lmbc_core.c

A sparse variant of the Levenberg-Marquardt algorithm implemented by levmar has been applied to bund

       *jacTe,      /* J^T e_i mx1 */
       *jac,        /* nxm */
       *jacTjac,    /* mxm */
       *Dp,         /* mx1 */
   *diag_jacTjac,   /* diagonal of J^T J, mx1 */
       *pDp;        /* p + Dp, mx1 */

register LM_REAL mu,  /* damping constant */
                tmp; /* mainly used in matrix & vector multiplications */
LM_REAL p_eL2, jacTe_inf, pDp_eL2; /* ||e(p)||_2, ||J^T e||_inf, ||e(p+Dp)||_2 */
LM_REAL p_L2, Dp_L2=LM_REAL_MAX, dF, dL;
LM_REAL tau, eps1, eps2, eps2_sq, eps3;
LM_REAL init_p_eL2;
int nu=2, nu2, stop=0, nfev, njev=0, nlss=0;
const int nm=n*m;

/* variables for constrained LM */
struct FUNC_STATE fstate;
LM_REAL alpha=LM_CNST(1e-4), beta=LM_CNST(0.9), gamma=LM_CNST(0.99995), gamma_sq=gamma*gamma, rho=LM_CNST(1e-8);
LM_REAL t, t0;
LM_REAL steptl=LM_CNST(1e3)*(LM_REAL)sqrt(LM_REAL_EPSILON), jacTeDp;
LM_REAL tmin=LM_CNST(1e-12), tming=LM_CNST(1e-18); /* minimum step length for LS and PG steps */
const LM_REAL tini=LM_CNST(1.0); /* initial step length for LS and PG steps */
int nLMsteps=0, nLSsteps=0, nPGsteps=0, gprevtaken=0;
int numactive;
int (*linsolver)(LM_REAL *A, LM_REAL *B, LM_REAL *x, int m)=NULL;

  mu=jacTe_inf=t=0.0;  tmin=tmin; /* -Wall */

  if(n<m){
    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(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.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, pDp[i], p[i]);

  /* compute e=x - f(p) and its L2 norm */
  (*func)(p, hx, m, n, adata); nfev=1;
  /* ### e=x-hx, p_eL2=||e|| */
#if 1
  p_eL2=LEVMAR_L2NRMXMY(e, x, hx, n);
#else
  for(i=0, p_eL2=0.0; i<n; ++i){
    e[i]=tmp=x[i]-hx[i];
    p_eL2+=tmp*tmp;
  }
#endif
  init_p_eL2=p_eL2;
  if(!LM_FINITE(p_eL2)) stop=7;

  for(k=0; k<itmax && !stop; ++k){
    /* Note that p and e have been updated at a previous iteration */

    if(p_eL2<=eps3){ /* error is small */
      stop=6;
      break;
    }

    /* Compute the Jacobian J at p,  J^T J,  J^T e,  ||J^T e||_inf and ||p||^2.
     * Since J^T J is symmetric, its computation can be sped up by computing
     * only its upper triangular part and copying it to the lower part
     */

    (*jacf)(p, jac, m, n, adata); ++njev;

    /* J^T J, J^T e */
    if(nm<__BLOCKSZ__SQ){ // this is a small problem
      /* J^T*J_ij = \sum_l J^T_il * J_lj = \sum_l J_li * J_lj.
       * Thus, the product J^T J can be computed using an outer loop for
       * l that adds J_li*J_lj to each element ij of the result. Note that
       * with this scheme, the accesses to J and JtJ are always along rows,
       * therefore induces less cache misses compared to the straightforward
       * algorithm for computing the product (i.e., l loop is innermost one).
       * A similar scheme applies to the computation of J^T e.
       * However, for large minimization problems (i.e., involving a large number
       * of unknowns and measurements) for which J/J^T J rows are too large to
       * fit in the L1 cache, even this scheme incures many cache misses. In
       * such cases, a cache-efficient blocking scheme is preferable.
       *
       * Thanks to John Nitao of Lawrence Livermore Lab for pointing out this
       * performance problem.
       *
       * Note that the non-blocking algorithm is faster on small
       * problems since in this case it avoids the overheads of blocking. 
       */
      register int l, im;
      register LM_REAL alpha, *jaclm;

