lin_svd_gen (complex)

Chapter 1: Linear Systems > lin_svd_gen (complex)

lin_svd_gen (complex)

Computes the SVD, A = USVH, of a complex rectangular matrix A. An approximate generalized inverse and rank of A also can be computed.

Synopsis

#include <imsl.h>

f_complex *imsl_c_lin_svd_gen (int m, int n, f_complex a[], …, 0)

The type d_complex function is imsl_z_lin_svd_gen.

Required Arguments

int m (Input)
Number of rows in the matrix.

int n (Input)
Number of columns in the matrix.

f_complex a[] (Input)
Array of size m × n containing the matrix.

Return Value

Using only required arguments, imsl_c_lin_svd_gen returns a pointer to a complex array of length min (m, n) containing the singular values of the matrix. To release this space, use imsl_free. If no value can be computed then NULL is returned.

Synopsis with Optional Arguments

#include <imsl.h>

f_complex *imsl_c_lin_svd_gen (int m, int n, f_complex a[],

IMSL_A_COL_DIM, int a_col_dim,

IMSL_RETURN_USER, f_complex s[],

IMSL_RANK, float tol, int *rank,

IMSL_U, f_complex **p_u,

IMSL_U_USER, f_complex u[],

IMSL_U_COL_DIM, int u_col_dim,

IMSL_V, f_complex **p_v,

IMSL_V_USER, f_complex v[],

IMSL_V_COL_DIM, int v_col_dim,

IMSL_INVERSE, f_complex **p_gen_inva,

IMSL_INVERSE_USER, f_complex gen_inva[],

IMSL_INV_COL_DIM, int gen_inva_col_dim,

Optional Arguments

IMSL_A_COL_DIM, int a_col_dim (Input)
The column dimension of the array a.
Default: a_col_dim = n

IMSL_RETURN_USER, f_complex s[] (Output)
A user-allocated array of length min (m, n) containing the singular values of A in its first min (m, n) positions in nonincreasing order. The complex entries are all real. If IMSL_RETURN_USER is used, the return value of imsl_c_lin_svd_gen is s.

IMSL_RANK, float tol, int *rank (Input/Output)

float tol (Input)
Scalar containing the tolerance used to determine when a singular value is negligible and replaced by the value zero. If tol > 0, then a singular value si_,i is considered negligible if si_,i ≤ tol. If tol < 0, then a singular value si_,i is considered negligible if si_,i ≤ |tol|*||A||_∝. In this case, should be an estimate of relative error or uncertainty in the data.

int *rank (Input/Output)
Integer containing an estimate of the rank of A.

IMSL_U, f_complex **p_u (Output)
The address of a pointer to an array of size m × min (m, n) containing the min (m, n) left-singular vectors of A. On return, the necessary space is allocated by imsl_c_lin_svd_gen. Typically, f_complex *p_u is declared, and &p_u is used as an argument.

IMSL_U_USER, f_complex u[] (Output)
The address of a pointer to an array of size m × min (m, n) containing the min (m, n) left-singular vectors of A. If m ≥ n, the left-singular vectors can be returned using the storage locations of the array a.

IMSL_U_COL_DIM, int u_col_dim (Input)
The column dimension of the array containing the left-singular vectors.
Default: u_col_dim = min (m, n)

IMSL_V, f_complex **p_v (Output)
The address of a pointer to an array of size n × n containing the right singular vectors of A. On return, the necessary space is allocated by imsl_c_lin_svd_gen. Typically, f_complex *p_v is declared, and &p_v is used as an argument.

IMSL_V_USER, f_complex v[] (Output)
The address of a pointer to an array of size n × n containing the right singular vectors of A. The right-singular vectors can be returned using the storage locations of the array a. Note that the return of the left and right-singular vectors cannot use the storage locations of a simultaneously.

