465 lines
12 KiB
C
465 lines
12 KiB
C
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#include <string.h>
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#include <float.h>
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#include <math.h>
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#include <glib.h>
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#include <assert.h>
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/* See Golub and Reinsch,
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* "Handbook for Automatic Computation vol II - Linear Algebra",
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* Springer, 1971
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*/
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#define MAX_ITERATION_COUNT 30
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/* Perform Householder reduction to bidiagonal form
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*
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* Input: Matrix A of size nrows x ncols
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*
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* Output: Matrices and vectors such that
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* A = U*Bidiag(diagonal, superdiagonal)*Vt
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*
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* All matrices are allocated by the caller
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*
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* Sizes:
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* A, U: nrows x ncols
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* diagonal, superdiagonal: ncols
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* V: ncols x ncols
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*/
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static void
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householder_reduction (double *A,
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int nrows,
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int ncols,
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double *U,
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double *V,
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double *diagonal,
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double *superdiagonal)
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{
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int i, j, k, ip1;
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double s, s2, si, scale;
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double *pu, *pui, *pv, *pvi;
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double half_norm_squared;
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assert (nrows >= 2);
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assert (ncols >= 2);
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memcpy (U, A, sizeof (double) * nrows * ncols);
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diagonal[0] = 0.0;
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s = 0.0;
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scale = 0.0;
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for (i = 0, pui = U, ip1 = 1;
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i < ncols;
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pui += ncols, i++, ip1++)
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{
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superdiagonal[i] = scale * s;
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for (j = i, pu = pui, scale = 0.0;
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j < nrows;
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j++, pu += ncols)
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scale += fabs( *(pu + i) );
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if (scale > 0.0)
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{
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for (j = i, pu = pui, s2 = 0.0; j < nrows; j++, pu += ncols)
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{
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*(pu + i) /= scale;
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s2 += *(pu + i) * *(pu + i);
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}
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s = *(pui + i) < 0.0 ? sqrt (s2) : -sqrt (s2);
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half_norm_squared = *(pui + i) * s - s2;
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*(pui + i) -= s;
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for (j = ip1; j < ncols; j++)
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{
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for (k = i, si = 0.0, pu = pui; k < nrows; k++, pu += ncols)
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si += *(pu + i) * *(pu + j);
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si /= half_norm_squared;
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for (k = i, pu = pui; k < nrows; k++, pu += ncols)
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*(pu + j) += si * *(pu + i);
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}
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}
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for (j = i, pu = pui; j < nrows; j++, pu += ncols)
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*(pu + i) *= scale;
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diagonal[i] = s * scale;
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s = 0.0;
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scale = 0.0;
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if (i >= nrows || i == ncols - 1)
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continue;
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for (j = ip1; j < ncols; j++)
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scale += fabs (*(pui + j));
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if (scale > 0.0)
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{
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for (j = ip1, s2 = 0.0; j < ncols; j++)
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{
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*(pui + j) /= scale;
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s2 += *(pui + j) * *(pui + j);
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}
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s = *(pui + ip1) < 0.0 ? sqrt (s2) : -sqrt (s2);
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half_norm_squared = *(pui + ip1) * s - s2;
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*(pui + ip1) -= s;
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for (k = ip1; k < ncols; k++)
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superdiagonal[k] = *(pui + k) / half_norm_squared;
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if (i < (nrows - 1))
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{
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for (j = ip1, pu = pui + ncols; j < nrows; j++, pu += ncols)
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{
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for (k = ip1, si = 0.0; k < ncols; k++)
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si += *(pui + k) * *(pu + k);
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for (k = ip1; k < ncols; k++)
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*(pu + k) += si * superdiagonal[k];
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}
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}
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for (k = ip1; k < ncols; k++)
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*(pui + k) *= scale;
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}
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}
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pui = U + ncols * (ncols - 2);
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pvi = V + ncols * (ncols - 1);
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*(pvi + ncols - 1) = 1.0;
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s = superdiagonal[ncols - 1];
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pvi -= ncols;
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for (i = ncols - 2, ip1 = ncols - 1;
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i >= 0;
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i--, pui -= ncols, pvi -= ncols, ip1--)
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{
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if (s != 0.0)
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{
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pv = pvi + ncols;
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for (j = ip1; j < ncols; j++, pv += ncols)
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*(pv + i) = ( *(pui + j) / *(pui + ip1) ) / s;
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for (j = ip1; j < ncols; j++)
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{
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si = 0.0;
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for (k = ip1, pv = pvi + ncols; k < ncols; k++, pv += ncols)
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si += *(pui + k) * *(pv + j);
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for (k = ip1, pv = pvi + ncols; k < ncols; k++, pv += ncols)
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*(pv + j) += si * *(pv + i);
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}
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}
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pv = pvi + ncols;
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for (j = ip1; j < ncols; j++, pv += ncols)
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{
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*(pvi + j) = 0.0;
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*(pv + i) = 0.0;
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}
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*(pvi + i) = 1.0;
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s = superdiagonal[i];
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}
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pui = U + ncols * (ncols - 1);
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for (i = ncols - 1, ip1 = ncols;
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i >= 0;
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ip1 = i, i--, pui -= ncols)
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{
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s = diagonal[i];
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for (j = ip1; j < ncols; j++)
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*(pui + j) = 0.0;
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if (s != 0.0)
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{
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for (j = ip1; j < ncols; j++)
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{
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si = 0.0;
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pu = pui + ncols;
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for (k = ip1; k < nrows; k++, pu += ncols)
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si += *(pu + i) * *(pu + j);
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si = (si / *(pui + i)) / s;
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for (k = i, pu = pui; k < nrows; k++, pu += ncols)
