public static int SVDecomp(PZMath_matrix A, PZMath_matrix V, PZMath_vector S, PZMath_vector work)
{
int a, b, i, j;
int M = A.RowCount;
int N = A.ColumnCount;
int K = System.Math.Min(M, N);
if (M < N)
{
PZMath_errno.ERROR ("PZMath_linalg::SVDDecomp(),svd of MxN matrix, M<N, is not implemented", PZMath_errno.PZMath_EUNIMPL);
}
else if (V.RowCount != N)
{
PZMath_errno.ERROR ("PZMath_linalg::SVDDecomp(), square matrix V must match second dimension of matrix A",
PZMath_errno.PZMath_EBADLEN);
}
else if (V.RowCount != V.ColumnCount)
{
PZMath_errno.ERROR ("PZMath_linalg::SVDDecomp(),matrix V must be square", PZMath_errno.PZMath_ENOTSQR);
}
else if (S.Size != N)
{
PZMath_errno.ERROR ("PZMath_linalg::SVDDecomp(),length of vector S must match second dimension of matrix A",
PZMath_errno.PZMath_EBADLEN);
}
else if (work.Size != N)
{
PZMath_errno.ERROR ("PZMath_linalg::SVDDecomp(),length of workspace must match second dimension of matrix A",
PZMath_errno.PZMath_EBADLEN);
}
/* Handle the case of N = 1 (SVD of a column vector) */
if (N == 1)
{
PZMath_vector column = A.Column(0);
double norm = PZMath_blas.Dnrm2(column);
S[0] = norm;
V[0, 0] = 1.0;
if (norm != 0.0)
{
PZMath_blas.Dscal(1.0 / norm, column);
}
return PZMath_errno.PZMath_SUCCESS;
}
{
PZMath_vector f = work.SubVector(0, K - 1);
/* bidiagonalize matrix A, unpack A into U S V */
PZMath_linalg.BidiagDecomp (A, S, f);
PZMath_linalg.BidiagUnpack2 (A, S, f, V);
/* apply reduction steps to B=(S,Sd) */
ChopSmallElements (S, f);
/* Progressively reduce the matrix until it is diagonal */
b = N - 1;
while (b > 0)
{
double fbm1 = f[b - 1];
if (fbm1 == 0.0 || PZMath_sys.Isnan (fbm1))
{
b--;
continue;
}
/* Find the largest unreduced block (a,b) starting from b
and working backwards */
a = b - 1;
while (a > 0)
{
double fam1 = f[a - 1];
if (fam1 == 0.0 || PZMath_sys.Isnan (fam1))
{
break;
}
a--;
}
{
int n_block = b - a + 1;
PZMath_vector S_block = S.SubVector(a, n_block);
PZMath_vector f_block = f.SubVector (a, n_block - 1);
PZMath_matrix U_block = A.Submatrix(0, a, A.RowCount, n_block);
PZMath_matrix V_block = V.Submatrix(0, a, V.RowCount, n_block);
QRStep (S_block, f_block, U_block, V_block);
/* remove any small off-diagonal elements */
ChopSmallElements (S_block, f_block);
}
}
}
/* Make singular values positive by reflections if necessary */
for (j = 0; j < K; j++)
{
double Sj = S[j];
if (Sj < 0.0)
{
for (i = 0; i < N; i++)
{
double Vij = V[i, j];
V[i, j] = -Vij;
}
S[j] = -Sj;
}
}
/* Sort singular values into decreasing order */
for (i = 0; i < K; i++)
{
double S_max = S[i];
int i_max = i;
for (j = i + 1; j < K; j++)
{
double Sj = S[j];
if (Sj > S_max)
{
S_max = Sj;
i_max = j;
}
}
if (i_max != i)
{
/* swap eigenvalues */
S.Swap(i, i_max);
/* swap eigenvectors */
A.SwapColumns(i, i_max);
V.SwapColumns(i, i_max);
}
}
return PZMath_errno.PZMath_SUCCESS;
}
public static int QRPTDecomp(PZMath_matrix A, PZMath_vector tau, PZMath_permutation p, out int signum, PZMath_vector norm)
{
int M = A.RowCount;
int N = A.ColumnCount;
signum = 0;
if (tau.Size != System.Math.Min(M, N))
{
PZMath_errno.ERROR("size of tau must be MIN(M,N)", PZMath_errno.PZMath_EBADLEN);
}
else if (p.Size != N)
{
PZMath_errno.ERROR("permutation size must be N", PZMath_errno.PZMath_EBADLEN);
}
else if (norm.Size != N)
{
PZMath_errno.ERROR("norm size must be N", PZMath_errno.PZMath_EBADLEN);
}
else
{
int i;
signum = 1;
p.Init(); /* set to identity */
/* Compute column norms and store in workspace */