IMSL_V_COL_DIM, int v_col_dim (Input)
The column dimension of the array containing the right-singular vectors.
Default: v_col_dim = n

IMSL_INVERSE, f_complex **p_gen_inva (Output)
The address of a pointer to an array of size n × m containing the generalized inverse of the matrix A. On return, the necessary space is allocated by imsl_c_lin_svd_gen. Typically, f_complex *p_gen_inva is declared, and &p_gen_inva is used as an argument.

IMSL_INVERSE_USER, f_complex gen_inva[] (Output)
A user-allocated array of size n × m containing the general inverse of the matrix A.

IMSL_INV_COL_DIM, int gen_inva_col_dim (Input)
The column dimension of the array containing the general inverse of the matrix A.
Default: gen_inva_col_dim = m

Description

The function imsl_c_lin_svd_gen computes the singular value decomposition of a complex matrix A. It first reduces the matrix A to a bidiagonal matrix B by pre- and post-multiplying Householder transformations. Then, the singular value decomposition of B is computed using the implicit-shifted QR algorithm. An estimate of the rank of the matrix A is obtained by finding the smallest integer k such that sk_,k ≤ tol or sk_,k ≤ |tol|*||A||_∝. Since si₊1_, ≤ si_,i, it follows that all the si_,i satisfy the same inequality for i = k, …, min (m, n) − 1. The rank is set to the value k − 1. If A = USVH, its generalized inverse is A⁺ = VS⁺ UT.

Here,

Only singular values that are not negligible are reciprocated. If IMSL_INVERSE or IMSL_INVERSE_USER is specified, the function first computes the singular value decomposition of the matrix A. The generalized inverse is then computed. The function imsl_c_lin_svd_gen fails if the QR algorithm does not converge after 30 iterations isolating an individual singular value.

Examples

Example 1

This example computes the singular values of a 6 × 3 complex matrix.

#include <imsl.h>

int main()

{

int m = 6, n = 3;

f_complex *s;

f_complex a[] = {{1.0, 2.0}, {3.0, 2.0}, {1.0,-4.0},

{3.0,-2.0}, {2.0,-4.0}, {1.0, 3.0},

{4.0, 3.0}, {-2.0,1.0}, {1.0, 4.0},

{2.0,-1.0}, {3.0, 0.0}, {3.0,-1.0},

{1.0,-5.0}, {2.0,-5.0}, {2.0, 2.0},

{1.0, 2.0}, {4.0,-2.0}, {2.0,-3.0}};

/* Compute singular values */

s = imsl_c_lin_svd_gen (m, n, a, 0);

/* Print singular values */

imsl_c_write_matrix ("Singular values", 1, n, s, 0);

}

Output

Singular values

1 2 3

( 11.77, 0.00) ( 9.30, 0.00) ( 4.99, 0.00)

Example 2

This example computes the singular value decomposition of the 6 × 3 complex matrix A. The singular values are returned in the user-provided array. The matrices U and V are returned in the space provided by the function imsl_c_lin_svd_gen.

#include <imsl.h>

int main()

{

int m = 6, n = 3;

f_complex s[3], *p_u, *p_v;

f_complex a[] = {{1.0, 2.0}, {3.0, 2.0}, {1.0,-4.0},

{3.0,-2.0}, {2.0,-4.0}, {1.0, 3.0},

{4.0, 3.0}, {-2.0,1.0}, {1.0, 4.0},

{2.0,-1.0}, {3.0, 0.0}, {3.0,-1.0},

{1.0,-5.0}, {2.0,-5.0}, {2.0, 2.0},

{1.0, 2.0}, {4.0,-2.0}, {2.0,-3.0}};

/* Compute SVD of a */

imsl_c_lin_svd_gen (m, n, a,

IMSL_RETURN_USER, s,

IMSL_U, &p_u,

IMSL_V, &p_v,

0);

/* Print decomposition factors */

imsl_c_write_matrix ("Singular values, S", 1, n, s, 0);

imsl_c_write_matrix ("Left singular vectors, U", m, n, p_u, 0);

imsl_c_write_matrix ("Right singular vectors, V", n, n, p_v, 0);

