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414 lines
17 KiB
C
414 lines
17 KiB
C
/////////////////////////////////////////////////////////////////////////////////
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//
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// Levenberg - Marquardt non-linear minimization algorithm
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// Copyright (C) 2004-06 Manolis Lourakis (lourakis at ics forth gr)
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// Institute of Computer Science, Foundation for Research & Technology - Hellas
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// Heraklion, Crete, Greece.
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//
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// This program is free software; you can redistribute it and/or modify
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// it under the terms of the GNU General Public License as published by
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// the Free Software Foundation; either version 2 of the License, or
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// (at your option) any later version.
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//
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// This program is distributed in the hope that it will be useful,
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// but WITHOUT ANY WARRANTY; without even the implied warranty of
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// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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// GNU General Public License for more details.
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//
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/////////////////////////////////////////////////////////////////////////////////
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/*******************************************************************************
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* This file implements combined box and linear equation constraints.
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*
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* Note that the algorithm implementing linearly constrained minimization does
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* so by a change in parameters that transforms the original program into an
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* unconstrained one. To employ the same idea for implementing box & linear
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* constraints would require the transformation of box constraints on the
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* original parameters to box constraints for the new parameter set. This
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* being impossible, a different approach is used here for finding the minimum.
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* The trick is to remove the box constraints by augmenting the function to
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* be fitted with penalty terms and then solve the resulting problem (which
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* involves linear constrains only) with the functions in lmlec.c
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*
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* More specifically, for the constraint a<=x[i]<=b to hold, the term C[i]=
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* (2*x[i]-(a+b))/(b-a) should be within [-1, 1]. This is enforced by adding
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* the penalty term w[i]*max((C[i])^2-1, 0) to the objective function, where
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* w[i] is a large weight. In the case of constraints of the form a<=x[i],
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* the term C[i]=a-x[i] has to be non positive, thus the penalty term is
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* w[i]*max(C[i], 0). If x[i]<=b, C[i]=x[i]-b has to be non negative and
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* the penalty is w[i]*max(C[i], 0). The derivatives needed for the Jacobian
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* are as follows:
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* For the constraint a<=x[i]<=b: 4*(2*x[i]-(a+b))/(b-a)^2 if x[i] not in [a, b],
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* 0 otherwise
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* For the constraint a<=x[i]: -1 if x[i]<=a, 0 otherwise
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* For the constraint x[i]<=b: 1 if b<=x[i], 0 otherwise
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*
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* Note that for the above to work, the weights w[i] should be large enough;
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* depending on your minimization problem, the default values might need some
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* tweaking (see arg "wghts" below).
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*******************************************************************************/
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#ifndef LM_REAL // not included by lmblec.c
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#error This file should not be compiled directly!
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#endif
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#define __MAX__(x, y) (((x)>=(y))? (x) : (y))
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#define __BC_WEIGHT__ LM_CNST(1E+04)
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#define __BC_INTERVAL__ 0
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#define __BC_LOW__ 1
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#define __BC_HIGH__ 2
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/* precision-specific definitions */
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#define LEVMAR_BOX_CHECK LM_ADD_PREFIX(levmar_box_check)
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#define LMBLEC_DATA LM_ADD_PREFIX(lmblec_data)
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#define LMBLEC_FUNC LM_ADD_PREFIX(lmblec_func)
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#define LMBLEC_JACF LM_ADD_PREFIX(lmblec_jacf)
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#define LEVMAR_LEC_DER LM_ADD_PREFIX(levmar_lec_der)
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#define LEVMAR_LEC_DIF LM_ADD_PREFIX(levmar_lec_dif)
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#define LEVMAR_BLEC_DER LM_ADD_PREFIX(levmar_blec_der)
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#define LEVMAR_BLEC_DIF LM_ADD_PREFIX(levmar_blec_dif)
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#define LEVMAR_COVAR LM_ADD_PREFIX(levmar_covar)
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struct LMBLEC_DATA{
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LM_REAL *x, *lb, *ub, *w;
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int *bctype;
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void (*func)(LM_REAL *p, LM_REAL *hx, int m, int n, void *adata);
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void (*jacf)(LM_REAL *p, LM_REAL *jac, int m, int n, void *adata);
