAL *a, *x, *work, max, sum, tmp;
/* calculate required memory size */
idx_sz=m;
a_sz=m*m;
x_sz=m;
work_sz=m;
tot_sz=idx_sz*sizeof(int) + (a_sz+x_sz+work_sz)*sizeof(LM_REAL);
buf_sz=tot_sz;
buf=(void *)malloc(tot_sz);
if(!buf){
fprintf(stderr, RCAT("memory allocation in ", LEVMAR_LUINVERSE) "() failed!\n");
exit(1);
}
idx=(int *)buf;
a=(LM_REAL *)(idx + idx_sz);
x=a + a_sz;
work=x + x_sz;
/* avoid destroying A by copying it to a */
for(i=0; i<a_sz; ++i) a[i]=A[i];
/* compute the LU decomposition of a row permutation of matrix a; the permutation itself is saved in idx[] */
for(i=0; i<m; ++i){
max=0.0;
for(j=0; j<m; ++j)
if((tmp=FABS(a[i*m+j]))>max)
max=tmp;
if(max==0.0){
fprintf(stderr, RCAT("Singular matrix A in ", LEVMAR_LUINVERSE) "()!\n");
free(buf);
return 0;
}
work[i]=CNST(1.0)/max;
}
for(j=0; j<m; ++j){
for(i=0; i<j; ++i){
sum=a[i*m+j];
for(k=0; k<i; ++k)
sum-=a[i*m+k]*a[k*m+j];
a[i*m+j]=sum;
}
max=0.0;
for(i=j; i<m; ++i){
sum=a[i*m+j];
for(k=0; k<j; ++k)
sum-=a[i*m+k]*a[k*m+j];
a[i*m+j]=sum;
if((tmp=work[i]*FABS(sum))>=max){
max=tmp;
maxi=i;
}
}
if(j!=maxi){
for(k=0; k<m; ++k){
tmp=a[maxi*m+k];
a[maxi*m+k]=a[j*m+k];
a[j*m+k]=tmp;
}
work[maxi]=work[j];
}
idx[j]=maxi;
if(a[j*m+j]==0.0)
a[j*m+j]=LM_REAL_EPSILON;
if(j!=m-1){
tmp=CNST(1.0)/(a[j*m+j]);
for(i=j+1; i<m; ++i)
a[i*m+j]*=tmp;
}
}
/* The decomposition has now replaced a. Solve the m linear systems using
* forward and back substitution
*/
for(l=0; l<m; ++l){
for(i=0; i<m; ++i) x[i]=0.0;
x[l]=CNST(1.0);
for(i=k=0; i<m; ++i){
j=idx[i];
sum=x[j];
x[j]=x[i];
if(k!=0)
for(j=k-1; j<i; ++j)
sum-=a[i*m+j]*x[j];
else
if(sum!=0.0)
k=i+1;
x[i]=sum;
}
for(i=m-1; i>=0; --i){
sum=x[i];
for(j=i+1; j<m; ++j)
sum-=a[i*m+j]*x[j];
x[i]=sum/a[i*m+i];
}
for(i=0; i<m; ++i)
B[i*m+l]=x[i];
}
free(buf);
return 1;
}
#endif /* HAVE_LAPACK */
/*
* This function computes in C the covariance matrix corresponding to a least
* squares fit. JtJ is the approximate Hessian at the solution (i.e. J^T*J, where
* J is the jacobian at the solution), sumsq is the sum of squared residuals
* (i.e. goodnes of fit) at the solution, m is the number of parameters (variables)
* and n the number of observations. JtJ can coincide with C.
*
* if JtJ is of full rank, C is computed as sumsq/(n-m)*(JtJ)^-1
* otherwise and if LAPACK is available, C=sumsq/(n-r)*(JtJ)^+
* where r is JtJ's rank and ^+ denotes the pseudoinverse
* The diagonal of C is made up from the estimates of the variances
* of the estimated regression coefficients.
* See the documentation of routine E04YCF from the NAG fortran lib
*
* The function returns the rank of JtJ if successful, 0 on error
*
* A and C are mxm
*
*/
int LEVMAR_COVAR(LM_REAL *JtJ, LM_REAL *C, LM_REAL sumsq, int m, int n)
{
register int i;
int rnk;
LM_REAL fact;
#ifdef HAVE_LAPACK
rnk=LEVMAR_PSEUDOINVERSE(JtJ, C, m);
if(!rnk) return 0;
#else
#ifdef _MSC_VER
#pragma message("LAPACK not available, LU will be used for matrix inversion when computing the covariance; this might be unstable at times")
#else
#warning LAPACK not available, LU will be used for matrix inversion when computing the covariance; this might be unstable at times
#endif // _MSC_VER
rnk=LEVMAR_LUINVERSE(JtJ, C, m);
if(!rnk) return 0;
rnk=m; /* assume full rank */
#endif /* HAVE_LAPACK */
fact=sumsq/(LM_REAL)(n-rnk);
for(i=0; i<m*m; ++i)
C[i]*=fact;
return rnk;
}
/* undefine everything. THIS MUST REMAIN AT THE END OF THE FILE */
#undef LEVMAR_PSEUDOINVERSE
#undef LEVMAR_LUINVERSE
#undef LEVMAR_COVAR
#undef LEVMAR_CHKJAC
#undef FDIF_FORW_JAC_APPROX
#undef FDIF_CENT_JAC_APPROX
#undef TRANS_MAT_MAT_MULT

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