/////////////////////////////////////////////////////////////////////////////////
//
// Levenberg - Marquardt non-linear minimization algorithm
// Copyright (C) 2004-05 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.
//
/////////////////////////////////////////////////////////////////////////////////
#ifndef LM_REAL // not included by lmbc.c
#error This file should not be compiled directly!
#endif
/* precision-specific definitions */
#define FUNC_STATE LM_ADD_PREFIX(func_state)
#define LNSRCH LM_ADD_PREFIX(lnsrch)
#define BOXPROJECT LM_ADD_PREFIX(boxProject)
#define LEVMAR_BOX_CHECK LM_ADD_PREFIX(levmar_box_check)
#define LEVMAR_BC_DER LM_ADD_PREFIX(levmar_bc_der)
#define LEVMAR_BC_DIF LM_ADD_PREFIX(levmar_bc_dif)
#define LEVMAR_FDIF_FORW_JAC_APPROX LM_ADD_PREFIX(levmar_fdif_forw_jac_approx)
#define LEVMAR_FDIF_CENT_JAC_APPROX LM_ADD_PREFIX(levmar_fdif_cent_jac_approx)
#define LEVMAR_TRANS_MAT_MAT_MULT LM_ADD_PREFIX(levmar_trans_mat_mat_mult)
#define LEVMAR_L2NRMXMY LM_ADD_PREFIX(levmar_L2nrmxmy)
#define LEVMAR_COVAR LM_ADD_PREFIX(levmar_covar)
#define LMBC_DIF_DATA LM_ADD_PREFIX(lmbc_dif_data)
#define LMBC_DIF_FUNC LM_ADD_PREFIX(lmbc_dif_func)
#define LMBC_DIF_JACF LM_ADD_PREFIX(lmbc_dif_jacf)
#ifdef HAVE_LAPACK
#define AX_EQ_B_LU LM_ADD_PREFIX(Ax_eq_b_LU)
#define AX_EQ_B_CHOL LM_ADD_PREFIX(Ax_eq_b_Chol)
#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_SVD LM_ADD_PREFIX(Ax_eq_b_SVD)
#else
#define AX_EQ_B_LU LM_ADD_PREFIX(Ax_eq_b_LU_noLapack)
#endif /* HAVE_LAPACK */
/* find the median of 3 numbers */
#define __MEDIAN3(a, b, c) ( ((a) >= (b))?\
( ((c) >= (a))? (a) : ( ((c) <= (b))? (b) : (c) ) ) : \
( ((c) >= (b))? (b) : ( ((c) <= (a))? (a) : (c) ) ) )
#define _POW_ LM_CNST(2.1)
#define __LSITMAX 150 // max #iterations for line search
struct FUNC_STATE{
int n, *nfev;
LM_REAL *hx, *x;
void *adata;
};
static void
LNSRCH(int m, LM_REAL *x, LM_REAL f, LM_REAL *g, LM_REAL *p, LM_REAL alpha, LM_REAL *xpls,
LM_REAL *ffpls, void (*func)(LM_REAL *p, LM_REAL *hx, int m, int n, void *adata), struct FUNC_STATE state,
int *mxtake, int *iretcd, LM_REAL stepmx, LM_REAL steptl, LM_REAL *sx)
{
/* Find a next newton iterate by backtracking line search.
* Specifically, finds a \lambda such that for a fixed alpha<0.5 (usually 1e-4),
* f(x + \lambda*p) <= f(x) + alpha * \lambda * g^T*p
*
* Translated (with minor changes) from Schnabel, Koontz & Weiss uncmin.f, v1.3
* PARAMETERS :
* m --> dimension of problem (i.e. number of variables)
* x(m) --> old iterate: x[k-1]
* f --> function value at old iterate, f(x)
* g(m) --> gradient at old iterate, g(x), or approximate
* p(m) --> non-zero newton step
* alpha --> fixed constant < 0.5 for line search (see above)
* xpls(m) <-- new iterate x[k]
* ffpls <-- function value at new iterate, f(xpls)
* func --> name of subroutine to evaluate function
* state <--> information other than x and m that func requires.
* state is not modified in xlnsrch (but can be modified by func).