      /* looping downwards saves a few computations */
      for(i=m*m; i-->0; )
        jacTjac[i]=0.0;
      for(i=m; i-->0; )
        jacTe[i]=0.0;

      for(l=n; l-->0; ){
        jaclm=jac+l*m;
        for(i=m; i-->0; ){
          im=i*m;
          alpha=jaclm[i]; //jac[l*m+i];
          for(j=i+1; j-->0; ) /* j<=i computes lower triangular part only */
            jacTjac[im+j]+=jaclm[j]*alpha; //jac[l*m+j]

          /* J^T e */
          jacTe[i]+=alpha*e[l];
        }
      }

      for(i=m; i-->0; ) /* copy to upper part */
        for(j=i+1; j<m; ++j)
          jacTjac[i*m+j]=jacTjac[j*m+i];
    }
    else{ // this is a large problem
      /* Cache efficient computation of J^T J based on blocking
       */
      LEVMAR_TRANS_MAT_MAT_MULT(jac, jacTjac, n, m);

      /* cache efficient computation of J^T e */
      for(i=0; i<m; ++i)
        jacTe[i]=0.0;

      for(i=0; i<n; ++i){
        register LM_REAL *jacrow;

        for(l=0, jacrow=jac+i*m, tmp=e[i]; l<m; ++l)
          jacTe[l]+=jacrow[l]*tmp;
      }
    }

	  /* Compute ||J^T e||_inf and ||p||^2. Note that ||J^T e||_inf
     * is computed for free (i.e. inactive) variables only. 
     * At a local minimum, if p[i]==ub[i] then g[i]>0;
     * if p[i]==lb[i] g[i]<0; otherwise g[i]=0 
     */
    for(i=j=numactive=0, p_L2=jacTe_inf=0.0; i<m; ++i){
      if(ub && p[i]==ub[i]){ ++numactive; if(jacTe[i]>0.0) ++j; }
      else if(lb && p[i]==lb[i]){ ++numactive; if(jacTe[i]<0.0) ++j; }
      else if(jacTe_inf < (tmp=FABS(jacTe[i]))) jacTe_inf=tmp;

      diag_jacTjac[i]=jacTjac[i*m+i]; /* save diagonal entries so that augmentation can be later canceled */
      p_L2+=p[i]*p[i];
    }
    //p_L2=sqrt(p_L2);

#if 0
if(!(k%100)){
  printf("Current estimate: ");
  for(i=0; i<m; ++i)
    printf("%.9g ", p[i]);
  printf("-- errors %.9g %0.9g, #active %d [%d]\n", jacTe_inf, p_eL2, numactive, j);
}
#endif

    /* check for convergence */
    if(j==numactive && (jacTe_inf <= eps1)){
      Dp_L2=0.0; /* no increment for p in this case */
      stop=1;
      break;
    }

   /* compute initial damping factor */
    if(k==0){
      if(!lb && !ub){ /* no bounds */
        for(i=0, tmp=LM_REAL_MIN; i<m; ++i)
          if(diag_jacTjac[i]>tmp) tmp=diag_jacTjac[i]; /* find max diagonal element */
        mu=tau*tmp;
      }
      else 
        mu=LM_CNST(0.5)*tau*p_eL2; /* use Kanzow's starting mu */
    }

    /* determine increment using a combination of adaptive damping, line search and projected gradient search */
    while(1){
      /* augment normal equations */
      for(i=0; i<m; ++i)
        jacTjac[i*m+i]+=mu;

