/* 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 */
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
* info[7]= # function evaluations
* info[8]= # jacobian evaluations
*/
LM_REAL *work, /* working memory, allocate 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;
//fprintf(stderr, RCAT("\nWarning: current implementation of ", LEVMAR_BC_DIF) "() does not use a secant approach!\n\n");
data.func=func;
data.hx=(LM_REAL *)malloc(2*n*sizeof(LM_REAL)); /* allocate a big chunk in one step */
if(!data.hx){
fprintf(stderr, LCAT(LEVMAR_BC_DIF, "(): memory allocation request failed\n"));
exit(1);
}
data.hxx=data.hx+n;
data.adata=adata;
data.delta=(opts)? FABS(opts[4]) : (LM_REAL)LM_DIFF_DELTA; // no central differences here...
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 */
info[7]+=info[8]*(m+1); /* each jacobian evaluation costs m+1 function calls */
free(data.hx);
return ret;
}
int LEVMAR_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 */
int m, /* I: parameter vector dimension (i.e. #unknowns) */
int n, /* I: measurement vector dimension */
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
*/
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
* info[7]= # function evaluations
* info[8]= # jacobian evaluations
*/
LM_REAL *work, /* working memory, allocate 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, nfev, njev=0;
const int nm=n*m;
mu=jacTe_inf=0.0; /* -Wall */
if(n<m){
fprintf(stderr, LCAT(LEVMAR_DER, "(): cannot solve a problem with fewer measurements [%d] than unknowns [%d]\n"), n, m);
return -1;
}
if(!jacf){
fprintf(stderr, RCAT("No function specified for computing the jacobian in ", LEVMAR_DER)
RCAT("().\nIf no such function is available, use ", LEVMAR_DIF) RCAT("() rather than ", LEVMAR_DER) "()\n");
return -1;
}
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=CNST(LM_INIT_MU);
eps1=CNST(LM_STOP_THRESH);
eps2=CNST(LM_STOP_THRESH);
eps2_sq=CNST(LM_STOP_THRESH)*CNST(LM_STOP_THRESH);
eps3=CNST(LM_STOP_THRESH);
}
if(!work){
worksz=LM_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){
fprintf(stderr, LCAT(LEVMAR_DER, "(): memory allocation request failed\n"));
return -1;
}
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;
/* compute e=x - f(p) and its L2 norm */
(*func)(p, hx, m, n, adata); nfev=1;
for(i=0, p_eL2=0.0; i<n; ++i){
e[i]=tmp=x[i]-hx[i];
p_eL2+=tmp*tmp;
}
init_p_eL2=p_eL2;
for(k=stop=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 speeded 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
/* This is the straightforward way to compute J^T J, J^T e. However, due to
* its noncontinuous memory access pattern, it incures many cache misses when
* applied to large minimization problems (i.e. problems involving a large
* number of free variables and measurements), in which J is too large to
* fit in the L1 cache. For such problems, a cache-efficient blocking scheme
* is preferable.
*
* Thanks to John Nitao of Lawrence Livermore Lab for pointing out this
* performance problem.
*
* On the other hand, the straightforward algorithm is faster on small
* problems since in this case it avoids the overheads of blocking.
*/
for(i=0; i<m; ++i){
for(j=i; j<m; ++j){
int lm;
for(l=0, tmp=0.0; l<n; ++l){
lm=l*m;
tmp+=jac[lm+i]*jac[lm+j];
}
/* store tmp in the corresponding upper and lower part elements */
jacTjac[i*m+j]=jacTjac[j*m+i]=tmp;
}
/* J^T e */
for(l=0, tmp=0.0; l<n; ++l)
tmp+=jac[l*m+i]*e[l];
jacTe[i]=tmp;
}
}
else{ // this is a large problem
/* Cache efficient computation of J^T J based on blocking
*/
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 */
for(i=0, p_L2=jacTe_inf=0.0; i<m; ++i){
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%10)){
printf("Iter: %d, estimate: ", k);
for(i=0; i<m; ++i)
printf("%.9g ", p[i]);
printf("-- errors %.9g %0.9g\n", jacTe_inf, p_eL2);
}
#endif
/* check for convergence */
if((jacTe_inf <= eps1)){
Dp_L2=0.0; /* no increment for p in this case */
stop=1;
break;
}
/* compute initial damping factor */
if(k==0){
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;
}
/* determine increment using adaptive damping */
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);
//issolved=AX_EQ_B_CHOL(jacTjac, jacTe, Dp, m);
//issolved=AX_EQ_B_QR(jacTjac, jacTe, Dp, m);
//issolved=AX_EQ_B_QRLS(jacTjac, jacTe, Dp, m, m);
//issolved=AX_EQ_B_SVD(jacTjac, jacTe, Dp, m);
#else
/* use the LU included with levmar */
issolved=AX_EQ_B_LU(jacTjac, jacTe, Dp, m);
#endif /* HAVE_LAPACK */
if(issolved){
/* compute p's new estimate and ||Dp||^2 */
for(i=0, Dp_L2=0.0; i<m; ++i){
pDp[i]=p[i] + (tmp=Dp[i]);
Dp_L2+=tmp*tmp;
}
//Dp_L2=sqrt(Dp_L2);
if(Dp_L2<=eps2_sq*p_L2){ /* relative change in p is small, stop */
//if(Dp_L2<=eps2*(p_L2 + eps2)){ /* relative change in p is small, stop */
stop=2;
break;
}
if(Dp_L2>=(p_L2+eps2)/(CNST(EPSILON)*CNST(EPSILON))){ /* almost singular */
//if(Dp_L2>=(p_L2+eps2)/CNST(EPSILON)){ /* almost singular */
stop=4;
break;
}