* iretcd <-- return code
* mxtake <-- boolean flag indicating step of maximum length used
* stepmx --> maximum allowable step size
* steptl --> relative step size at which successive iterates
* considered close enough to terminate algorithm
* sx(m) --> diagonal scaling matrix for x, can be NULL
* internal variables
* sln newton length
* rln relative length of newton step
*/
register int i, j;
int firstback = 1;
LM_REAL disc;
LM_REAL a3, b;
LM_REAL t1, t2, t3, lambda, tlmbda, rmnlmb;
LM_REAL scl, rln, sln, slp;
LM_REAL tmp1, tmp2;
LM_REAL fpls, pfpls = 0., plmbda = 0.; /* -Wall */
f*=LM_CNST(0.5);
*mxtake = 0;
*iretcd = 2;
tmp1 = 0.;
if(!sx) /* no scaling */
for (i = 0; i < m; ++i)
tmp1 += p[i] * p[i];
else
for (i = 0; i < m; ++i)
tmp1 += sx[i] * sx[i] * p[i] * p[i];
sln = (LM_REAL)sqrt(tmp1);
if (sln > stepmx) {
/* newton step longer than maximum allowed */
scl = stepmx / sln;
for(i=0; i<m; ++i) /* p * scl */
p[i]*=scl;
sln = stepmx;
}
for(i=0, slp=0.; i<m; ++i) /* g^T * p */
slp+=g[i]*p[i];
rln = 0.;
if(!sx) /* no scaling */
for (i = 0; i < m; ++i) {
tmp1 = (FABS(x[i])>=LM_CNST(1.))? FABS(x[i]) : LM_CNST(1.);
tmp2 = FABS(p[i])/tmp1;
if(rln < tmp2) rln = tmp2;
}
else
for (i = 0; i < m; ++i) {
tmp1 = (FABS(x[i])>=LM_CNST(1.)/sx[i])? FABS(x[i]) : LM_CNST(1.)/sx[i];
tmp2 = FABS(p[i])/tmp1;
if(rln < tmp2) rln = tmp2;
}
rmnlmb = steptl / rln;
lambda = LM_CNST(1.0);
/* check if new iterate satisfactory. generate new lambda if necessary. */
for(j=__LSITMAX; j>=0; --j) {
for (i = 0; i < m; ++i)
xpls[i] = x[i] + lambda * p[i];
/* evaluate function at new point */
(*func)(xpls, state.hx, m, state.n, state.adata); ++(*(state.nfev));
/* ### state.hx=state.x-state.hx, tmp1=||state.hx|| */
#if 1
tmp1=LEVMAR_L2NRMXMY(state.hx, state.x, state.hx, state.n);
#else
for(i=0, tmp1=0.0; i<state.n; ++i){
state.hx[i]=tmp2=state.x[i]-state.hx[i];
tmp1+=tmp2*tmp2;
}
#endif
fpls=LM_CNST(0.5)*tmp1; *ffpls=tmp1;
if (fpls <= f + slp * alpha * lambda) { /* solution found */
*iretcd = 0;
if (lambda == LM_CNST(1.) && sln > stepmx * LM_CNST(.99)) *mxtake = 1;
return;
}
/* else : solution not (yet) found */
/* First find a point with a finite value */
if (lambda < rmnlmb) {
/* no satisfactory xpls found sufficiently distinct from x */
*iretcd = 1;
return;
}
else { /* calculate new lambda */
/* modifications to cover non-finite values */
if (!LM_FINITE(fpls)) {
lambda *= LM_CNST(0.1);
firstback = 1;
}
else {
if (firstback) { /* first backtrack: quadratic fit */
tlmbda = -lambda * slp / ((fpls - f - slp) * LM_CNST(2.));
firstback = 0;
}
else { /* all subsequent backtracks: cubic fit */
t1 = fpls - f - lambda * slp;
t2 = pfpls - f - plmbda * slp;
t3 = LM_CNST(1.) / (lambda - plmbda);
a3 = LM_CNST(3.) * t3 * (t1 / (lambda * lambda)
- t2 / (plmbda * plmbda));
b = t3 * (t2 * lambda / (plmbda * plmbda)
- t1 * plmbda / (lambda * lambda));
disc = b * b - a3 * slp;
if (disc > b * b)
/* only one positive critical point, must be minimum */
tlmbda = (-b + ((a3 < 0)? -(LM_REAL)sqrt(disc): (LM_REAL)sqrt(disc))) /a3;
else
/* both critical points positive, first is minimum */
tlmbda = (-b + ((a3 < 0)? (LM_REAL)sqrt(disc): -(LM_REAL)sqrt(disc))) /a3;
if (tlmbda > lambda * LM_CNST(.5))
tlmbda = lambda * LM_CNST(.5);
}
plmbda = lambda;
pfpls = fpls;
if (tlmbda < lambda * LM_CNST(.1))
lambda *= LM_CNST(.1);
else
lambda = tlmbda;
}
}
}
/* this point is reached when the iterations limit is exceeded */
*iretcd = 1; /* failed */
return;
} /* LNSRCH */
/* Projections to feasible set \Omega: P_{\Omega}(y) := arg min { ||x - y|| : x \in \Omega}, y \in R^m */
/* project vector p to a box shaped feasible set. p is a mx1 vector.
* Either lb, ub can be NULL. If not NULL, they are mx1 vectors
*/
static void BOXPROJECT(LM_REAL *p, LM_REAL *lb, LM_REAL *ub, int m)
{
register int i;
if(!lb){ /* no lower bounds */
if(!ub) /* no upper bounds */
return;
else{ /* upper bounds only */
for(i=0; i<m; ++i)
if(p[i]>ub[i]) p[i]=ub[i];
}
}
else
if(!ub){ /* lower bounds only */
for(i=0; i<m; ++i)
if(p[i]<lb[i]) p[i]=lb[i];
}
else /* box bounds */
for(i=0; i<m; ++i)
p[i]=__MEDIAN3(lb[i], p[i], ub[i]);
}
/*
* This function seeks the parameter vector p that best describes the measurements
* vector x under box constraints.
* More precisely, given a vector function func : R^m --> R^n with n>=m,
* it finds p s.t. func(p) ~= x, i.e. the squared second order (i.e. L2) norm of
* e=x-func(p) is minimized under the constraints lb[i]<=p[i]<=ub[i].
