linSvdGenComplex¶
Computes the SVD, \(A = USV^H\), of a complex rectangular matrix A. An approximate generalized inverse and rank of A also can be computed.
Synopsis¶
linSvdGenComplex (a)
Required Arguments¶
- complex
a[[]](Input) - Array of size
m×ncontaining the matrix.
Return Value¶
Using only required arguments, linSvdGenComplex returns a complex array
of length min (m, n) containing the singular values of the matrix.
If no value can be computed then None is returned.
Optional Arguments¶
rank, float tol, int rank (Input/Output)
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 \(s_{i,i}\) is considered negligible if \(s_{i,i}\) ≤tol. Iftol< 0, then a singular value \(s_{i,i}\) is considered negligible if \(s_{i,i} \leq |\mathtt{tol}| * \|A\|_\infty\). In this case, should be an estimate of relative error or uncertainty in the data.rank(Input/Output)- Integer containing an estimate of the rank of A.
u(Output)- An array of size
m× min (m,n) containing the min (m,n) left-singular vectors of A. v(Output)- An array of size
n×ncontaining the right singular vectors of A. inverse(Output)- An array of size
n×mcontaining the generalized inverse of the matrix A.
Description¶
The function linSvdGenComplex 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 \(s_{k,k}\) ≤
tol or \(s_{k,k} \leq |\mathtt{tol}| * \|A\|_\infty\). Since
\(s_{i+1,i+1} \leq s_{i,i}\), it follows that all the
\(s_{i,i}\) satisfy the same inequality for i = k, …, min (m,
n) − 1. The rank is set to the value k − 1. If \(A = USV^H\), its
generalized inverse is \(A^+ = VS^+ U^T\).
Here,
Only singular values that are not negligible are reciprocated. If
inverse is specified, the function first computes the singular value
decomposition of the matrix A. The generalized inverse is then computed.
The function linSvdGenComplex 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.
from numpy import *
from pyimsl.math.linSvdGenComplex import linSvdGenComplex
from pyimsl.math.writeMatrixComplex import writeMatrixComplex
a = array([[1.0 + 2.0j, 3.0 + 2.0j, 1.0 - 4.0j],
[3.0 - 2.0j, 2.0 - 4.0j, 1.0 + 3.0j],
[4.0 + 3.0j, -2.0 + 1.0j, 1.0 + 4.0j],
[2.0 - 1.0j, 3.0 + 0.0j, 3.0 - 1.0j],
[1.0 - 5.0j, 2.0 - 5.0j, 2.0 + 2.0j],
[1.0 + 2.0j, 4.0 - 2.0j, 2.0 - 3.0j]])
# Compute SVD of a
s = linSvdGenComplex(a)
# Print decomposition
writeMatrixComplex("Singular values, S", s, column=True)
Output¶
Singular values, S
1 ( 11.77, 0.00)
2 ( 9.30, 0.00)
3 ( 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
linSvdGenComplex.