(*func)(pDp, hx, m, n, adata); ++nfev; /* evaluate function at p + 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;
}
for(i=0, dL=0.0; i<m; ++i)
dL+=Dp[i]*(mu*Dp[i]+jacTe[i]);
dF=p_eL2-pDp_eL2;
if(dL>0.0 && dF>0.0){ /* reduction in error, increment is accepted */
tmp=(CNST(2.0)*dF/dL-CNST(1.0));
tmp=CNST(1.0)-tmp*tmp*tmp;
mu=mu*( (tmp>=CNST(ONE_THIRD))? tmp : CNST(ONE_THIRD) );
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;
break;
}
}
/* if this point is reached, either the linear system could not be solved or
* the error did not reduce; in any case, the increment must be rejected
*/
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];
} /* inner loop */
}
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;
}
/* covariance matrix */
if(covar){
LEVMAR_COVAR(jacTjac, covar, p_eL2, m, n);
}
if(freework) free(work);
return (stop!=4)? k : -1;
}
/* Similar to the LEVMAR_LEC_DER() function above, except that the jacobian is approximated
* with the aid of finite differences (forward or central, see the comment for the opts argument)
*/
int LEVMAR_LEC_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 */
int m, /* I: parameter vector dimension (i.e. #unknowns) */
int n, /* I: measurement vector dimension */
LM_REAL *A, /* I: constraints matrix, kxm */
LM_REAL *b, /* I: right hand constraints vector, kx1 */
int k, /* I: number of contraints (i.e. A's #rows) */
int itmax, /* I: maximum number of iterations */
LM_REAL opts[5], /* I: opts[0-3] = 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
* info[7]= # function evaluations
* info[8]= # jacobian evaluations
*/
LM_REAL *work, /* working memory, allocate 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 LMLEC_DATA data;
LM_REAL *ptr, *Z, *pp, *p0, *Zimm; /* Z is mxmm */
int mm, ret;
register int i, j;
register LM_REAL tmp;
LM_REAL locinfo[LM_INFO_SZ];
mm=m-k;
ptr=(LM_REAL *)malloc((2*m + m*mm + mm)*sizeof(LM_REAL));
if(!ptr){
fprintf(stderr, LCAT(LEVMAR_LEC_DIF, "(): memory allocation request failed\n"));
exit(1);
}
data.p=p;
p0=ptr;
data.c=p0+m;
data.Z=Z=data.c+m;
data.jac=NULL;
pp=data.Z+m*mm;
data.ncnstr=k;
data.func=func;
data.jacf=NULL;
data.adata=adata;
LMLEC_ELIM(A, b, data.c, NULL, Z, k, m); // compute c, Z
/* compute pp s.t. p = c + Z*pp or (Z^T Z)*pp=Z^T*(p-c)
* Due to orthogonality, Z^T Z = I and the last equation
* becomes pp=Z^T*(p-c). Also, save the starting p in p0
*/
for(i=0; i<m; ++i){
p0[i]=p[i];
p[i]-=data.c[i];
}
/* Z^T*(p-c) */
for(i=0; i<mm; ++i){
for(j=0, tmp=0.0; j<m; ++j)
tmp+=Z[j*mm+i]*p[j];
pp[i]=tmp;
}
/* compute the p corresponding to pp (i.e. c + Z*pp) and compare with p0 */
for(i=0; i<m; ++i){
Zimm=Z+i*mm;
for(j=0, tmp=data.c[i]; j<mm; ++j)
tmp+=Zimm[j]*pp[j]; // tmp+=Z[i*mm+j]*pp[j];
if(FABS(tmp-p0[i])>CNST(1E-03))
fprintf(stderr, RCAT("Warning: component %d of starting point not feasible in ", LEVMAR_LEC_DIF) "()! [%.10g reset to %.10g]\n",
i, p0[i], tmp);
}
if(!info) info=locinfo; /* make sure that LEVMAR_DIF() is called with non-null info */
/* note that covariance computation is not requested from LEVMAR_DIF() */
ret=LEVMAR_DIF(LMLEC_FUNC, pp, x, mm, n, itmax, opts, info, work, NULL, (void *)&data);
/* p=c + Z*pp */
for(i=0; i<m; ++i){
Zimm=Z+i*mm;
for(j=0, tmp=data.c[i]; j<mm; ++j)
tmp+=Zimm[j]*pp[j]; // tmp+=Z[i*mm+j]*pp[j];
p[i]=tmp;
}
/* compute the jacobian with finite differences and use it to estimate the covariance */
if(covar){
LM_REAL *hx, *wrk, *jac;
hx=(LM_REAL *)malloc((2*n+n*m)*sizeof(LM_REAL));
if(!work){
fprintf(stderr, LCAT(LEVMAR_LEC_DIF, "(): memory allocation request failed\n"));
exit(1);
}
wrk=hx+n;
jac=wrk+n;
(*func)(p, hx, m, n, adata); /* evaluate function at p */
FDIF_FORW_JAC_APPROX(func, p, hx, wrk, (LM_REAL)LM_DIFF_DELTA, jac, m, n, adata); /* compute the jacobian at p */
TRANS_MAT_MAT_MULT(jac, covar, n, m, __BLOCKSZ__); /* covar = J^T J */
LEVMAR_COVAR(covar, covar, info[1], m, n);
free(hx);
}
free(ptr);
return ret;
}
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 */
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
* info[7]= # function evaluations
* info[8]= # jacobian evaluations
*/
LM_REAL *work, /* working memory, allocate 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, nfev, njev=0;
const int nm=n*m;
/* variables for constrained LM */
struct FUNC_STATE fstate;
LM_REAL alpha=CNST(1e-4), beta=CNST(0.9), gamma=CNST(0.99995), gamma_sq=gamma*gamma, rho=CNST(1e-8);
LM_REAL t, t0;
LM_REAL steptl=CNST(1e3)*(LM_REAL)sqrt(LM_REAL_EPSILON), jacTeDp;
LM_REAL tmin=CNST(1e-12), tming=CNST(1e-18); /* minimum step length for LS and PG steps */
const LM_REAL tini=CNST(1.0); /* initial step length for LS and PG steps */
int nLMsteps=0, nLSsteps=0, nPGsteps=0, gprevtaken=0;
int numactive;
mu=jacTe_inf=t=0.0; tmin=tmin; /* -Wall */
if(n<m){
fprintf(stderr, LCAT(LEVMAR_BC_DER, "(): cannot solve a problem with fewer measurements [%d] than unknowns [%d]\n"), n, m);
exit(1);
}
if(!jacf){
fprintf(stderr, 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");
exit(1);
}
if(!BOXCHECK(lb, ub, m)){