* If no lower bound constraint applies for p[i], use -DBL_MAX/-FLT_MAX for lb[i];
* If no upper bound constraint applies for p[i], use DBL_MAX/FLT_MAX for ub[i].
*
* This function requires an analytic Jacobian. In case the latter is unavailable,
* use LEVMAR_BC_DIF() bellow
*
* Returns the number of iterations (>=0) if successful, or an error code (<0) on failure.
*
* For details, see C. Kanzow, N. Yamashita and M. Fukushima: "Levenberg-Marquardt
* methods for constrained nonlinear equations with strong local convergence properties",
* Journal of Computational and Applied Mathematics 172, 2004, pp. 375-397.
* Also, see K. Madsen, H.B. Nielsen and O. Tingleff's lecture notes on
* unconstrained Levenberg-Marquardt at http://www.imm.dtu.dk/pubdb/views/edoc_download.php/3215/pdf/imm3215.pdf
*/
int LEVMAR_BC_DER(
void (*func)(LM_REAL *p, LM_REAL *hx, int m, int n, void *adata), /* functional relation describing measurements. A p \in R^m yields a \hat{x} \in R^n */
void (*jacf)(LM_REAL *p, LM_REAL *j, int m, int n, void *adata), /* function to evaluate the Jacobian \part x / \part p */
LM_REAL *p, /* I/O: initial parameter estimates. On output has the estimated solution */
LM_REAL *x, /* I: measurement vector. NULL implies a zero vector */
int m, /* I: parameter vector dimension (i.e. #unknowns) */
int n, /* I: measurement vector dimension */
LM_REAL *lb, /* I: vector of lower bounds. If NULL, no lower bounds apply */
LM_REAL *ub, /* I: vector of upper bounds. If NULL, no upper bounds apply */
int itmax, /* I: maximum number of iterations */
LM_REAL opts[4], /* I: minim. options [\mu, \epsilon1, \epsilon2, \epsilon3]. Respectively the scale factor for initial \mu,
* stopping thresholds for ||J^T e||_inf, ||Dp||_2 and ||e||_2. Set to NULL for defaults to be used.
* Note that ||J^T e||_inf is computed on free (not equal to lb[i] or ub[i]) variables only.
*/
LM_REAL info[LM_INFO_SZ],
/* O: information regarding the minimization. Set to NULL if don't care
* info[0]= ||e||_2 at initial p.
* info[1-4]=[ ||e||_2, ||J^T e||_inf, ||Dp||_2, mu/max[J^T J]_ii ], all computed at estimated p.
* info[5]= # iterations,
* info[6]=reason for terminating: 1 - stopped by small gradient J^T e
* 2 - stopped by small Dp
* 3 - stopped by itmax
* 4 - singular matrix. Restart from current p with increased mu
* 5 - no further error reduction is possible. Restart with increased mu
* 6 - stopped by small ||e||_2
* 7 - stopped by invalid (i.e. NaN or Inf) "func" values. This is a user error
* info[7]= # function evaluations
* info[8]= # Jacobian evaluations
* info[9]= # linear systems solved, i.e. # attempts for reducing error
*/
LM_REAL *work, /* working memory at least LM_BC_DER_WORKSZ() reals large, allocated if NULL */
LM_REAL *covar, /* O: Covariance matrix corresponding to LS solution; mxm. Set to NULL if not needed. */
void *adata) /* pointer to possibly additional data, passed uninterpreted to func & jacf.
* Set to NULL if not needed
*/
{
register int i, j, k, l;
int worksz, freework=0, issolved;
/* temp work arrays */
LM_REAL *e, /* nx1 */
*hx, /* \hat{x}_i, nx1 */
*jacTe, /* J^T e_i mx1 */
*jac, /* nxm */
*jacTjac, /* mxm */
*Dp, /* mx1 */
*diag_jacTjac, /* diagonal of J^T J, mx1 */
*pDp; /* p + Dp, mx1 */
register LM_REAL mu, /* damping constant */
tmp; /* mainly used in matrix & vector multiplications */
LM_REAL p_eL2, jacTe_inf, pDp_eL2; /* ||e(p)||_2, ||J^T e||_inf, ||e(p+Dp)||_2 */
LM_REAL p_L2, Dp_L2=LM_REAL_MAX, dF, dL;
LM_REAL tau, eps1, eps2, eps2_sq, eps3;
LM_REAL init_p_eL2;
int nu=2, nu2, stop=0, nfev, njev=0, nlss=0;
const int nm=n*m;
/* variables for constrained LM */
struct FUNC_STATE fstate;
LM_REAL alpha=LM_CNST(1e-4), beta=LM_CNST(0.9), gamma=LM_CNST(0.99995), gamma_sq=gamma*gamma, rho=LM_CNST(1e-8);
LM_REAL t, t0;
LM_REAL steptl=LM_CNST(1e3)*(LM_REAL)sqrt(LM_REAL_EPSILON), jacTeDp;