      /* solve augmented equations */
#ifdef HAVE_LAPACK
      /* 5 alternatives are available: LU, Cholesky, 2 variants of QR decomposition and SVD.
       * Cholesky is the fastest but might be inaccurate; QR is slower but more accurate;
       * SVD is the slowest but most accurate; LU offers a tradeoff between accuracy and speed
       */

      issolved=AX_EQ_B_LU(jacTjac, jacTe, Dp, m); ++nlss; linsolver=AX_EQ_B_LU;
      //issolved=AX_EQ_B_CHOL(jacTjac, jacTe, Dp, m); ++nlss; linsolver=AX_EQ_B_CHOL;
      //issolved=AX_EQ_B_QR(jacTjac, jacTe, Dp, m); ++nlss; linsolver=AX_EQ_B_QR;
      //issolved=AX_EQ_B_QRLS(jacTjac, jacTe, Dp, m, m); ++nlss; linsolver=(int (*)(LM_REAL *A, LM_REAL *B, LM_REAL *x, int m))AX_EQ_B_QRLS;
      //issolved=AX_EQ_B_SVD(jacTjac, jacTe, Dp, m); ++nlss; linsolver=AX_EQ_B_SVD;

#else
      /* use the LU included with levmar */
      issolved=AX_EQ_B_LU(jacTjac, jacTe, Dp, m); ++nlss; linsolver=AX_EQ_B_LU;
#endif /* HAVE_LAPACK */

      if(issolved){
        for(i=0; i<m; ++i)
          pDp[i]=p[i] + Dp[i];

        /* compute p's new estimate and ||Dp||^2 */
        BOXPROJECT(pDp, lb, ub, m); /* project to feasible set */
        for(i=0, Dp_L2=0.0; i<m; ++i){
          Dp[i]=tmp=pDp[i]-p[i];
          Dp_L2+=tmp*tmp;
        }
        //Dp_L2=sqrt(Dp_L2);

        if(Dp_L2<=eps2_sq*p_L2){ /* relative change in p is small, stop */
          stop=2;
          break;
        }

        if(Dp_L2>=(p_L2+eps2)/(LM_CNST(EPSILON)*LM_CNST(EPSILON))){ /* almost singular */
          stop=4;
          break;
        }

        (*func)(pDp, hx, m, n, adata); ++nfev; /* evaluate function at p + Dp */
        /* ### hx=x-hx, pDp_eL2=||hx|| */
#if 1
        pDp_eL2=LEVMAR_L2NRMXMY(hx, x, hx, n);
#else
        for(i=0, pDp_eL2=0.0; i<n; ++i){ /* compute ||e(pDp)||_2 */
          hx[i]=tmp=x[i]-hx[i];
          pDp_eL2+=tmp*tmp;
        }
#endif
        if(!LM_FINITE(pDp_eL2)){
          stop=7;
          break;
        }

        if(pDp_eL2<=gamma_sq*p_eL2){
          for(i=0, dL=0.0; i<m; ++i)
            dL+=Dp[i]*(mu*Dp[i]+jacTe[i]);

#if 1
          if(dL>0.0){
            dF=p_eL2-pDp_eL2;
            tmp=(LM_CNST(2.0)*dF/dL-LM_CNST(1.0));
            tmp=LM_CNST(1.0)-tmp*tmp*tmp;
            mu=mu*( (tmp>=LM_CNST(ONE_THIRD))? tmp : LM_CNST(ONE_THIRD) );
          }
          else
            mu=(mu>=pDp_eL2)? pDp_eL2 : mu; /* pDp_eL2 is the new pDp_eL2 */
#else

          mu=(mu>=pDp_eL2)? pDp_eL2 : mu; /* pDp_eL2 is the new pDp_eL2 */
#endif

          nu=2;

          for(i=0 ; i<m; ++i) /* update p's estimate */
            p[i]=pDp[i];

          for(i=0; i<n; ++i) /* update e and ||e||_2 */
            e[i]=hx[i];
          p_eL2=pDp_eL2;