fprintf(stderr, LCAT(LEVMAR_BC_DER, "(): at least one lower bound exceeds the upper one\n"));
exit(1);
}
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=CNST(LM_INIT_MU);
eps1=CNST(LM_STOP_THRESH);
eps2=CNST(LM_STOP_THRESH);
eps2_sq=CNST(LM_STOP_THRESH)*CNST(LM_STOP_THRESH);
eps3=CNST(LM_STOP_THRESH);
}
if(!work){
worksz=LM_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){
fprintf(stderr, LCAT(LEVMAR_BC_DER, "(): memory allocation request failed\n"));
exit(1);
}
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])
fprintf(stderr, RCAT("Warning: component %d of starting point not feasible in ", LEVMAR_BC_DER) "()! [%g projected to %g]\n",
i, p[i], pDp[i]);
/* compute e=x - f(p) and its L2 norm */
(*func)(p, hx, m, n, adata); nfev=1;
for(i=0, p_eL2=0.0; i<n; ++i){
e[i]=tmp=x[i]-hx[i];
p_eL2+=tmp*tmp;
}
init_p_eL2=p_eL2;
for(k=stop=0; k<itmax && !stop; ++k){
//printf("%d %.15g\n", k, 0.5*p_eL2);
/* 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 speeded 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
/* This is the straightforward way to compute J^T J, J^T e. However, due to
* its noncontinuous memory access pattern, it incures many cache misses when
* applied to large minimization problems (i.e. problems involving a large
* number of free variables and measurements), in which J is too large to
* fit in the L1 cache. For such problems, a cache-efficient blocking scheme
* is preferable.
*
* Thanks to John Nitao of Lawrence Livermore Lab for pointing out this
* performance problem.
*
* On the other hand, the straightforward algorithm is faster on small
* problems since in this case it avoids the overheads of blocking.
*/
for(i=0; i<m; ++i){
for(j=i; j<m; ++j){
int lm;
for(l=0, tmp=0.0; l<n; ++l){
lm=l*m;
tmp+=jac[lm+i]*jac[lm+j];
}
/* store tmp in the corresponding upper and lower part elements */
jacTjac[i*m+j]=jacTjac[j*m+i]=tmp;
}
/* J^T e */
for(l=0, tmp=0.0; l<n; ++l)
tmp+=jac[l*m+i]*e[l];
jacTe[i]=tmp;
}
}
else{ // this is a large problem
/* Cache efficient computation of J^T J based on blocking
*/
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=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 */
/* 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);
//issolved=AX_EQ_B_CHOL(jacTjac, jacTe, Dp, m);
//issolved=AX_EQ_B_QR(jacTjac, jacTe, Dp, m);
//issolved=AX_EQ_B_QRLS(jacTjac, jacTe, Dp, m, m);
//issolved=AX_EQ_B_SVD(jacTjac, jacTe, Dp, m);
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)/(CNST(EPSILON)*CNST(EPSILON))){ /* almost singular */
stop=4;
break;
}
(*func)(pDp, hx, m, n, adata); ++nfev; /* evaluate function at p + 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(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=(CNST(2.0)*dF/dL-CNST(1.0));
tmp=CNST(1.0)-tmp*tmp*tmp;
mu=mu*( (tmp>=CNST(ONE_THIRD))? tmp : 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_/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=CNST(1e3)*( (tmp>=CNST(1.0))? tmp : 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 */
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(CNST(0.5)*pDp_eL2<=CNST(0.5)*p_eL2 + t*alpha*jacTeDp) break;
if(pDp_eL2<=p_eL2 + 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=CNST(100.0)/(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 */
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;
}
for(i=0, tmp=0.0; i<m; ++i) /* compute ||g^T * Dp|| */
tmp+=jacTe[i]*Dp[i];
if(gprevtaken && pDp_eL2<=p_eL2 + CNST(2.0)*CNST(0.99999)*tmp){ /* starting t too small */
t=t0;
gprevtaken=0;
continue;
}
//if(CNST(0.5)*pDp_eL2<=CNST(0.5)*p_eL2 + alpha*tmp) break;
if(pDp_eL2<=p_eL2 + 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 */
}
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;
}
/* covariance matrix */
if(covar){
LEVMAR_COVAR(jacTjac, covar, p_eL2, m, n);
}
if(freework) free(work);
#if 0
printf("%d LM steps, %d line search, %d projected gradient\n", nLMsteps, nLSsteps, nPGsteps);
#endif
return (stop!=4)? k : -1;
}
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 */
LM_REAL *dscl, /* I: diagonal scaling constants. NULL implies no scaling */
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 */
*sp_pDp=NULL; /* dscl*p or dscl*pDp, 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), rho=LM_CNST(1e-8);
LM_REAL t, t0, 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; /* -Wall */
if(n<m){
fprintf(stderr, LCAT(LEVMAR_BC_DER, "(): cannot solve a problem with fewer measurements [%d] than unknowns [%d]\n"), n, m);
return LM_ERROR;
}
if(!jacf){
fprintf(stderr, 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;
}
if(!LEVMAR_BOX_CHECK(lb, ub, m)){
fprintf(stderr, LCAT(LEVMAR_BC_DER, "(): at least one lower bound exceeds the upper one\n"));
return LM_ERROR;
}
if(dscl){ /* check that scaling consts are valid */
for(i=m; i-->0; )
if(dscl[i]<=0.0){
fprintf(stderr, LCAT(LEVMAR_BC_DER, "(): scaling constants should be positive (scale %d: %g <= 0)\n"), i, dscl[i]);
return LM_ERROR;
}
sp_pDp=(LM_REAL *)malloc(m*sizeof(LM_REAL));
