misc_core.c

levenberg marquit算法的.C源码

        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;
}

#ifdef HAVE_LAPACK
/*
 * This function computes the pseudoinverse of a square matrix A
 * into B using SVD. A and B can coincide
 * 
 * The function returns 0 in case of error (e.g. A is singular),
 * the rank of A if successfull
 *
 * A, B are mxm
 *
 */
static int LEVMAR_PSEUDOINVERSE(LM_REAL *A, LM_REAL *B, int m)
{
LM_REAL *buf=NULL;
int buf_sz=0;
static LM_REAL eps=CNST(-1.0);

register int i, j;
LM_REAL *a, *u, *s, *vt, *work;
int a_sz, u_sz, s_sz, vt_sz, tot_sz;
LM_REAL thresh, one_over_denom;
int info, rank, worksz, *iwork, iworksz;
   
  /* calculate required memory size */
  worksz=16*m; /* more than needed */
  iworksz=8*m;
  a_sz=m*m;
  u_sz=m*m; s_sz=m; vt_sz=m*m;

  tot_sz=iworksz*sizeof(int) + (a_sz + u_sz + s_sz + vt_sz + worksz)*sizeof(LM_REAL);

    buf_sz=tot_sz;
    buf=(LM_REAL *)malloc(buf_sz);
    if(!buf){
      fprintf(stderr, RCAT("memory allocation in ", LEVMAR_PSEUDOINVERSE) "() failed!\n");
      exit(1);
    }

  iwork=(int *)buf;
  a=(LM_REAL *)(iwork+iworksz);
  /* store A (column major!) into a */
  for(i=0; i<m; i++)
    for(j=0; j<m; j++)
      a[i+j*m]=A[i*m+j];

  u=a + a_sz;
  s=u+u_sz;
  vt=s+s_sz;
  work=vt+vt_sz;

  /* SVD decomposition of A */
  GESVD("A", "A", (int *)&m, (int *)&m, a, (int *)&m, s, u, (int *)&m, vt, (int *)&m, work, (int *)&worksz, &info);
  //GESDD("A", (int *)&m, (int *)&m, a, (int *)&m, s, u, (int *)&m, vt, (int *)&m, work, (int *)&worksz, iwork, &info);

  /* error treatment */
  if(info!=0){
    if(info<0){
      fprintf(stderr, RCAT(RCAT(RCAT("LAPACK error: illegal value for argument %d of ", GESVD), "/" GESDD) " in ", LEVMAR_PSEUDOINVERSE) "()\n", -info);
      exit(1);
    }
    else{
      fprintf(stderr, RCAT("LAPACK error: dgesdd (dbdsdc)/dgesvd (dbdsqr) failed to converge in ", LEVMAR_PSEUDOINVERSE) "() [info=%d]\n", info);
      free(buf);

      return 0;
    }
  }

  if(eps<0.0){
    LM_REAL aux;

    /* compute machine epsilon */
    for(eps=CNST(1.0); aux=eps+CNST(1.0), aux-CNST(1.0)>0.0; eps*=CNST(0.5))
                                          ;
    eps*=CNST(2.0);
  }

  /* compute the pseudoinverse in B */
	for(i=0; i<a_sz; i++) B[i]=0.0; /* initialize to zero */
  for(rank=0, thresh=eps*s[0]; rank<m && s[rank]>thresh; rank++){
    one_over_denom=CNST(1.0)/s[rank];

    for(j=0; j<m; j++)
      for(i=0; i<m; i++)
        B[i*m+j]+=vt[rank+i*m]*u[j+rank*m]*one_over_denom;
  }

  free(buf);

	return rank;
}
#else // no LAPACK

/*
 * This function computes the inverse of A in B. A and B can coincide
 *
 * The function employs LAPACK-free LU decomposition of A to solve m linear
 * systems A*B_i=I_i, where B_i and I_i are the i-th columns of B and I.
 *
 * A and B are mxm
 *
 * The function returns 0 in case of error,
 * 1 if successfull
 *
 */
static int LEVMAR_LUINVERSE(LM_REAL *A, LM_REAL *B, int m)
{
void *buf=NULL;
int buf_sz=0;

register int i, j, k, l;
int *idx, maxi=-1, idx_sz, a_sz, x_sz, work_sz, tot_sz;
LM_REAL *a, *x, *work, max, sum, tmp;

  /* calculate required memory size */
  idx_sz=m;
  a_sz=m*m;
  x_sz=m;
  work_sz=m;
  tot_sz=idx_sz*sizeof(int) + (a_sz+x_sz+work_sz)*sizeof(LM_REAL);

  buf_sz=tot_sz;
  buf=(void *)malloc(tot_sz);
  if(!buf){
    fprintf(stderr, RCAT("memory allocation in ", LEVMAR_LUINVERSE) "() failed!\n");
    exit(1);
  }

  idx=(int *)buf;
  a=(LM_REAL *)(idx + idx_sz);
  x=a + a_sz;
  work=x + x_sz;

  /* avoid destroying A by copying it to a */
  for(i=0; i<a_sz; ++i) a[i]=A[i];