}

Output

Singular values, S

1 2 3

( 11.77, 0.00) ( 9.30, 0.00) ( 4.99, 0.00)

Left singular vectors, U

1 2 3

1 ( 0.1968, 0.2186) ( 0.5011, 0.0217) ( -0.2007, -0.1003)

2 ( 0.3443, -0.3542) ( -0.2933, 0.0248) ( 0.1155, -0.2338)

3 ( 0.1457, 0.2307) ( -0.5424, 0.1381) ( -0.4361, -0.4407)

4 ( 0.3016, -0.0844) ( 0.2157, 0.2659) ( -0.0523, -0.0894)

5 ( 0.2283, -0.6008) ( -0.1325, 0.1433) ( 0.3152, -0.0090)

6 ( 0.2876, -0.0350) ( 0.4377, -0.0400) ( 0.0458, -0.6205)

Right singular vectors, V

1 2 3

1 ( 0.6616, 0.0000) ( -0.2651, 0.0000) ( -0.7014, 0.0000)

2 ( 0.7355, 0.0379) ( 0.3850, -0.0707) ( 0.5482, 0.0624)

3 ( 0.0507, -0.1317) ( 0.1724, 0.8642) ( -0.0173, -0.4509)

Example 3

This example computes the rank and generalized inverse of a 6 × 4 matrix A. The rank and the 4 × 6 generalized inverse matrix A⁺ are printed.

#include <imsl.h>

int main()

{

int m = 6, n = 4;

int *rank;

float tol;

f_complex gen_inv[24], *s;

f_complex a[] = {{1.0, 2.0}, {3.0, 2.0}, {1.0,-4.0}, {1.0,0.0},

{3.0,-2.0}, {2.0,-4.0}, {1.0, 3.0}, {0.0,1.0},

{4.0, 3.0}, {-2.0,1.0}, {1.0, 4.0}, {0.0,0.0},

{2.0,-1.0}, {3.0, 0.0}, {3.0,-1.0}, {2.0,1.0},

{1.0,-5.0}, {2.0,-5.0}, {2.0, 2.0}, {1.0,3.1},

{1.0, 2.0}, {4.0,-2.0}, {2.0,-3.0}, {1.4,1.9}};

/* Factor a */

tol = 1.e-4;

s = imsl_c_lin_svd_gen (m, n, a,

IMSL_RANK, tol, &rank,

IMSL_INVERSE_USER, gen_inv,

IMSL_INV_COL_DIM, m,

0);

/* Print rank and generalized */

/* inverse matrix */

printf ("Rank = %2d", rank);

imsl_c_write_matrix ("Singular values", 1, n, s, 0);

imsl_c_write_matrix ("Generalized inverse", n, m, gen_inv,

IMSL_A_COL_DIM, m, 0);

}

Output

Rank = 4

Singular values

1 2 3

( 12.13, 0.00) ( 9.53, 0.00) ( 5.67, 0.00)

( 1.74, 0.00)

Generalized inverse

1 2 3

1 ( 0.0266, 0.0164) ( -0.0185, 0.0453) ( 0.0720, 0.0700)

2 ( 0.0061, 0.0280) ( 0.0820, -0.1156) ( -0.0410, -0.0242)

3 ( -0.0019, -0.0572) ( 0.1174, 0.0812) ( 0.0499, 0.0463)

4 ( 0.0380, 0.0298) ( -0.0758, -0.2158) ( 0.0356, -0.0557)

4 5 6

1 ( -0.0220, -0.0428) ( -0.0003, -0.0709) ( 0.0254, 0.1050)

2 ( 0.0959, 0.0885) ( -0.0187, 0.0287) ( -0.0218, -0.1109)

3 ( -0.0234, 0.1033) ( -0.0769, 0.0103) ( 0.0810, -0.1074)

4 ( 0.2918, -0.0763) ( 0.0881, 0.2070) ( -0.1531, 0.0814)

Warning Errors

IMSL_SLOWCONVERGENT_MATRIX Convergence cannot be reached after 30 iterations.

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