if(!sp_pDp){
fprintf(stderr, LCAT(LEVMAR_BC_DER, "(): memory allocation request failed\n"));
return LM_ERROR;
}
}
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){
fprintf(stderr, LCAT(LEVMAR_BC_DER, "(): memory allocation request failed\n"));
return LM_ERROR;
}
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.lb=lb;
fstate.ub=ub;
fstate.adata=adata;
fstate.nfev=&nfev;
/* see if starting point is within the feasible 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])
fprintf(stderr, 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;
if(dscl){
/* scale starting point and constraints */
for(i=m; i-->0; ) p[i]/=dscl[i];
BOXSCALE(lb, ub, dscl, m, 1);
}
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
*/
if(!dscl){
(*jacf)(p, jac, m, n, adata); ++njev;
}
else{
for(i=m; i-->0; ) sp_pDp[i]=p[i]*dscl[i];
(*jacf)(sp_pDp, jac, m, n, adata); ++njev;
/* compute jac*D */
for(i=n; i-->0; ){
register LM_REAL *jacim;
jacim=jac+i*m;
for(j=m; j-->0; )
jacim[j]*=dscl[j]; // jac[i*m+j]*=dscl[j];
}
}
/* 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 LM_REAL alpha, *jaclm, *jacTjacim;
/* 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; ){
jacTjacim=jacTjac+i*m;
alpha=jaclm[i]; //jac[l*m+i];
for(j=i+1; j-->0; ) /* j<=i computes lower triangular part only */
jacTjacim[j]+=jaclm[j]*alpha; //jacTjac[i*m+j]+=jac[l*m+j]*alpha
/* 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
/* 7 alternatives are available: LU, Cholesky + Cholesky with PLASMA, LDLt, 2 variants of QR decomposition and SVD.
* For matrices with dimensions of at least a few hundreds, the PLASMA implementation of Cholesky is the fastest.
* From the serial solvers, Cholesky is the fastest but might occasionally be inapplicable due to numerical round-off;
* QR is slower but more robust; SVD is the slowest but most robust; LU is quite robust but
* slower than LDLt; LDLt offers a good tradeoff between robustness and speed
*/
issolved=AX_EQ_B_BK(jacTjac, jacTe, Dp, m); ++nlss; linsolver=AX_EQ_B_BK;
//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;
#ifdef HAVE_PLASMA
//issolved=AX_EQ_B_PLASMA_CHOL(jacTjac, jacTe, Dp, m); ++nlss; linsolver=AX_EQ_B_PLASMA_CHOL;
#endif
//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;
}
if(!dscl){
(*func)(pDp, hx, m, n, adata); ++nfev; /* evaluate function at p + Dp */
}
else{
for(i=m; i-->0; ) sp_pDp[i]=pDp[i]*dscl[i];
(*func)(sp_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
/* the following test ensures that the computation of pDp_eL2 has not overflowed.
* Such an overflow does no harm here, thus it is not signalled as an error
*/
if(!LM_FINITE(pDp_eL2) && !LM_FINITE(VECNORM(hx, n))){
stop=7;
break;
}
if(pDp_eL2<=gamma*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{
tmp=LM_CNST(0.1)*pDp_eL2; /* pDp_eL2 is the new p_eL2 */
mu=(mu>=tmp)? tmp : mu;
}
#else
tmp=LM_CNST(0.1)*pDp_eL2; /* pDp_eL2 is the new p_eL2 */
mu=(mu>=tmp)? tmp : mu;
#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;
}
/* note that if the LM step is not taken, code falls through to the LM line search below */
}
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, LM_CNST(_POW_)/LM_CNST(2.0))){
/* Dp is a descent direction; do a line search along it */
#if 1
/* use Schnabel's backtracking line search; it requires fewer "func" evaluations */
{
int mxtake, iretcd;
LM_REAL stepmx, steptl=LM_CNST(1e3)*(LM_REAL)sqrt(LM_REAL_EPSILON);
tmp=(LM_REAL)sqrt(p_L2); stepmx=LM_CNST(1e3)*( (tmp>=LM_CNST(1.0))? tmp : LM_CNST(1.0) );
LNSRCH(m, p, p_eL2, jacTe, Dp, alpha, pDp, &pDp_eL2, func, &fstate,
&mxtake, &iretcd, stepmx, steptl, dscl); /* NOTE: LNSRCH() updates hx */
if(iretcd!=0 || !LM_FINITE(pDp_eL2)) 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];
BOXPROJECT(pDp, lb, ub, m); /* project to feasible set */
if(!dscl){
(*func)(pDp, hx, m, n, adata); ++nfev; /* evaluate function at p + t*Dp */
}
else{
for(i=m; i-->0; ) sp_pDp[i]=pDp[i]*dscl[i];
(*func)(sp_pDp, hx, m, n, adata); ++nfev; /* evaluate function at p + t*Dp */
}
/* 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 /* ||e(pDp)||_2 */
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 /* line search alternatives */
++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{
/* Note that this point can also be reached via a goto when LNSRCH() fails. */
gradproj:
/* jacTe has been negated above. Being a descent direction, it is next used
* to make a projected gradient 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 */
/* if the previous step was along the gradient descent, try to use the t employed in that step */
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, Dp_L2=0.0; i<m; ++i){
Dp[i]=tmp=pDp[i]-p[i];
Dp_L2+=tmp*tmp;
}
if(!dscl){
(*func)(pDp, hx, m, n, adata); ++nfev; /* evaluate function at p - t*g */
}
else{
for(i=m; i-->0; ) sp_pDp[i]=pDp[i]*dscl[i];
(*func)(sp_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
/* the following test ensures that the computation of pDp_eL2 has not overflowed.