LM_REAL tmin=LM_CNST(1e-12), tming=LM_CNST(1e-18); /* minimum step length for LS and PG steps */
const LM_REAL tini=LM_CNST(1.0); /* initial step length for LS and PG steps */
int nLMsteps=0, nLSsteps=0, nPGsteps=0, gprevtaken=0;
int numactive;
int (*linsolver)(LM_REAL *A, LM_REAL *B, LM_REAL *x, int m)=NULL;
mu=jacTe_inf=t=0.0; tmin=tmin; /* -Wall */
if(n<m){
PRINT_ERROR(LCAT(LEVMAR_BC_DER, "(): cannot solve a problem with fewer measurements [%d] than unknowns [%d]\n"), n, m);
return LM_ERROR_TOO_FEW_MEASUREMENTS;
}
if(!jacf){
PRINT_ERROR(RCAT("No function specified for computing the Jacobian in ", LEVMAR_BC_DER)
RCAT("().\nIf no such function is available, use ", LEVMAR_BC_DIF) RCAT("() rather than ", LEVMAR_BC_DER) "()\n");
return LM_ERROR_NO_JACOBIAN;
}
if(!LEVMAR_BOX_CHECK(lb, ub, m)){
PRINT_ERROR(LCAT(LEVMAR_BC_DER, "(): at least one lower bound exceeds the upper one\n"));
return LM_ERROR_FAILED_BOX_CHECK;
}
if(opts){
tau=opts[0];
eps1=opts[1];
eps2=opts[2];
eps2_sq=opts[2]*opts[2];
eps3=opts[3];
}
else{ // use default values
tau=LM_CNST(LM_INIT_MU);
eps1=LM_CNST(LM_STOP_THRESH);
eps2=LM_CNST(LM_STOP_THRESH);
eps2_sq=LM_CNST(LM_STOP_THRESH)*LM_CNST(LM_STOP_THRESH);
eps3=LM_CNST(LM_STOP_THRESH);
}
if(!work){
worksz=LM_BC_DER_WORKSZ(m, n); //2*n+4*m + n*m + m*m;
work=(LM_REAL *)malloc(worksz*sizeof(LM_REAL)); /* allocate a big chunk in one step */
if(!work){
PRINT_ERROR(LCAT(LEVMAR_BC_DER, "(): memory allocation request failed\n"));
return LM_ERROR_MEMORY_ALLOCATION_FAILURE;
}
freework=1;
}
/* set up work arrays */
e=work;
hx=e + n;
jacTe=hx + n;
jac=jacTe + m;
jacTjac=jac + nm;
Dp=jacTjac + m*m;
diag_jacTjac=Dp + m;
pDp=diag_jacTjac + m;
fstate.n=n;
fstate.hx=hx;
fstate.x=x;
fstate.adata=adata;
fstate.nfev=&nfev;
/* see if starting point is within the feasile set */
for(i=0; i<m; ++i)
pDp[i]=p[i];
BOXPROJECT(p, lb, ub, m); /* project to feasible set */
for(i=0; i<m; ++i)
if(pDp[i]!=p[i])
PRINT_ERROR(RCAT("Warning: component %d of starting point not feasible in ", LEVMAR_BC_DER) "()! [%g projected to %g]\n",
i, pDp[i], p[i]);
/* compute e=x - f(p) and its L2 norm */
(*func)(p, hx, m, n, adata); nfev=1;
/* ### e=x-hx, p_eL2=||e|| */
#if 1
p_eL2=LEVMAR_L2NRMXMY(e, x, hx, n);
#else
for(i=0, p_eL2=0.0; i<n; ++i){
e[i]=tmp=x[i]-hx[i];
p_eL2+=tmp*tmp;
}
#endif
init_p_eL2=p_eL2;
if(!LM_FINITE(p_eL2)) stop=7;
for(k=0; k<itmax && !stop; ++k){
/* Note that p and e have been updated at a previous iteration */
if(p_eL2<=eps3){ /* error is small */
stop=6;
break;
}
/* Compute the Jacobian J at p, J^T J, J^T e, ||J^T e||_inf and ||p||^2.
* Since J^T J is symmetric, its computation can be sped up by computing
* only its upper triangular part and copying it to the lower part
*/
(*jacf)(p, jac, m, n, adata); ++njev;
/* J^T J, J^T e */
if(nm<__BLOCKSZ__SQ){ // this is a small problem
/* J^T*J_ij = \sum_l J^T_il * J_lj = \sum_l J_li * J_lj.
* Thus, the product J^T J can be computed using an outer loop for
* l that adds J_li*J_lj to each element ij of the result. Note that
* with this scheme, the accesses to J and JtJ are always along rows,
* therefore induces less cache misses compared to the straightforward
* algorithm for computing the product (i.e., l loop is innermost one).
* A similar scheme applies to the computation of J^T e.
* However, for large minimization problems (i.e., involving a large number
* of unknowns and measurements) for which J/J^T J rows are too large to
* fit in the L1 cache, even this scheme incures many cache misses. In
* such cases, a cache-efficient blocking scheme is preferable.
*
* Thanks to John Nitao of Lawrence Livermore Lab for pointing out this
* performance problem.
*
* Note that the non-blocking algorithm is faster on small
* problems since in this case it avoids the overheads of blocking.