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void *adata;
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};
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/* augmented measurements */
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static void LMBLEC_FUNC(LM_REAL *p, LM_REAL *hx, int m, int n, void *adata)
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{
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struct LMBLEC_DATA *data=(struct LMBLEC_DATA *)adata;
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int nn;
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register int i, j, *typ;
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register LM_REAL *lb, *ub, *w, tmp;
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nn=n-m;
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lb=data->lb;
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ub=data->ub;
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w=data->w;
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typ=data->bctype;
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(*(data->func))(p, hx, m, nn, data->adata);
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for(i=nn, j=0; i<n; ++i, ++j){
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switch(typ[j]){
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case __BC_INTERVAL__:
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tmp=(LM_CNST(2.0)*p[j]-(lb[j]+ub[j]))/(ub[j]-lb[j]);
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hx[i]=w[j]*__MAX__(tmp*tmp-LM_CNST(1.0), LM_CNST(0.0));
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break;
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case __BC_LOW__:
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hx[i]=w[j]*__MAX__(lb[j]-p[j], LM_CNST(0.0));
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break;
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case __BC_HIGH__:
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hx[i]=w[j]*__MAX__(p[j]-ub[j], LM_CNST(0.0));
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break;
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}
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}
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}
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/* augmented Jacobian */
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static void LMBLEC_JACF(LM_REAL *p, LM_REAL *jac, int m, int n, void *adata)
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{
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struct LMBLEC_DATA *data=(struct LMBLEC_DATA *)adata;
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int nn, *typ;
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register int i, j;
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register LM_REAL *lb, *ub, *w, tmp;
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nn=n-m;
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lb=data->lb;
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ub=data->ub;
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w=data->w;
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typ=data->bctype;
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(*(data->jacf))(p, jac, m, nn, data->adata);
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/* clear all extra rows */
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for(i=nn*m; i<n*m; ++i)
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jac[i]=0.0;
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for(i=nn, j=0; i<n; ++i, ++j){
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switch(typ[j]){
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case __BC_INTERVAL__:
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if(lb[j]<=p[j] && p[j]<=ub[j])
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continue; // corresp. jac element already 0
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/* out of interval */
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tmp=ub[j]-lb[j];
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tmp=LM_CNST(4.0)*(LM_CNST(2.0)*p[j]-(lb[j]+ub[j]))/(tmp*tmp);
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jac[i*m+j]=w[j]*tmp;
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break;
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case __BC_LOW__: // (lb[j]<=p[j])? 0.0 : -1.0;
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if(lb[j]<=p[j])
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continue; // corresp. jac element already 0
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/* smaller than lower bound */
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jac[i*m+j]=-w[j];
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break;
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case __BC_HIGH__: // (p[j]<=ub[j])? 0.0 : 1.0;
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if(p[j]<=ub[j])
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continue; // corresp. jac element already 0
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/* greater than upper bound */
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jac[i*m+j]=w[j];
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break;
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}
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}
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}
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/*
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* This function seeks the parameter vector p that best describes the measurements
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* vector x under box & linear constraints.
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* More precisely, given a vector function func : R^m --> R^n with n>=m,
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* it finds p s.t. func(p) ~= x, i.e. the squared second order (i.e. L2) norm of
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* e=x-func(p) is minimized under the constraints lb[i]<=p[i]<=ub[i] and A p=b;
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* A is kxm, b kx1. Note that this function DOES NOT check the satisfiability of
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* the specified box and linear equation constraints.
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* If no lower bound constraint applies for p[i], use -DBL_MAX/-FLT_MAX for lb[i];
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* If no upper bound constraint applies for p[i], use DBL_MAX/FLT_MAX for ub[i].
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*
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* This function requires an analytic Jacobian. In case the latter is unavailable,
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* use LEVMAR_BLEC_DIF() bellow
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*
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* Returns the number of iterations (>=0) if successful, LM_ERROR if failed
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*
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* For more details on the algorithm implemented by this function, please refer to
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* the comments in the top of this file.