  /* compute the LU decomposition of a row permutation of matrix a; the permutation itself is saved in idx[] */
	for(i=0; i<m; ++i){
		max=0.0;
		for(j=0; j<m; ++j)
			if((tmp=FABS(a[i*m+j]))>max)
        max=tmp;
		  if(max==0.0){
        fprintf(stderr, RCAT("Singular matrix A in ", LEVMAR_LUINVERSE) "()!\n");
        free(buf);

        return 0;
      }
		  work[i]=CNST(1.0)/max;
	}

	for(j=0; j<m; ++j){
		for(i=0; i<j; ++i){
			sum=a[i*m+j];
			for(k=0; k<i; ++k)
        sum-=a[i*m+k]*a[k*m+j];
			a[i*m+j]=sum;
		}
		max=0.0;
		for(i=j; i<m; ++i){
			sum=a[i*m+j];
			for(k=0; k<j; ++k)
        sum-=a[i*m+k]*a[k*m+j];
			a[i*m+j]=sum;
			if((tmp=work[i]*FABS(sum))>=max){
				max=tmp;
				maxi=i;
			}
		}
		if(j!=maxi){
			for(k=0; k<m; ++k){
				tmp=a[maxi*m+k];
				a[maxi*m+k]=a[j*m+k];
				a[j*m+k]=tmp;
			}
			work[maxi]=work[j];
		}
		idx[j]=maxi;
		if(a[j*m+j]==0.0)
      a[j*m+j]=LM_REAL_EPSILON;
		if(j!=m-1){
			tmp=CNST(1.0)/(a[j*m+j]);
			for(i=j+1; i<m; ++i)
        a[i*m+j]*=tmp;
		}
	}

  /* The decomposition has now replaced a. Solve the m linear systems using
   * forward and back substitution
   */
  for(l=0; l<m; ++l){
    for(i=0; i<m; ++i) x[i]=0.0;
    x[l]=CNST(1.0);

	  for(i=k=0; i<m; ++i){
		  j=idx[i];
		  sum=x[j];
		  x[j]=x[i];
		  if(k!=0)
			  for(j=k-1; j<i; ++j)
          sum-=a[i*m+j]*x[j];
		  else
        if(sum!=0.0)
			    k=i+1;
		  x[i]=sum;
	  }

	  for(i=m-1; i>=0; --i){
		  sum=x[i];
		  for(j=i+1; j<m; ++j)
        sum-=a[i*m+j]*x[j];
		  x[i]=sum/a[i*m+i];
	  }

    for(i=0; i<m; ++i)
      B[i*m+l]=x[i];
  }

  free(buf);

  return 1;
}
#endif /* HAVE_LAPACK */

/*
 * This function computes in C the covariance matrix corresponding to a least
 * squares fit. JtJ is the approximate Hessian at the solution (i.e. J^T*J, where
 * J is the jacobian at the solution), sumsq is the sum of squared residuals
 * (i.e. goodnes of fit) at the solution, m is the number of parameters (variables)
 * and n the number of observations. JtJ can coincide with C.
 * 
 * if JtJ is of full rank, C is computed as sumsq/(n-m)*(JtJ)^-1
 * otherwise and if LAPACK is available, C=sumsq/(n-r)*(JtJ)^+
 * where r is JtJ's rank and ^+ denotes the pseudoinverse
 * The diagonal of C is made up from the estimates of the variances
 * of the estimated regression coefficients.
 * See the documentation of routine E04YCF from the NAG fortran lib
 *
 * The function returns the rank of JtJ if successful, 0 on error
 *
 * A and C are mxm
 *
 */
int LEVMAR_COVAR(LM_REAL *JtJ, LM_REAL *C, LM_REAL sumsq, int m, int n)
{
register int i;
int rnk;
LM_REAL fact;

#ifdef HAVE_LAPACK
   rnk=LEVMAR_PSEUDOINVERSE(JtJ, C, m);
   if(!rnk) return 0;
#else
#ifdef _MSC_VER
#pragma message("LAPACK not available, LU will be used for matrix inversion when computing the covariance; this might be unstable at times")
#else
#warning LAPACK not available, LU will be used for matrix inversion when computing the covariance; this might be unstable at times
#endif // _MSC_VER

   rnk=LEVMAR_LUINVERSE(JtJ, C, m);
   if(!rnk) return 0;

   rnk=m; /* assume full rank */
#endif /* HAVE_LAPACK */

   fact=sumsq/(LM_REAL)(n-rnk);
   for(i=0; i<m*m; ++i)
     C[i]*=fact;

   return rnk;
}

/* undefine everything. THIS MUST REMAIN AT THE END OF THE FILE */
#undef LEVMAR_PSEUDOINVERSE
#undef LEVMAR_LUINVERSE
#undef LEVMAR_COVAR
#undef LEVMAR_CHKJAC
#undef FDIF_FORW_JAC_APPROX
#undef FDIF_CENT_JAC_APPROX
#undef TRANS_MAT_MAT_MULT