static void LMBC_DIF_JACF(LM_REAL *p, LM_REAL *jac, int m, int n, void *data) { struct LMBC_DIF_DATA *dta=(struct LMBC_DIF_DATA *)data; if(dta->ffdif){ /* evaluate user-supplied function at p */ (*(dta->func))(p, dta->hx, m, n, dta->adata); LEVMAR_FDIF_FORW_JAC_APPROX(dta->func, p, dta->hx, dta->hxx, dta->delta, jac, m, n, dta->adata); } else LEVMAR_FDIF_CENT_JAC_APPROX(dta->func, p, dta->hx, dta->hxx, dta->delta, jac, m, n, dta->adata); }
/* 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 * info[9]= # linear systems solved, i.e. # attempts for reducing error */ 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, nlss=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")); 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; 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 /* J^T*J_ij = \sum_l J^T_il * J_lj = \sum_l J_li * J_lj. * Thus, the product J^T J can be computed using an outer loop for * l that adds J_li*J_lj to each element ij of the result. Note that * with this scheme, the accesses to J and JtJ are always along rows, * therefore induces less cache misses compared to the straightforward * algorithm for computing the product (i.e., l loop is innermost one). * A similar scheme applies to the computation of J^T e. * However, for large minimization problems (i.e., involving a large number * of unknowns and measurements) for which J/J^T J rows are too large to * fit in the L1 cache, even this scheme incures many cache misses. In * such cases, a cache-efficient blocking scheme is preferable. * * Thanks to John Nitao of Lawrence Livermore Lab for pointing out this * performance problem. * * Note that the non-blocking algorithm is faster on small * problems since in this case it avoids the overheads of blocking. */ register int l; 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 */ 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 /* 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, 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; 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; }