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*(pu + j) += si * *(pu + i);
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}
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for (j = i, pu = pui; j < nrows; j++, pu += ncols)
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*(pu + i) /= s;
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}
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else
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for (j = i, pu = pui; j < nrows; j++, pu += ncols)
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*(pu + i) = 0.0;
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*(pui + i) += 1.0;
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}
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}
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/* Perform Givens reduction
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*
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* Input: Matrices such that
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* A = U*Bidiag(diagonal,superdiagonal)*Vt
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*
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* Output: The same, with superdiagonal = 0
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*
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* All matrices are allocated by the caller
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*
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* Sizes:
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* U: nrows x ncols
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* diagonal, superdiagonal: ncols
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* V: ncols x ncols
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*/
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static int
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givens_reduction (int nrows,
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int ncols,
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double *U,
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double *V,
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double *diagonal,
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double *superdiagonal)
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{
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double epsilon;
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double c, s;
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double f,g,h;
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double x,y,z;
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double *pu, *pv;
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int i,j,k,m;
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int rotation_test;
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int iteration_count;
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assert (nrows >= 2);
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assert (ncols >= 2);
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for (i = 0, x = 0.0; i < ncols; i++)
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{
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y = fabs (diagonal[i]) + fabs (superdiagonal[i]);
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if (x < y)
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x = y;
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}
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epsilon = x * DBL_EPSILON;
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for (k = ncols - 1; k >= 0; k--)
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{
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iteration_count = 0;
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while (1)
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{
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rotation_test = 1;
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for (m = k; m >= 0; m--)
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{
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if (fabs (superdiagonal[m]) <= epsilon)
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{
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rotation_test = 0;
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break;
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}
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if (fabs (diagonal[m-1]) <= epsilon)
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break;
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}
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if (rotation_test)
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{
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c = 0.0;
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s = 1.0;
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for (i = m; i <= k; i++)
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{
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f = s * superdiagonal[i];
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superdiagonal[i] *= c;
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if (fabs (f) <= epsilon)
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break;
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g = diagonal[i];
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h = sqrt (f*f + g*g);
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diagonal[i] = h;
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c = g / h;
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s = -f / h;
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for (j = 0, pu = U; j < nrows; j++, pu += ncols)
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{
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y = *(pu + m - 1);
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z = *(pu + i);
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*(pu + m - 1 ) = y * c + z * s;
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*(pu + i) = -y * s + z * c;
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}
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}
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}
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z = diagonal[k];
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if (m == k)
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{
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if (z < 0.0)
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{
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diagonal[k] = -z;
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for (j = 0, pv = V; j < ncols; j++, pv += ncols)
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*(pv + k) = - *(pv + k);
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}
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break;
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}
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else
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{
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if (iteration_count >= MAX_ITERATION_COUNT)
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return -1;
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iteration_count++;
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x = diagonal[m];
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y = diagonal[k-1];
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g = superdiagonal[k-1];
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h = superdiagonal[k];
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f = ((y - z) * ( y + z ) + (g - h) * (g + h))/(2.0 * h * y);
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g = sqrt (f * f + 1.0);
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if (f < 0.0)
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g = -g;
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f = ((x - z) * (x + z) + h * (y / (f + g) - h)) / x;
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c = 1.0;
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s = 1.0;
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for (i = m + 1; i <= k; i++)
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{
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g = superdiagonal[i];
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y = diagonal[i];
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h = s * g;
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g *= c;
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z = sqrt (f * f + h * h);
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superdiagonal[i-1] = z;
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c = f / z;
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s = h / z;
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f = x * c + g * s;
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g = -x * s + g * c;
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h = y * s;
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y *= c;
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for (j = 0, pv = V; j < ncols; j++, pv += ncols)
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{
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x = *(pv + i - 1);
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z = *(pv + i);
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*(pv + i - 1) = x * c + z * s;
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*(pv + i) = -x * s + z * c;
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}
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z = sqrt (f * f + h * h);
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diagonal[i - 1] = z;
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if (z != 0.0)
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{
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c = f / z;
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s = h / z;
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}
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f = c * g + s * y;
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x = -s * g + c * y;
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for (j = 0, pu = U; j < nrows; j++, pu += ncols)
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{
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y = *(pu + i - 1);
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z = *(pu + i);
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*(pu + i - 1) = c * y + s * z;
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*(pu + i) = -s * y + c * z;
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}
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}
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superdiagonal[m] = 0.0;
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superdiagonal[k] = f;
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diagonal[k] = x;
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}
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}
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}
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return 0;
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}
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/* Given a singular value decomposition
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* of an nrows x ncols matrix A = U*Diag(S)*Vt,
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* sort the values of S by decreasing value,
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* permuting V to match.