for (i = 0; i < N; i++)
{
PZMath_vector c = A.Column(i);
double x = PZMath_blas.Dnrm2(c);
norm[i] = x;
}
for (i = 0; i < System.Math.Min(M, N); i++)
{
/* Bring the column of largest norm into the pivot position */
double max_norm = norm[i];
int j;
int kmax = i;
for (j = i + 1; j < N; j++)
{
double x = norm[j];
if (x > max_norm)
{
max_norm = x;
kmax = j;
}
}
if (kmax != i)
{
A.SwapColumns(i, kmax);
p.Swap(i, kmax);
norm.Swap(i, kmax);
signum = - signum;
}
/* Compute the Householder transformation to reduce the j-th
column of the matrix to a multiple of the j-th unit vector */
{
PZMath_vector c_full = A.Column(i);
PZMath_vector c = c_full.SubVector(i, M - i);
double tau_i = HouseholderTransform(c);
tau[i] = tau_i;
/* Apply the transformation to the remaining columns */
if (i + 1 < N)
{
PZMath_matrix m = A.Submatrix(i, i + 1, M - i, N - (i + 1));
HouseholderHM (tau_i, c, m);
}
}
/* Update the norms of the remaining columns too */
if (i + 1 < M)
{
for (j = i + 1; j < N; j++)
{
double x = norm[j];
if (x > 0.0)
{
double y = 0;
double temp= A[i, j] / x;
if (System.Math.Abs(temp) >= 1)
y = 0.0;
else
y = x * System.Math.Sqrt(1 - temp * temp);
/* recompute norm to prevent loss of accuracy */
if (System.Math.Abs(y / x) < System.Math.Sqrt(20.0) * PZMath_machine.PZMath_SQRT_DBL_EPSILON)
{
PZMath_vector c_full = A.Column(j);
PZMath_vector c = c_full.SubVector(i + 1, M - (i + 1));
y = PZMath_blas.Dnrm2(c);
}
norm[j] = y;
}
}
}
}
}
return PZMath_errno.PZMath_SUCCESS;
}
public static void SVD2(PZMath_vector d, PZMath_vector f, PZMath_matrix U, PZMath_matrix V)
{
int i;
double c, s, a11, a12, a21, a22;
int M = U.RowCount;
int N = V.RowCount;
double d0 = d[0];
double f0 = f[0];
double d1 = d[1];
if (d0 == 0.0)
{
/* Eliminate off-diagonal element in [0,f0;0,d1] to make [d,0;0,0] */
CreateGivens (f0, d1, out c, out s);
/* compute B <= G^T B X, where X = [0,1;1,0] */
d[0] = c * f0 - s * d1;
f[0] = s * f0 + c * d1;
d[1] = 0.0;
/* Compute U <= U G */
for (i = 0; i < M; i++)
{
double Uip = U[i, 0];
double Uiq = U[i, 1];
U[i, 0] = c * Uip - s * Uiq;
U[i, 1] = s * Uip + c * Uiq;
}
/* Compute V <= V X */
V.SwapColumns(0, 1);
return;
}
else if (d1 == 0.0)
{
/* Eliminate off-diagonal element in [d0,f0;0,0] */
CreateGivens (d0, f0, out c, out s);
/* compute B <= B G */
d[0] = d0 * c - f0 * s;
f[0] = 0.0;
/* Compute V <= V G */
for (i = 0; i < N; i++)
{
double Vip = V[i, 0];
double Viq = V[i, 1];
V[i, 0] = c * Vip - s * Viq;
V[i, 1] = s * Vip + c * Viq;
}
return;
}
else
{
/* Make columns orthogonal, A = [d0, f0; 0, d1] * G */
CreateSchur (d0, f0, d1, out c, out s);
/* compute B <= B G */
a11 = c * d0 - s * f0;
a21 = - s * d1;
a12 = s * d0 + c * f0;
a22 = c * d1;
/* Compute V <= V G */
for (i = 0; i < N; i++)
{
double Vip = V[i, 0];
double Viq = V[i, 1];
V[i, 0] = c * Vip - s * Viq;
V[i, 1] = s * Vip + c * Viq;
}
/* Eliminate off-diagonal elements, bring column with largest
norm to first column */
if (PZMath_sys.Hypot (a11, a21) < PZMath_sys.Hypot (a12,a22))
{
double t1, t2;
/* B <= B X */
t1 = a11; a11 = a12; a12 = t1;
t2 = a21; a21 = a22; a22 = t2;
/* V <= V X */
V.SwapColumns(0, 1);
}
CreateGivens (a11, a21, out c, out s);
/* compute B <= G^T B */
d[0] = c * a11 - s * a21;
f[0] = c * a12 - s * a22;
d[1] = s * a12 + c * a22;
/* Compute U <= U G */
for (i = 0; i < M; i++)
{
double Uip = U[i, 0];
double Uiq = U[i, 1];
U[i, 0] = c * Uip - s * Uiq;
U[i, 1] = s * Uip + c * Uiq;
}
return;
}
}