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