*/
register int l, im;
register LM_REAL alpha, *jaclm;
/* looping downwards saves a few computations */
for(i=m*m; i-->0; )
jacTjac[i]=0.0;
for(i=m; i-->0; )
jacTe[i]=0.0;
for(l=n; l-->0; ){
jaclm=jac+l*m;
for(i=m; i-->0; ){
im=i*m;
alpha=jaclm[i]; //jac[l*m+i];
for(j=i+1; j-->0; ) /* j<=i computes lower triangular part only */
jacTjac[im+j]+=jaclm[j]*alpha; //jac[l*m+j]
/* J^T e */
jacTe[i]+=alpha*e[l];
}
}
for(i=m; i-->0; ) /* copy to upper part */
for(j=i+1; j<m; ++j)
jacTjac[i*m+j]=jacTjac[j*m+i];
}
else{ // this is a large problem
/* Cache efficient computation of J^T J based on blocking
*/
LEVMAR_TRANS_MAT_MAT_MULT(jac, jacTjac, n, m);
/* cache efficient computation of J^T e */
for(i=0; i<m; ++i)
jacTe[i]=0.0;
for(i=0; i<n; ++i){
register LM_REAL *jacrow;
for(l=0, jacrow=jac+i*m, tmp=e[i]; l<m; ++l)
jacTe[l]+=jacrow[l]*tmp;
}
}
/* Compute ||J^T e||_inf and ||p||^2. Note that ||J^T e||_inf
* is computed for free (i.e. inactive) variables only.
* At a local minimum, if p[i]==ub[i] then g[i]>0;
* if p[i]==lb[i] g[i]<0; otherwise g[i]=0
*/
for(i=j=numactive=0, p_L2=jacTe_inf=0.0; i<m; ++i){
if(ub && p[i]==ub[i]){ ++numactive; if(jacTe[i]>0.0) ++j; }
else if(lb && p[i]==lb[i]){ ++numactive; if(jacTe[i]<0.0) ++j; }
else if(jacTe_inf < (tmp=FABS(jacTe[i]))) jacTe_inf=tmp;
diag_jacTjac[i]=jacTjac[i*m+i]; /* save diagonal entries so that augmentation can be later canceled */
p_L2+=p[i]*p[i];
}
//p_L2=sqrt(p_L2);
#if 0
if(!(k%100)){
printf("Current estimate: ");
for(i=0; i<m; ++i)
printf("%.9g ", p[i]);
printf("-- errors %.9g %0.9g, #active %d [%d]\n", jacTe_inf, p_eL2, numactive, j);
}
#endif
/* check for convergence */
if(j==numactive && (jacTe_inf <= eps1)){
Dp_L2=0.0; /* no increment for p in this case */
stop=1;
break;
}
/* compute initial damping factor */
if(k==0){
if(!lb && !ub){ /* no bounds */
for(i=0, tmp=LM_REAL_MIN; i<m; ++i)
if(diag_jacTjac[i]>tmp) tmp=diag_jacTjac[i]; /* find max diagonal element */
mu=tau*tmp;
}
else
mu=LM_CNST(0.5)*tau*p_eL2; /* use Kanzow's starting mu */
}
/* determine increment using a combination of adaptive damping, line search and projected gradient search */
while(1){
/* augment normal equations */
for(i=0; i<m; ++i)
jacTjac[i*m+i]+=mu;
/* solve augmented equations */
#ifdef HAVE_LAPACK
/* 5 alternatives are available: LU, Cholesky, 2 variants of QR decomposition and SVD.
* Cholesky is the fastest but might be inaccurate; QR is slower but more accurate;
* SVD is the slowest but most accurate; LU offers a tradeoff between accuracy and speed
*/
issolved=AX_EQ_B_LU(jacTjac, jacTe, Dp, m); ++nlss; linsolver=AX_EQ_B_LU;
//issolved=AX_EQ_B_CHOL(jacTjac, jacTe, Dp, m); ++nlss; linsolver=AX_EQ_B_CHOL;
//issolved=AX_EQ_B_QR(jacTjac, jacTe, Dp, m); ++nlss; linsolver=AX_EQ_B_QR;
//issolved=AX_EQ_B_QRLS(jacTjac, jacTe, Dp, m, m); ++nlss; linsolver=(int (*)(LM_REAL *A, LM_REAL *B, LM_REAL *x, int m))AX_EQ_B_QRLS;
//issolved=AX_EQ_B_SVD(jacTjac, jacTe, Dp, m); ++nlss; linsolver=AX_EQ_B_SVD;
#else
/* use the LU included with levmar */
issolved=AX_EQ_B_LU(jacTjac, jacTe, Dp, m); ++nlss; linsolver=AX_EQ_B_LU;
#endif /* HAVE_LAPACK */
if(issolved){
for(i=0; i<m; ++i)
pDp[i]=p[i] + Dp[i];