* Such an overflow does no harm here, thus it is not signalled as an error
*/
if(!LM_FINITE(pDp_eL2) && !LM_FINITE(VECNORM(hx, n))){
stop=7;
goto breaknested;
}
/* compute ||g^T * Dp||. Note that if pDp has not been altered by projection
* (i.e. BOXPROJECT), jacTeDp=-t*||g||^2
*/
for(i=0, jacTeDp=0.0; i<m; ++i)
jacTeDp+=jacTe[i]*Dp[i];
if(gprevtaken && pDp_eL2<=p_eL2 + LM_CNST(2.0)*LM_CNST(0.99999)*jacTeDp){ /* starting t too small */
t=t0;
gprevtaken=0;
continue;
}
//if(LM_CNST(0.5)*pDp_eL2<=LM_CNST(0.5)*p_eL2 + alpha*jacTeDp) terminatePGLS;
if(pDp_eL2<=p_eL2 + LM_CNST(2.0)*alpha*jacTeDp) goto terminatePGLS;
//if(pDp_eL2<=p_eL2 - LM_CNST(2.0)*alpha/t*Dp_L2) goto terminatePGLS; // sufficient decrease condition proposed by Kelley in (5.13)
}
/* if this point is reached then the gradient line search has failed */
gprevtaken=0;
break;
terminatePGLS:
++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(dscl){ /* correct for the scaling */
for(i=m; i-->0; )
for(j=m; j-->0; )
covar[i*m+j]*=(dscl[i]*dscl[j]);
}
}
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
if(dscl){
/* scale final point and constraints */
for(i=0; i<m; ++i) p[i]*=dscl[i];
BOXSCALE(lb, ub, dscl, m, 0);
free(sp_pDp);
}
return (stop!=4 && stop!=7)? k : LM_ERROR;
}
/* Similar to the LEVMAR_BLEC_DER() function above, except that the Jacobian is approximated
* with the aid of finite differences (forward or central, see the comment for the opts argument)
*/
int LEVMAR_BLEC_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 */
LM_REAL *A, /* I: constraints matrix, kxm */
LM_REAL *b, /* I: right hand constraints vector, kx1 */
int k, /* I: number of constraints (i.e. A's #rows) */
LM_REAL *wghts, /* mx1 weights for penalty terms, defaults used if NULL */
int itmax, /* I: maximum number of iterations */
LM_REAL opts[5], /* I: opts[0-3] = 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_BLEC_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 LMBLEC_DATA data;
int ret;
register int i;
LM_REAL locinfo[LM_INFO_SZ];
if(!lb && !ub){
PRINT_ERROR(RCAT(LCAT(LEVMAR_BLEC_DIF, "(): lower and upper bounds for box constraints cannot be both NULL, use "),
LEVMAR_LEC_DIF) "() in this case!\n");
return LM_ERROR_NO_BOX_CONSTRAINTS;
}
if(!LEVMAR_BOX_CHECK(lb, ub, m)){
PRINT_ERROR(LCAT(LEVMAR_BLEC_DER, "(): at least one lower bound exceeds the upper one\n"));
return LM_ERROR_FAILED_BOX_CHECK;
}
/* measurement vector needs to be extended by m */
if(x){ /* nonzero x */
data.x=(LM_REAL *)malloc((n+m)*sizeof(LM_REAL));
if(!data.x){
PRINT_ERROR(LCAT(LEVMAR_BLEC_DER, "(): memory allocation request #1 failed\n"));
return LM_ERROR_MEMORY_ALLOCATION_FAILURE;
}
for(i=0; i<n; ++i)
data.x[i]=x[i];
for(i=n; i<n+m; ++i)
data.x[i]=0.0;
}
else
data.x=NULL;
data.w=(LM_REAL *)malloc(m*sizeof(LM_REAL) + m*sizeof(int)); /* should be arranged in that order for proper doubles alignment */
if(!data.w){
PRINT_ERROR(LCAT(LEVMAR_BLEC_DER, "(): memory allocation request #2 failed\n"));
if(data.x) free(data.x);
return LM_ERROR_MEMORY_ALLOCATION_FAILURE;
}
data.bctype=(int *)(data.w+m);
/* note: at this point, one of lb, ub are not NULL */
for(i=0; i<m; ++i){
data.w[i]=(!wghts)? __BC_WEIGHT__ : wghts[i];
if(!lb) data.bctype[i]=__BC_HIGH__;
else if(!ub) data.bctype[i]=__BC_LOW__;
else if(ub[i]!=LM_REAL_MAX && lb[i]!=LM_REAL_MIN) data.bctype[i]=__BC_INTERVAL__;
else if(lb[i]!=LM_REAL_MIN) data.bctype[i]=__BC_LOW__;
else data.bctype[i]=__BC_HIGH__;
}
data.lb=lb;
data.ub=ub;
data.func=func;
data.jacf=NULL;
data.adata=adata;
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);
if(data.x) free(data.x);
free(data.w);
return ret;
}
int LEVMAR_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 */
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
*/
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_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;
int (*linsolver)(LM_REAL *A, LM_REAL *B, LM_REAL *x, int m)=NULL;
mu=jacTe_inf=0.0; /* -Wall */
if(n<m){
fprintf(stderr, LCAT(LEVMAR_DER, "(): cannot solve a problem with fewer measurements [%d] than unknowns [%d]\n"), n, m);
return LM_ERROR;
}
if(!jacf){
fprintf(stderr, RCAT("No function specified for computing the Jacobian in ", LEVMAR_DER)
RCAT("().\nIf no such function is available, use ", LEVMAR_DIF) RCAT("() rather than ", LEVMAR_DER) "()\n");
return LM_ERROR;
}
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_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){
fprintf(stderr, LCAT(LEVMAR_DER, "(): memory allocation request failed\n"));
return LM_ERROR;
}
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;
/* 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.