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*
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*/
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int LEVMAR_BLEC_DER(
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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 */
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void (*jacf)(LM_REAL *p, LM_REAL *j, int m, int n, void *adata), /* function to evaluate the Jacobian \part x / \part p */
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LM_REAL *p, /* I/O: initial parameter estimates. On output has the estimated solution */
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LM_REAL *x, /* I: measurement vector. NULL implies a zero vector */
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int m, /* I: parameter vector dimension (i.e. #unknowns) */
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int n, /* I: measurement vector dimension */
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LM_REAL *lb, /* I: vector of lower bounds. If NULL, no lower bounds apply */
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LM_REAL *ub, /* I: vector of upper bounds. If NULL, no upper bounds apply */
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LM_REAL *A, /* I: constraints matrix, kxm */
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LM_REAL *b, /* I: right hand constraints vector, kx1 */
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int k, /* I: number of constraints (i.e. A's #rows) */
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LM_REAL *wghts, /* mx1 weights for penalty terms, defaults used if NULL */
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int itmax, /* I: maximum number of iterations */
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LM_REAL opts[4], /* I: minim. options [\mu, \epsilon1, \epsilon2, \epsilon3]. Respectively the scale factor for initial \mu,
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* stopping thresholds for ||J^T e||_inf, ||Dp||_2 and ||e||_2. Set to NULL for defaults to be used
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*/
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LM_REAL info[LM_INFO_SZ],
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/* O: information regarding the minimization. Set to NULL if don't care
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* info[0]= ||e||_2 at initial p.
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* info[1-4]=[ ||e||_2, ||J^T e||_inf, ||Dp||_2, mu/max[J^T J]_ii ], all computed at estimated p.
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* info[5]= # iterations,
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* info[6]=reason for terminating: 1 - stopped by small gradient J^T e
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* 2 - stopped by small Dp
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* 3 - stopped by itmax
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* 4 - singular matrix. Restart from current p with increased mu
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* 5 - no further error reduction is possible. Restart with increased mu
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* 6 - stopped by small ||e||_2
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* 7 - stopped by invalid (i.e. NaN or Inf) "func" values. This is a user error
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* info[7]= # function evaluations
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* info[8]= # Jacobian evaluations
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* info[9]= # linear systems solved, i.e. # attempts for reducing error
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*/
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LM_REAL *work, /* working memory at least LM_BLEC_DER_WORKSZ() reals large, allocated if NULL */
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LM_REAL *covar, /* O: Covariance matrix corresponding to LS solution; mxm. Set to NULL if not needed. */
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void *adata) /* pointer to possibly additional data, passed uninterpreted to func & jacf.
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* Set to NULL if not needed
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*/
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{
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struct LMBLEC_DATA data;
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int ret;
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LM_REAL locinfo[LM_INFO_SZ];
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register int i;
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if(!jacf){
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fprintf(stderr, RCAT("No function specified for computing the Jacobian in ", LEVMAR_BLEC_DER)
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RCAT("().\nIf no such function is available, use ", LEVMAR_BLEC_DIF) RCAT("() rather than ", LEVMAR_BLEC_DER) "()\n");
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return LM_ERROR;
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}
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if(!lb && !ub){
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fprintf(stderr, RCAT(LCAT(LEVMAR_BLEC_DER, "(): lower and upper bounds for box constraints cannot be both NULL, use "),
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LEVMAR_LEC_DER) "() in this case!\n");
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return LM_ERROR;
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}
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if(!LEVMAR_BOX_CHECK(lb, ub, m)){
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fprintf(stderr, LCAT(LEVMAR_BLEC_DER, "(): at least one lower bound exceeds the upper one\n"));
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return LM_ERROR;
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}
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/* measurement vector needs to be extended by m */
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if(x){ /* nonzero x */
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data.x=(LM_REAL *)malloc((n+m)*sizeof(LM_REAL));