/* compute p's new estimate and ||Dp||^2 */
BOXPROJECT(pDp, lb, ub, m); /* project to feasible set */
for(i=0, Dp_L2=0.0; i<m; ++i){
Dp[i]=tmp=pDp[i]-p[i];
Dp_L2+=tmp*tmp;
}
//Dp_L2=sqrt(Dp_L2);
if(Dp_L2<=eps2_sq*p_L2){ /* relative change in p is small, stop */
stop=2;
break;
}
if(Dp_L2>=(p_L2+eps2)/(LM_CNST(EPSILON)*LM_CNST(EPSILON))){ /* almost singular */
stop=4;
break;
}
(*func)(pDp, hx, m, n, adata); ++nfev; /* evaluate function at p + Dp */
/* ### hx=x-hx, pDp_eL2=||hx|| */
#if 1
pDp_eL2=LEVMAR_L2NRMXMY(hx, x, hx, n);
#else
for(i=0, pDp_eL2=0.0; i<n; ++i){ /* compute ||e(pDp)||_2 */
hx[i]=tmp=x[i]-hx[i];
pDp_eL2+=tmp*tmp;
}
#endif
if(!LM_FINITE(pDp_eL2)){
stop=7;
break;
}
if(pDp_eL2<=gamma_sq*p_eL2){
for(i=0, dL=0.0; i<m; ++i)
dL+=Dp[i]*(mu*Dp[i]+jacTe[i]);
#if 1
if(dL>0.0){
dF=p_eL2-pDp_eL2;
tmp=(LM_CNST(2.0)*dF/dL-LM_CNST(1.0));
tmp=LM_CNST(1.0)-tmp*tmp*tmp;
mu=mu*( (tmp>=LM_CNST(ONE_THIRD))? tmp : LM_CNST(ONE_THIRD) );
}
else
mu=(mu>=pDp_eL2)? pDp_eL2 : mu; /* pDp_eL2 is the new pDp_eL2 */
#else
mu=(mu>=pDp_eL2)? pDp_eL2 : mu; /* pDp_eL2 is the new pDp_eL2 */
#endif
nu=2;
for(i=0 ; i<m; ++i) /* update p's estimate */
p[i]=pDp[i];
for(i=0; i<n; ++i) /* update e and ||e||_2 */
e[i]=hx[i];
p_eL2=pDp_eL2;
++nLMsteps;
gprevtaken=0;
break;
}
}
else{
/* the augmented linear system could not be solved, increase mu */
mu*=nu;
nu2=nu<<1; // 2*nu;
if(nu2<=nu){ /* nu has wrapped around (overflown). Thanks to Frank Jordan for spotting this case */
stop=5;
break;
}
nu=nu2;
for(i=0; i<m; ++i) /* restore diagonal J^T J entries */
jacTjac[i*m+i]=diag_jacTjac[i];
continue; /* solve again with increased nu */
}
/* if this point is reached, the LM step did not reduce the error;
* see if it is a descent direction
*/
/* negate jacTe (i.e. g) & compute g^T * Dp */
for(i=0, jacTeDp=0.0; i<m; ++i){
jacTe[i]=-jacTe[i];
jacTeDp+=jacTe[i]*Dp[i];
}
if(jacTeDp<=-rho*pow(Dp_L2, _POW_/LM_CNST(2.0))){
/* Dp is a descent direction; do a line search along it */
int mxtake, iretcd;
LM_REAL stepmx;
tmp=(LM_REAL)sqrt(p_L2); stepmx=LM_CNST(1e3)*( (tmp>=LM_CNST(1.0))? tmp : LM_CNST(1.0) );
#if 1
/* use Schnabel's backtracking line search; it requires fewer "func" evaluations */
LNSRCH(m, p, p_eL2, jacTe, Dp, alpha, pDp, &pDp_eL2, func, fstate,
&mxtake, &iretcd, stepmx, steptl, NULL); /* NOTE: LNSRCH() updates hx */
if(iretcd!=0) goto gradproj; /* rather inelegant but effective way to handle LNSRCH() failures... */
#else
/* use the simpler (but slower!) line search described by Kanzow et al */
for(t=tini; t>tmin; t*=beta){
for(i=0; i<m; ++i){
pDp[i]=p[i] + t*Dp[i];
//pDp[i]=__MEDIAN3(lb[i], pDp[i], ub[i]); /* project to feasible set */
}
(*func)(pDp, hx, m, n, adata); ++nfev; /* evaluate function at p + t*Dp */
for(i=0, pDp_eL2=0.0; i<n; ++i){ /* compute ||e(pDp)||_2 */
hx[i]=tmp=x[i]-hx[i];
pDp_eL2+=tmp*tmp;
}
if(!LM_FINITE(pDp_eL2)) goto gradproj; /* treat as line search failure */
//if(LM_CNST(0.5)*pDp_eL2<=LM_CNST(0.5)*p_eL2 + t*alpha*jacTeDp) break;
if(pDp_eL2<=p_eL2 + LM_CNST(2.0)*t*alpha*jacTeDp) break;
}
#endif
++nLSsteps;
gprevtaken=0;
/* NOTE: new estimate for p is in pDp, associated error in hx and its norm in pDp_eL2.