*/
/* looping downwards saves a few computations */
register int l;
register LM_REAL alpha, *jaclm, *jacTjacim;
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; ){
jacTjacim=jacTjac+i*m;
alpha=jaclm[i]; //jac[l*m+i];
for(j=i+1; j-->0; ) /* j<=i computes lower triangular part only */
jacTjacim[j]+=jaclm[j]*alpha; //jacTjac[i*m+j]+=jac[l*m+j]*alpha
/* 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 */
for(i=0, p_L2=jacTe_inf=0.0; i<m; ++i){
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\n", jacTe_inf, p_eL2);
}
#endif
/* check for convergence */
if((jacTe_inf <= eps1)){
Dp_L2=0.0; /* no increment for p in this case */
stop=1;
break;
}
/* compute initial damping factor */
if(k==0){
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;
}
/* determine increment using adaptive damping */
while(1){
/* augment normal equations */
for(i=0; i<m; ++i)
jacTjac[i*m+i]+=mu;
/* solve augmented equations */
#ifdef HAVE_LAPACK
/* 7 alternatives are available: LU, Cholesky + Cholesky with PLASMA, LDLt, 2 variants of QR decomposition and SVD.
* For matrices with dimensions of at least a few hundreds, the PLASMA implementation of Cholesky is the fastest.
* From the serial solvers, Cholesky is the fastest but might occasionally be inapplicable due to numerical round-off;
* QR is slower but more robust; SVD is the slowest but most robust; LU is quite robust but
* slower than LDLt; LDLt offers a good tradeoff between robustness and speed
*/
issolved=AX_EQ_B_BK(jacTjac, jacTe, Dp, m); ++nlss; linsolver=AX_EQ_B_BK;
//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;
#ifdef HAVE_PLASMA
//issolved=AX_EQ_B_PLASMA_CHOL(jacTjac, jacTe, Dp, m); ++nlss; linsolver=AX_EQ_B_PLASMA_CHOL;
#endif
//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){
/* compute p's new estimate and ||Dp||^2 */
for(i=0, Dp_L2=0.0; i<m; ++i){
pDp[i]=p[i] + (tmp=Dp[i]);
Dp_L2+=tmp*tmp;
}
//Dp_L2=sqrt(Dp_L2);
if(Dp_L2<=eps2_sq*p_L2){ /* relative change in p is small, stop */
//if(Dp_L2<=eps2*(p_L2 + eps2)){ /* relative change in p is small, stop */
stop=2;
break;
}
if(Dp_L2>=(p_L2+eps2)/(LM_CNST(EPSILON)*LM_CNST(EPSILON))){ /* almost singular */
//if(Dp_L2>=(p_L2+eps2)/LM_CNST(EPSILON)){ /* almost singular */
stop=4;
break;
}
(*func)(pDp, hx, m, n, adata); ++nfev; /* evaluate function at p + Dp */
/* 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)){ /* sum of squares is not finite, most probably due to a user error.
* This check makes sure that the inner loop does not run indefinitely.
* Thanks to Steve Danauskas for reporting such cases
*/
stop=7;
break;
}
for(i=0, dL=0.0; i<m; ++i)
dL+=Dp[i]*(mu*Dp[i]+jacTe[i]);
dF=p_eL2-pDp_eL2;
if(dL>0.0 && dF>0.0){ /* reduction in error, increment is accepted */
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) );
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;
break;
}
}
/* if this point is reached, either the linear system could not be solved or
* the error did not reduce; in any case, the increment must be rejected
*/
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];
} /* inner loop */
}
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
return (stop!=4 && stop!=7)? k : LM_ERROR;
}
/* Secant version of the LEVMAR_DER() function above: the Jacobian is approximated with
* the aid of finite differences (forward or central, see the comment for the opts argument)
*/
int LEVMAR_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 */
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
*/
LM_REAL *work, /* working memory at least LM_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
*/
{
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 */
*wrk, /* nx1 */
*wrk2; /* nx1, used only for holding a temporary e vector and when differentiating with central differences */
int using_ffdif=1;
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, delta;
LM_REAL init_p_eL2;
int nu, nu2, stop=0, nfev, njap=0, K=(m>=10)? m: 10, updjac, updp=1, newjac;
const int nm=n*m;
int (*linsolver)(LM_REAL *A, LM_REAL *B, LM_REAL *x, int m)=NULL;
mu=jacTe_inf=p_L2=0.0; /* -Wall */
updjac=newjac=0; /* -Wall */
if(n<m){
fprintf(stderr, LCAT(LEVMAR_DIF, "(): cannot solve a problem with fewer measurements [%d] than unknowns [%d]\n"), n, m);
return LM_ERROR;
}
if(opts){
tau=opts[0];
eps1=opts[1];
eps2=opts[2];
eps2_sq=opts[2]*opts[2];
eps3=opts[3];
delta=opts[4];
if(delta<0.0){
delta=-delta; /* make positive */
using_ffdif=0; /* use central differencing */
}
}
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);
delta=LM_CNST(LM_DIFF_DELTA);
}
if(!work){
worksz=LM_DIF_WORKSZ(m, n); //4*n+4*m + n*m + m*m;
work=(LM_REAL *)malloc(worksz*sizeof(LM_REAL)); /* allocate a big chunk in one step */
if(!work){
fprintf(stderr, LCAT(LEVMAR_DIF, "(): memory allocation request failed\n"));
exit(1);
}
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;
wrk=pDp + m;
wrk2=wrk + n;
/* 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;
nu=20; /* force computation of J */
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.