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if(!data.x){
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fprintf(stderr, LCAT(LEVMAR_BLEC_DER, "(): memory allocation request #1 failed\n"));
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return LM_ERROR;
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}
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for(i=0; i<n; ++i)
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data.x[i]=x[i];
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for(i=n; i<n+m; ++i)
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data.x[i]=0.0;
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}
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else
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data.x=NULL;
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data.w=(LM_REAL *)malloc(m*sizeof(LM_REAL) + m*sizeof(int)); /* should be arranged in that order for proper doubles alignment */
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if(!data.w){
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fprintf(stderr, LCAT(LEVMAR_BLEC_DER, "(): memory allocation request #2 failed\n"));
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if(data.x) free(data.x);
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return LM_ERROR;
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}
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data.bctype=(int *)(data.w+m);
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/* note: at this point, one of lb, ub are not NULL */
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for(i=0; i<m; ++i){
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data.w[i]=(!wghts)? __BC_WEIGHT__ : wghts[i];
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if(!lb) data.bctype[i]=__BC_HIGH__;
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else if(!ub) data.bctype[i]=__BC_LOW__;
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else if(ub[i]!=LM_REAL_MAX && lb[i]!=LM_REAL_MIN) data.bctype[i]=__BC_INTERVAL__;
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else if(lb[i]!=LM_REAL_MIN) data.bctype[i]=__BC_LOW__;
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else data.bctype[i]=__BC_HIGH__;
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}
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data.lb=lb;
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data.ub=ub;
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data.func=func;
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data.jacf=jacf;
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data.adata=adata;
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if(!info) info=locinfo; /* make sure that LEVMAR_LEC_DER() is called with non-null info */
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ret=LEVMAR_LEC_DER(LMBLEC_FUNC, LMBLEC_JACF, p, data.x, m, n+m, A, b, k, itmax, opts, info, work, covar, (void *)&data);
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if(data.x) free(data.x);
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free(data.w);
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return ret;
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}
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/* Similar to the LEVMAR_BLEC_DER() function above, except that the Jacobian is approximated
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* with the aid of finite differences (forward or central, see the comment for the opts argument)
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*/
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int LEVMAR_BLEC_DIF(
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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 */
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LM_REAL *p, /* I/O: initial parameter estimates. On output has the estimated solution */
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LM_REAL *x, /* I: measurement vector. NULL implies a zero vector */
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int m, /* I: parameter vector dimension (i.e. #unknowns) */
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int n, /* I: measurement vector dimension */
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LM_REAL *lb, /* I: vector of lower bounds. If NULL, no lower bounds apply */
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LM_REAL *ub, /* I: vector of upper bounds. If NULL, no upper bounds apply */
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LM_REAL *A, /* I: constraints matrix, kxm */
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LM_REAL *b, /* I: right hand constraints vector, kx1 */
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int k, /* I: number of constraints (i.e. A's #rows) */
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LM_REAL *wghts, /* mx1 weights for penalty terms, defaults used if NULL */
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int itmax, /* I: maximum number of iterations */
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LM_REAL opts[5], /* I: opts[0-3] = minim. options [\mu, \epsilon1, \epsilon2, \epsilon3, \delta]. Respectively the
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* scale factor for initial \mu, stopping thresholds for ||J^T e||_inf, ||Dp||_2 and ||e||_2 and
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* the step used in difference approximation to the Jacobian. Set to NULL for defaults to be used.
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* If \delta<0, the Jacobian is approximated with central differences which are more accurate
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* (but slower!) compared to the forward differences employed by default.
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*/
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LM_REAL info[LM_INFO_SZ],
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/* O: information regarding the minimization. Set to NULL if don't care
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* info[0]= ||e||_2 at initial p.
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* info[1-4]=[ ||e||_2, ||J^T e||_inf, ||Dp||_2, mu/max[J^T J]_ii ], all computed at estimated p.