* These values are used below to update their corresponding variables
*/
}
else{
gradproj: /* Note that this point can also be reached via a goto when LNSRCH() fails */
/* jacTe is a descent direction; make a projected gradient step */
/* if the previous step was along the gradient descent, try to use the t employed in that step */
/* compute ||g|| */
for(i=0, tmp=0.0; i<m; ++i)
tmp+=jacTe[i]*jacTe[i];
tmp=(LM_REAL)sqrt(tmp);
tmp=LM_CNST(100.0)/(LM_CNST(1.0)+tmp);
t0=(tmp<=tini)? tmp : tini; /* guard against poor scaling & large steps; see (3.50) in C.T. Kelley's book */
for(t=(gprevtaken)? t : t0; t>tming; t*=beta){
for(i=0; i<m; ++i)
pDp[i]=p[i] - t*jacTe[i];
BOXPROJECT(pDp, lb, ub, m); /* project to feasible set */
for(i=0; i<m; ++i)
Dp[i]=pDp[i]-p[i];
(*func)(pDp, hx, m, n, adata); ++nfev; /* evaluate function at p - t*g */
/* compute ||e(pDp)||_2 */
/* ### hx=x-hx, pDp_eL2=||hx|| */
#if 1
pDp_eL2=LEVMAR_L2NRMXMY(hx, x, hx, n);
#else
for(i=0, pDp_eL2=0.0; i<n; ++i){
hx[i]=tmp=x[i]-hx[i];
pDp_eL2+=tmp*tmp;
}
#endif
if(!LM_FINITE(pDp_eL2)){
stop=7;
goto breaknested;
}
for(i=0, tmp=0.0; i<m; ++i) /* compute ||g^T * Dp|| */
tmp+=jacTe[i]*Dp[i];
if(gprevtaken && pDp_eL2<=p_eL2 + LM_CNST(2.0)*LM_CNST(0.99999)*tmp){ /* starting t too small */
t=t0;
gprevtaken=0;
continue;
}
//if(LM_CNST(0.5)*pDp_eL2<=LM_CNST(0.5)*p_eL2 + alpha*tmp) break;
if(pDp_eL2<=p_eL2 + LM_CNST(2.0)*alpha*tmp) break;
}
++nPGsteps;
gprevtaken=1;
/* NOTE: new estimate for p is in pDp, associated error in hx and its norm in pDp_eL2 */
}
/* update using computed values */
for(i=0, Dp_L2=0.0; i<m; ++i){
tmp=pDp[i]-p[i];
Dp_L2+=tmp*tmp;
}
//Dp_L2=sqrt(Dp_L2);
if(Dp_L2<=eps2_sq*p_L2){ /* relative change in p is small, stop */
stop=2;
break;
}
for(i=0 ; i<m; ++i) /* update p's estimate */
p[i]=pDp[i];
for(i=0; i<n; ++i) /* update e and ||e||_2 */
e[i]=hx[i];
p_eL2=pDp_eL2;
break;
} /* inner loop */
}
breaknested: /* NOTE: this point is also reached via an explicit goto! */
if(k>=itmax) stop=3;
for(i=0; i<m; ++i) /* restore diagonal J^T J entries */
jacTjac[i*m+i]=diag_jacTjac[i];
if(info){
info[0]=init_p_eL2;
info[1]=p_eL2;
info[2]=jacTe_inf;
info[3]=Dp_L2;
for(i=0, tmp=LM_REAL_MIN; i<m; ++i)
if(tmp<jacTjac[i*m+i]) tmp=jacTjac[i*m+i];
info[4]=mu/tmp;
info[5]=(LM_REAL)k;
info[6]=(LM_REAL)stop;
info[7]=(LM_REAL)nfev;
info[8]=(LM_REAL)njev;
info[9]=(LM_REAL)nlss;
}
/* covariance matrix */
if(covar){
LEVMAR_COVAR(jacTjac, covar, p_eL2, m, n);
}
if(freework) free(work);
#ifdef LINSOLVERS_RETAIN_MEMORY
if(linsolver) (*linsolver)(NULL, NULL, NULL, 0);
#endif
#if 0
printf("%d LM steps, %d line search, %d projected gradient\n", nLMsteps, nLSsteps, nPGsteps);
#endif
switch (stop) {
case 4: return LM_ERROR_SINGULAR_MATRIX;
case 7: return LM_ERROR_SUM_OF_SQUARES_NOT_FINITE;
default: return k;
}
}
/* following struct & LMBC_DIF_XXX functions won't be necessary if a true secant
* version of LEVMAR_BC_DIF() is implemented...