* The symmetry of J^T J is again exploited for speed
*/
if((updp && nu>16) || updjac==K){ /* compute difference approximation to J */
if(using_ffdif){ /* use forward differences */
LEVMAR_FDIF_FORW_JAC_APPROX(func, p, hx, wrk, delta, jac, m, n, adata);
++njap; nfev+=m;
}
else{ /* use central differences */
LEVMAR_FDIF_CENT_JAC_APPROX(func, p, wrk, wrk2, delta, jac, m, n, adata);
++njap; nfev+=2*m;
}
nu=2; updjac=0; updp=0; newjac=1;
}
if(newjac){ /* Jacobian has changed, recompute J^T J, J^t e, etc */
newjac=0;
/* J^T J, J^T e */
if(nm<=__BLOCKSZ__SQ){ // this is a small problem
/* This is the straightforward way to compute J^T J, J^T e. However, due to
* its noncontinuous memory access pattern, it incures many cache misses when
* applied to large minimization problems (i.e. problems involving a large
* number of free variables and measurements), in which J is too large to
* fit in the L1 cache. For such problems, a cache-efficient blocking scheme
* is preferable.
*
* Thanks to John Nitao of Lawrence Livermore Lab for pointing out this
* performance problem.
*
* On the other hand, the straightforward algorithm is faster on small
* problems since in this case it avoids the overheads of blocking.
*/
for(i=0; i<m; ++i){
for(j=i; j<m; ++j){
int lm;
for(l=0, tmp=0.0; l<n; ++l){
lm=l*m;
tmp+=jac[lm+i]*jac[lm+j];
}
jacTjac[i*m+j]=jacTjac[j*m+i]=tmp;
}
/* J^T e */
for(l=0, tmp=0.0; l<n; ++l)
tmp+=jac[l*m+i]*e[l];
jacTe[i]=tmp;
}
}
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 */
for(i=0, p_L2=jacTe_inf=0.0; i<m; ++i){
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\n", jacTe_inf, p_eL2);
}
#endif
/* check for convergence */
if((jacTe_inf <= eps1)){
Dp_L2=0.0; /* no increment for p in this case */
stop=1;
break;
}
/* compute initial damping factor */
if(k==0){
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;
}
/* determine increment using adaptive damping */
/* 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); linsolver=AX_EQ_B_LU;
//issolved=AX_EQ_B_CHOL(jacTjac, jacTe, Dp, m); linsolver=AX_EQ_B_CHOL;
//issolved=AX_EQ_B_QR(jacTjac, jacTe, Dp, m); linsolver=AX_EQ_B_QR;
//issolved=AX_EQ_B_QRLS(jacTjac, jacTe, Dp, m, m); linsolver=AX_EQ_B_QRLS;
//issolved=AX_EQ_B_SVD(jacTjac, jacTe, Dp, m); linsolver=AX_EQ_B_SVD;
#else
/* use the LU included with levmar */
issolved=AX_EQ_B_LU(jacTjac, jacTe, Dp, m); linsolver=AX_EQ_B_LU;
#endif /* HAVE_LAPACK */
if(issolved){
/* compute p's new estimate and ||Dp||^2 */
for(i=0, Dp_L2=0.0; i<m; ++i){
pDp[i]=p[i] + (tmp=Dp[i]);
Dp_L2+=tmp*tmp;
}
//Dp_L2=sqrt(Dp_L2);
if(Dp_L2<=eps2_sq*p_L2){ /* relative change in p is small, stop */
//if(Dp_L2<=eps2*(p_L2 + eps2)){ /* relative change in p is small, stop */
stop=2;
break;
}
if(Dp_L2>=(p_L2+eps2)/(LM_CNST(EPSILON)*LM_CNST(EPSILON))){ /* almost singular */
//if(Dp_L2>=(p_L2+eps2)/LM_CNST(EPSILON)){ /* almost singular */
stop=4;
break;
}
(*func)(pDp, wrk, m, n, adata); ++nfev; /* evaluate function at p + Dp */
/* compute ||e(pDp)||_2 */
/* ### wrk2=x-wrk, pDp_eL2=||wrk2|| */
#if 1
pDp_eL2=LEVMAR_L2NRMXMY(wrk2, x, wrk, n);
#else
for(i=0, pDp_eL2=0.0; i<n; ++i){
wrk2[i]=tmp=x[i]-wrk[i];
pDp_eL2+=tmp*tmp;
}
#endif
if(!LM_FINITE(pDp_eL2)){ /* sum of squares is not finite, most probably due to a user error.