from numpy import *
from pyimsl.math.linSvdGenComplex import linSvdGenComplex
from pyimsl.math.writeMatrixComplex import writeMatrixComplex
m = 6
n = 3
p_u = []
p_v = []
a = array([[1.0 + 2.0j, 3.0 + 2.0j, 1.0 - 4.0j],
[3.0 - 2.0j, 2.0 - 4.0j, 1.0 + 3.0j],
[4.0 + 3.0j, -2.0 + 1.0j, 1.0 + 4.0j],
[2.0 - 1.0j, 3.0 + 0.0j, 3.0 - 1.0j],
[1.0 - 5.0j, 2.0 - 5.0j, 2.0 + 2.0j],
[1.0 + 2.0j, 4.0 - 2.0j, 2.0 - 3.0j]])
# Compute SVD of a
s = linSvdGenComplex(a, u=p_u, v=p_v)
# Print decomposition
writeMatrixComplex("Singular values, S", s, column=True)
writeMatrixComplex("Left singular vectors, U", p_u)
writeMatrixComplex("Right singular vectors, V", p_v)
Output¶
Singular values, S
1 ( 11.77, 0.00)
2 ( 9.30, 0.00)
3 ( 4.99, 0.00)
Left singular vectors, U
1 2
1 ( 0.1968, 0.2186) ( 0.5011, 0.0217)
2 ( 0.3443, -0.3542) ( -0.2933, 0.0248)
3 ( 0.1457, 0.2307) ( -0.5424, 0.1381)
4 ( 0.3016, -0.0844) ( 0.2157, 0.2659)
5 ( 0.2283, -0.6008) ( -0.1325, 0.1433)
6 ( 0.2876, -0.0350) ( 0.4377, -0.0400)
3
1 ( 0.2007, 0.1003)
2 ( -0.1155, 0.2338)
3 ( 0.4361, 0.4407)
4 ( 0.0523, 0.0894)
5 ( -0.3152, 0.0090)
6 ( -0.0458, 0.6205)
Right singular vectors, V
1 2
1 ( 0.6616, 0.0000) ( -0.2651, 0.0000)
2 ( 0.7355, 0.0379) ( 0.3850, -0.0707)
3 ( 0.0507, -0.1317) ( 0.1724, 0.8642)
3
1 ( 0.7014, 0.0000)
2 ( -0.5482, -0.0624)
3 ( 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.
from __future__ import print_function
from numpy import *
from pyimsl.math.linSvdGenComplex import linSvdGenComplex
from pyimsl.math.writeMatrixComplex import writeMatrixComplex
a = array([[1.0 + 2.0j, 3.0 + 2.0j, 1.0 - 4.0j, 1.0 + 0.0j],
[3.0 - 2.0j, 2.0 - 4.0j, 1.0 + 3.0j, 0.0 + 1.0j],
[4.0 + 3.0j, -2.0 + 1.0j, 1.0 + 4.0j, 0.0 + 0.0j],
[2.0 - 1.0j, 3.0 + 0.0j, 3.0 - 1.0j, 2.0 + 1.0j],
[1.0 - 5.0j, 2.0 - 5.0j, 2.0 + 2.0j, 1.0 + 3.1j],
[1.0 + 2.0j, 4.0 - 2.0j, 2.0 - 3.0j, 1.4 + 1.9j]])
# Factor a
rank = {'tol': 1.e-4}
gen_inva = []
s = linSvdGenComplex(a, rank=rank,
inverse=gen_inva)
# Print rank and generalized inverse matrix
print("Rank = %2d" % rank['rank'])
writeMatrixComplex("Singular values", s, column=True)
writeMatrixComplex("Generalized inverse", gen_inva)
Output¶
Rank = 4
Singular values
1 ( 12.13, 0.00)
2 ( 9.53, 0.00)
3 ( 5.67, 0.00)
4 ( 1.74, 0.00)
Generalized inverse
1 2
1 ( 0.0266, -0.0164) ( -0.0185, -0.0453)
2 ( 0.0061, -0.0280) ( 0.0820, 0.1156)
3 ( -0.0019, 0.0572) ( 0.1174, -0.0812)
4 ( 0.0380, -0.0298) ( -0.0758, 0.2158)
3 4
1 ( 0.0720, -0.0700) ( -0.0220, 0.0428)
2 ( -0.0410, 0.0242) ( 0.0959, -0.0885)
3 ( 0.0499, -0.0463) ( -0.0234, -0.1033)
4 ( 0.0356, 0.0557) ( 0.2918, 0.0763)
5 6
1 ( -0.0003, 0.0709) ( 0.0254, -0.1050)
2 ( -0.0187, -0.0287) ( -0.0218, 0.1109)
3 ( -0.0769, -0.0103) ( 0.0810, 0.1074)
4 ( 0.0881, -0.2070) ( -0.1531, -0.0814)
Warning Errors¶
IMSL_SLOWCONVERGENT_MATRIX |
Convergence cannot be reached after 30 iterations. |