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*/
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static void
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sort_singular_values (int nrows,
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int ncols,
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double *S,
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double *U,
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double *V)
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{
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int i, j, max_index;
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double temp;
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double *p1, *p2;
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assert (nrows >= 2);
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assert (ncols >= 2);
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for (i = 0; i < ncols - 1; i++)
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{
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max_index = i;
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for (j = i + 1; j < ncols; j++)
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if (S[j] > S[max_index])
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max_index = j;
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if (max_index == i)
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continue;
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temp = S[i];
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S[i] = S[max_index];
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S[max_index] = temp;
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p1 = U + max_index;
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p2 = U + i;
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for (j = 0; j < nrows; j++, p1 += ncols, p2 += ncols)
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{
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temp = *p1;
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*p1 = *p2;
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*p2 = temp;
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}
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p1 = V + max_index;
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p2 = V + i;
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for (j = 0; j < ncols; j++, p1 += ncols, p2 += ncols)
|
||
|
|
{
|
||
|
|
temp = *p1;
|
||
|
|
*p1 = *p2;
|
||
|
|
*p2 = temp;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
/* Compute a singular value decomposition of A,
|
||
|
|
* A = U*Diag(S)*Vt
|
||
|
|
*
|
||
|
|
* All matrices are allocated by the caller
|
||
|
|
*
|
||
|
|
* Sizes:
|
||
|
|
* A, U: nrows x ncols
|
||
|
|
* S: ncols
|
||
|
|
* V: ncols x ncols
|
||
|
|
*/
|
||
|
|
int
|
||
|
|
singular_value_decomposition (double *A,
|
||
|
|
int nrows,
|
||
|
|
int ncols,
|
||
|
|
double *U,
|
||
|
|
double *S,
|
||
|
|
double *V)
|
||
|
|
{
|
||
|
|
double *superdiagonal;
|
||
|
|
|
||
|
|
superdiagonal = g_alloca (sizeof (double) * ncols);
|
||
|
|
|
||
|
|
if (nrows < ncols)
|
||
|
|
return -1;
|
||
|
|
|
||
|
|
householder_reduction (A, nrows, ncols, U, V, S, superdiagonal);
|
||
|
|
|
||
|
|
if (givens_reduction (nrows, ncols, U, V, S, superdiagonal) < 0)
|
||
|
|
return -1;
|
||
|
|
|
||
|
|
sort_singular_values (nrows, ncols, S, U, V);
|
||
|
|
|
||
|
|
return 0;
|
||
|
|
}
|
||
|
|
|
||
|
|
/*
|
||
|
|
* Given a singular value decomposition of A = U*Diag(S)*Vt,
|
||
|
|
* compute the best approximation x to A*x = B.
|
||
|
|
*
|
||
|
|
* All matrices are allocated by the caller
|
||
|
|
*
|
||
|
|
* Sizes:
|
||
|
|
* U: nrows x ncols
|
||
|
|
* S: ncols
|
||
|
|
* V: ncols x ncols
|
||
|
|
* B, x: ncols
|
||
|
|
*/
|
||
|
|
void
|
||
|
|
singular_value_decomposition_solve (double *U,
|
||
|
|
double *S,
|
||
|
|
double *V,
|
||
|
|
int nrows,
|
||
|
|
int ncols,
|
||
|
|
double *B,
|
||
|
|
double *x)
|
||
|
|
{
|
||
|
|
int i, j, k;
|
||
|
|
double *pu, *pv;
|
||
|
|
double d;
|
||
|
|
double tolerance;
|
||
|
|
|
||
|
|
assert (nrows >= 2);
|
||
|
|
assert (ncols >= 2);
|
||
|
|
|
||
|
|
tolerance = DBL_EPSILON * S[0] * (double) ncols;
|
||
|
|
|
||
|
|
for (i = 0, pv = V; i < ncols; i++, pv += ncols)
|
||
|
|
{
|
||
|
|
x[i] = 0.0;
|
||
|
|
for (j = 0; j < ncols; j++)
|
||
|
|
{
|
||
|
|
if (S[j] > tolerance)
|
||
|
|
{
|
||
|
|
for (k = 0, d = 0.0, pu = U; k < nrows; k++, pu += ncols)
|
||
|
|
d += *(pu + j) * B[k];
|
||
|
|
x[i] += d * *(pv + j) / S[j];
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|