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* info[5]= # iterations,
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* info[6]=reason for terminating: 1 - stopped by small gradient J^T e
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* 2 - stopped by small Dp
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* 3 - stopped by itmax
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* 4 - singular matrix. Restart from current p with increased mu
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* 5 - no further error reduction is possible. Restart with increased mu
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* 6 - stopped by small ||e||_2
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* 7 - stopped by invalid (i.e. NaN or Inf) "func" values. This is a user error
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* info[7]= # function evaluations
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* info[8]= # Jacobian evaluations
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* info[9]= # linear systems solved, i.e. # attempts for reducing error
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*/
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LM_REAL *work, /* working memory at least LM_BLEC_DIF_WORKSZ() reals large, allocated if NULL */
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LM_REAL *covar, /* O: Covariance matrix corresponding to LS solution; mxm. Set to NULL if not needed. */
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void *adata) /* pointer to possibly additional data, passed uninterpreted to func.
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* Set to NULL if not needed
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*/
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{
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struct LMBLEC_DATA data;
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int ret;
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register int i;
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LM_REAL locinfo[LM_INFO_SZ];
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if(!lb && !ub){
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fprintf(stderr, RCAT(LCAT(LEVMAR_BLEC_DIF, "(): lower and upper bounds for box constraints cannot be both NULL, use "),
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LEVMAR_LEC_DIF) "() in this case!\n");
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return LM_ERROR;
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}
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if(!LEVMAR_BOX_CHECK(lb, ub, m)){
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fprintf(stderr, LCAT(LEVMAR_BLEC_DER, "(): at least one lower bound exceeds the upper one\n"));
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return LM_ERROR;
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}
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/* measurement vector needs to be extended by m */
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if(x){ /* nonzero x */
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data.x=(LM_REAL *)malloc((n+m)*sizeof(LM_REAL));
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if(!data.x){
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fprintf(stderr, LCAT(LEVMAR_BLEC_DER, "(): memory allocation request #1 failed\n"));
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return LM_ERROR;
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}
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for(i=0; i<n; ++i)
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data.x[i]=x[i];
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for(i=n; i<n+m; ++i)
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data.x[i]=0.0;
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}
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else
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data.x=NULL;
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data.w=(LM_REAL *)malloc(m*sizeof(LM_REAL) + m*sizeof(int)); /* should be arranged in that order for proper doubles alignment */
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if(!data.w){
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fprintf(stderr, LCAT(LEVMAR_BLEC_DER, "(): memory allocation request #2 failed\n"));
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if(data.x) free(data.x);
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return LM_ERROR;
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}
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data.bctype=(int *)(data.w+m);
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/* note: at this point, one of lb, ub are not NULL */
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for(i=0; i<m; ++i){
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data.w[i]=(!wghts)? __BC_WEIGHT__ : wghts[i];
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if(!lb) data.bctype[i]=__BC_HIGH__;
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else if(!ub) data.bctype[i]=__BC_LOW__;
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else if(ub[i]!=LM_REAL_MAX && lb[i]!=LM_REAL_MIN) data.bctype[i]=__BC_INTERVAL__;
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else if(lb[i]!=LM_REAL_MIN) data.bctype[i]=__BC_LOW__;
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else data.bctype[i]=__BC_HIGH__;
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}
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data.lb=lb;
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data.ub=ub;
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data.func=func;
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data.jacf=NULL;
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data.adata=adata;
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if(!info) info=locinfo; /* make sure that LEVMAR_LEC_DIF() is called with non-null info */
|
|
ret=LEVMAR_LEC_DIF(LMBLEC_FUNC, p, data.x, m, n+m, A, b, k, itmax, opts, info, work, covar, (void *)&data);
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|
|
|
if(data.x) free(data.x);
|
|
free(data.w);
|
|
|
|
return ret;
|
|
}
|
|
|
|
/* undefine all. THIS MUST REMAIN AT THE END OF THE FILE */
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|
#undef LEVMAR_BOX_CHECK
|
|
#undef LMBLEC_DATA
|
|
#undef LMBLEC_FUNC
|
|
#undef LMBLEC_JACF
|
|
#undef LEVMAR_COVAR
|
|
#undef LEVMAR_LEC_DER
|
|
#undef LEVMAR_LEC_DIF
|
|
#undef LEVMAR_BLEC_DER
|
|
#undef LEVMAR_BLEC_DIF
|