*/
struct LMBC_DIF_DATA{
int ffdif; // nonzero if forward differencing is used
void (*func)(LM_REAL *p, LM_REAL *hx, int m, int n, void *adata);
LM_REAL *hx, *hxx;
void *adata;
LM_REAL delta;
};
static void LMBC_DIF_FUNC(LM_REAL *p, LM_REAL *hx, int m, int n, void *data)
{
struct LMBC_DIF_DATA *dta=(struct LMBC_DIF_DATA *)data;
/* call user-supplied function passing it the user-supplied data */
(*(dta->func))(p, hx, m, n, dta->adata);
}
static void LMBC_DIF_JACF(LM_REAL *p, LM_REAL *jac, int m, int n, void *data)
{
struct LMBC_DIF_DATA *dta=(struct LMBC_DIF_DATA *)data;
if(dta->ffdif){
/* evaluate user-supplied function at p */
(*(dta->func))(p, dta->hx, m, n, dta->adata);
LEVMAR_FDIF_FORW_JAC_APPROX(dta->func, p, dta->hx, dta->hxx, dta->delta, jac, m, n, dta->adata);
}
else
LEVMAR_FDIF_CENT_JAC_APPROX(dta->func, p, dta->hx, dta->hxx, dta->delta, jac, m, n, dta->adata);
}
/* No Jacobian version of the LEVMAR_BC_DER() function above: the Jacobian is approximated with
* the aid of finite differences (forward or central, see the comment for the opts argument)
* Ideally, this function should be implemented with a secant approach. Currently, it just calls
* LEVMAR_BC_DER()
*/
int LEVMAR_BC_DIF(
void (*func)(LM_REAL *p, LM_REAL *hx, int m, int n, void *adata), /* functional relation describing measurements. A p \in R^m yields a \hat{x} \in R^n */
LM_REAL *p, /* I/O: initial parameter estimates. On output has the estimated solution */
LM_REAL *x, /* I: measurement vector. NULL implies a zero vector */
int m, /* I: parameter vector dimension (i.e. #unknowns) */
int n, /* I: measurement vector dimension */
LM_REAL *lb, /* I: vector of lower bounds. If NULL, no lower bounds apply */
LM_REAL *ub, /* I: vector of upper bounds. If NULL, no upper bounds apply */
int itmax, /* I: maximum number of iterations */
LM_REAL opts[5], /* I: opts[0-4] = minim. options [\mu, \epsilon1, \epsilon2, \epsilon3, \delta]. Respectively the
* scale factor for initial \mu, stopping thresholds for ||J^T e||_inf, ||Dp||_2 and ||e||_2 and
* the step used in difference approximation to the Jacobian. Set to NULL for defaults to be used.
* If \delta<0, the Jacobian is approximated with central differences which are more accurate
* (but slower!) compared to the forward differences employed by default.
*/
LM_REAL info[LM_INFO_SZ],
/* O: information regarding the minimization. Set to NULL if don't care
* info[0]= ||e||_2 at initial p.
* info[1-4]=[ ||e||_2, ||J^T e||_inf, ||Dp||_2, mu/max[J^T J]_ii ], all computed at estimated p.
* info[5]= # iterations,
* info[6]=reason for terminating: 1 - stopped by small gradient J^T e
* 2 - stopped by small Dp
* 3 - stopped by itmax
* 4 - singular matrix. Restart from current p with increased mu
* 5 - no further error reduction is possible. Restart with increased mu
* 6 - stopped by small ||e||_2
* 7 - stopped by invalid (i.e. NaN or Inf) "func" values. This is a user error
* info[7]= # function evaluations
* info[8]= # Jacobian evaluations
* info[9]= # linear systems solved, i.e. # attempts for reducing error
*/
LM_REAL *work, /* working memory at least LM_BC_DIF_WORKSZ() reals large, allocated if NULL */
LM_REAL *covar, /* O: Covariance matrix corresponding to LS solution; mxm. Set to NULL if not needed. */
void *adata) /* pointer to possibly additional data, passed uninterpreted to func.
* Set to NULL if not needed
*/
{
struct LMBC_DIF_DATA data;
int ret;
//PRINT_ERROR(RCAT("\nWarning: current implementation of ", LEVMAR_BC_DIF) "() does not use a secant approach!\n\n");
data.ffdif=!opts || opts[4]>=0.0;
data.func=func;
data.hx=(LM_REAL *)malloc(2*n*sizeof(LM_REAL)); /* allocate a big chunk in one step */
if(!data.hx){
PRINT_ERROR(LCAT(LEVMAR_BC_DIF, "(): memory allocation request failed\n"));
return LM_ERROR_MEMORY_ALLOCATION_FAILURE;
}
data.hxx=data.hx+n;
data.adata=adata;
data.delta=(opts)? FABS(opts[4]) : (LM_REAL)LM_DIFF_DELTA;
ret=LEVMAR_BC_DER(LMBC_DIF_FUNC, LMBC_DIF_JACF, p, x, m, n, lb, ub, itmax, opts, info, work, covar, (void *)&data);
if(info){ /* correct the number of function calls */
if(data.ffdif)
info[7]+=info[8]*(m+1); /* each Jacobian evaluation costs m+1 function calls */
else
info[7]+=info[8]*(2*m); /* each Jacobian evaluation costs 2*m function calls */
}
free(data.hx);
return ret;
}
/* undefine everything. THIS MUST REMAIN AT THE END OF THE FILE */
#undef FUNC_STATE
#undef LNSRCH
#undef BOXPROJECT
#undef LEVMAR_BOX_CHECK
#undef LEVMAR_BC_DER
#undef LMBC_DIF_DATA
#undef LMBC_DIF_FUNC
#undef LMBC_DIF_JACF
#undef LEVMAR_BC_DIF
#undef LEVMAR_FDIF_FORW_JAC_APPROX
#undef LEVMAR_FDIF_CENT_JAC_APPROX
#undef LEVMAR_COVAR
#undef LEVMAR_TRANS_MAT_MAT_MULT
#undef LEVMAR_L2NRMXMY
#undef AX_EQ_B_LU
#undef AX_EQ_B_CHOL
#undef AX_EQ_B_QR
#undef AX_EQ_B_QRLS
#undef AX_EQ_B_SVD