* This check makes sure that the loop terminates early in the case
* of invalid input. Thanks to Steve Danauskas for suggesting it
*/
stop=7;
break;
}
dF=p_eL2-pDp_eL2;
if(updp || dF>0){ /* update jac */
for(i=0; i<n; ++i){
for(l=0, tmp=0.0; l<m; ++l)
tmp+=jac[i*m+l]*Dp[l]; /* (J * Dp)[i] */
tmp=(wrk[i] - hx[i] - tmp)/Dp_L2; /* (f(p+dp)[i] - f(p)[i] - (J * Dp)[i])/(dp^T*dp) */
for(j=0; j<m; ++j)
jac[i*m+j]+=tmp*Dp[j];
}
++updjac;
newjac=1;
}
for(i=0, dL=0.0; i<m; ++i)
dL+=Dp[i]*(mu*Dp[i]+jacTe[i]);
if(dL>0.0 && dF>0.0){ /* reduction in error, increment is accepted */
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) );
nu=2;
for(i=0 ; i<m; ++i) /* update p's estimate */
p[i]=pDp[i];
for(i=0; i<n; ++i){ /* update e, hx and ||e||_2 */
e[i]=wrk2[i]; //x[i]-wrk[i];
hx[i]=wrk[i];
}
p_eL2=pDp_eL2;
updp=1;
continue;
}
}
/* if this point is reached, either the linear system could not be solved or
* the error did not reduce; in any case, the increment must be rejected
*/
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];
}
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)njap;
}
/* 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
return (stop!=4 && stop!=7)? k : LM_ERROR;
}
/*
* This function seeks the parameter vector p that best describes the measurements
* vector x under box & linear 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] and A p=b;
* A is kxm, b kx1. Note that this function DOES NOT check the satisfiability of
* the specified box and linear equation constraints.
* 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_BLEC_DIF() bellow
*
* Returns the number of iterations (>=0) if successfull, LM_ERROR if failed
*
* For more details on the algorithm implemented by this function, please refer to
* the comments in the top of this file.
*
*/
int LEVMAR_BLEC_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 */
LM_REAL *A, /* I: constraints matrix, kxm */
LM_REAL *b, /* I: right hand constraints vector, kx1 */
int k, /* I: number of constraints (i.e. A's #rows) */
LM_REAL *wghts, /* mx1 weights for penalty terms, defaults used if NULL */
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
*/
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
*/
LM_REAL *work, /* working memory at least LM_BLEC_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
*/
{
struct LMBLEC_DATA data;
int ret;
LM_REAL locinfo[LM_INFO_SZ];
register int i;
if(!jacf){
fprintf(stderr, RCAT("No function specified for computing the Jacobian in ", LEVMAR_BLEC_DER)
RCAT("().\nIf no such function is available, use ", LEVMAR_BLEC_DIF) RCAT("() rather than ", LEVMAR_BLEC_DER) "()\n");
return LM_ERROR;
}
if(!LEVMAR_BOX_CHECK(lb, ub, m)){
fprintf(stderr, LCAT(LEVMAR_BLEC_DER, "(): at least one lower bound exceeds the upper one\n"));
return LM_ERROR;
}
/* measurement vector needs to be extended by m */
if(x){ /* nonzero x */
data.x=(LM_REAL *)malloc((n+m)*sizeof(LM_REAL));
if(!data.x){
fprintf(stderr, LCAT(LEVMAR_BLEC_DER, "(): memory allocation request #1 failed\n"));
exit(1);
}
for(i=0; i<n; ++i)
data.x[i]=x[i];
for(i=n; i<n+m; ++i)
data.x[i]=0.0;
}
else
data.x=NULL;
data.w=(LM_REAL *)malloc(m*sizeof(LM_REAL) + m*sizeof(int));
if(!data.w){
fprintf(stderr, LCAT(LEVMAR_BLEC_DER, "(): memory allocation request #2 failed\n"));
exit(1);
}
data.bctype=(int *)(data.w+m);
for(i=0; i<m; ++i){
data.w[i]=(!wghts)? __BC_WEIGHT__ : wghts[i];
if(ub[i]!=LM_REAL_MAX && lb[i]!=LM_REAL_MIN) data.bctype[i]=__BC_INTERVAL__;
else if(lb[i]!=LM_REAL_MIN) data.bctype[i]=__BC_LOW__;
else data.bctype[i]=__BC_HIGH__;
}
data.lb=lb;
data.ub=ub;
data.func=func;
data.jacf=jacf;
data.adata=adata;
if(!info) info=locinfo; /* make sure that LEVMAR_LEC_DER() is called with non-null info */
ret=LEVMAR_LEC_DER(LMBLEC_FUNC, LMBLEC_JACF, p, data.x, m, n+m, A, b, k, itmax, opts, info, work, covar, (void *)&data);
if(data.x) free(data.x);
free(data.w);
return ret;
}
/* compute the Cholesky decompostion of C in W, s.t. C=W^t W and W is upper triangular */
int LEVMAR_CHOLESKY(LM_REAL *C, LM_REAL *W, int m)
{
register int i, j;
int info;
/* copy weights array C to W (in column-major order!) so that LAPACK won't destroy it */
for(i=0; i<m; i++)
for(j=0; j<m; j++)
W[i+j*m]=C[i*m+j];
/* cholesky decomposition */
POTF2("U", (int *)&m, W, (int *)&m, (int *)&info);
/* error treatment */
if(info!=0){
if(info<0){
fprintf(stderr, "LAPACK error: illegal value for argument %d of dpotf2 in %s\n", -info, LCAT(LEVMAR_DER, "()"));
exit(1);
}
else{
fprintf(stderr, "LAPACK error: the leading minor of order %d is not positive definite,\n%s()\n", info,
RCAT("and the cholesky factorization could not be completed in ", LEVMAR_CHOLESKY));
return LM_ERROR;
}
}
/* the decomposition is in the upper part of W (in column-major order!).
* copying it to the lower part and zeroing the upper transposes
* W in row-major order
*/
for(i=0; i<m; i++)
for(j=0; j<i; j++){
W[i+j*m]=W[j+i*m];
W[j+i*m]=0.0;
}
return 0;
}
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;
}
