linSolPosdefComplex¶

Solves a complex Hermitian positive definite system of linear equations $Ax = b$ . Using optional arguments, any of several related computations can be performed. These extra tasks include computing the Cholesky factor, L, of A such that $A = LL^H$ or computing the solution to $Ax = b$ given the Cholesky factor, L.

Synopsis¶

linSolPosdefComplex (a,  b)

Required Arguments¶

complex a[[]] (Input): Array of size n × n containing the matrix.
complex b[] (Input): Array of size n containing the right-hand side.

Return Value¶

The solution x of the Hermitian positive definite linear system Ax = b. If no solution was computed, then None is returned.

Optional Arguments¶

factor, complex pFactor (Output): An array of size n × n containing the $LL^H$ factorization of A. The lower‑triangular part of this array contains L, and the upper-triangular part contains $L^H$ .
condition (Output): A scalar containing an estimate of the $L_1$ norm condition number of the matrix A. Do not use this option with solveOnly.
factorOnly: Compute the Cholesky factorization $LL^H$ of A. If factorOnly is used, factor is required. The argument b is then ignored, and the returned value of linSolPosdefComplex is None.
solveOnly: Solve $Ax = b$ given the $LL^H$ factorization previously computed by linSolPosdefComplex. By default, the solution to $Ax = b$ is pointed to by linSolPosdefComplex. If solveOnly is used, argument factor is required and argument a is ignored.

Description¶

The function linSolPosdefComplex solves a system of linear algebraic equations having a Hermitian positive definite coefficient matrix A. The function first computes the $LL^H$ factorization of A. The solution of the linear system is then found by solving the two simpler systems, $y = L^{-1}b$ and $x = L^{-H}y$ . When the solution to the linear system is required, an estimate of the $L_1$ condition number of A is computed using the algorithm in Dongarra et al. (1979). If the estimated condition number is greater than 1∕ɛ (where ɛ is the machine precision), a warning message is issued. This indicates that very small changes in A may produce large changes in the solution x. The function linSolPosdefComplex fails if L, the lower-triangular matrix in the factorization, has a zero diagonal element.

Examples¶

Example 1¶

A system of five linear equations with a Hermitian positive definite coefficient matrix is solved in this example. The equations are as follows:

$2x_1 +(−1 + i)x_2 = 1 +5i$

$(−1 − i)x_1 +4x_2 + (1 + 2i)x_3 = 12 − 6i$

$(1 − 2i)x_2 +10x_3 + 4ix_4 = 1 − 16i$

$−4ix_3 + 6x_4 + (1 + i)x_5 = −3 − 3i$

$(1 − i)x_4 + 9x_5 = 25 + 16i$

from numpy import *
from pyimsl.math.linSolPosdefComplex import linSolPosdefComplex
from pyimsl.math.writeMatrixComplex import writeMatrixComplex

factor = []
a = array([[2.0 + 0.0j, -1.0 + 1.0j, 0.0 + 0.0j, 0.0 + 0.0j, 0.0 + 0.0j],
           [-1.0 - 1.0j, 4.0 + 0.0j, 1.0 + 2.0j, 0.0 + 0.0j, 0.0 + 0.0j],
           [0.0 + 0.0j, 1.0 - 2.0j, 10.0 + 0.0j, 0.0 + 4.0j, 0.0 + 0.0j],
           [0.0 + 0.0j, 0.0 + 0.0j, 0.0 - 4.0j, 6.0 + 0.0j, 1.0 + 1.0j],
           [0.0 + 0.0j, 0.0 + 0.0j, 0.0 + 0.0j, 1.0 - 1.0j, 9.0 + 0.0j]])
b = array([1.0 + 5.0j, 12.0 - 6.0j, 1.0 - 16.0j, -3.0 - 3.0j, 25.0 + 16.0j])

# Solve Ax=b for x
x = linSolPosdefComplex(a, b)
writeMatrixComplex("Solution, x, of Ax = b", x, column=True)

Output¶

 
   Solution, x, of Ax = b
(          2,          1)
(          3,          0)
(         -1,         -1)
(          0,         -2)
(          3,          2)

Example 2¶

This example solves the same system of five linear equations as in the first example. This time, the $LL^H$ factorization of A and the solution x are returned.

from numpy import *
from pyimsl.math.linSolPosdefComplex import linSolPosdefComplex
from pyimsl.math.writeMatrixComplex import writeMatrixComplex

factor = []
a = array([[2.0 + 0.0j, -1.0 + 1.0j, 0.0 + 0.0j, 0.0 + 0.0j, 0.0 + 0.0j],
           [-1.0 - 1.0j, 4.0 + 0.0j, 1.0 + 2.0j, 0.0 + 0.0j, 0.0 + 0.0j],
           [0.0 + 0.0j, 1.0 - 2.0j, 10.0 + 0.0j, 0.0 + 4.0j, 0.0 + 0.0j],
           [0.0 + 0.0j, 0.0 + 0.0j, 0.0 - 4.0j, 6.0 + 0.0j, 1.0 + 1.0j],
           [0.0 + 0.0j, 0.0 + 0.0j, 0.0 + 0.0j, 1.0 - 1.0j, 9.0 + 0.0j]])
b = array([1.0 + 5.0j, 12.0 + -6.0j, 1.0 - 16.0j, -3.0 + -3.0j, 25.0 + 16.0j])

# Solve Ax=b for x
x = linSolPosdefComplex(a, b, factor=factor)
writeMatrixComplex("Solution, x, of Ax = b", x, column=True)
writeMatrixComplex("Cholesky factor L, and ctrans(L), of A", factor)

Output¶

 
   Solution, x, of Ax = b
(          2,          1)
(          3,          0)
(         -1,         -1)
(          0,         -2)
(          3,          2)
 
        Cholesky factor L, and ctrans(L), of A
                           1                          2
(      1.414,      0.000)  (     -0.707,      0.707)
(     -0.707,     -0.707)  (      1.732,      0.000)
(      0.000,      0.000)  (      0.577,     -1.155)
(      0.000,      0.000)  (      0.000,      0.000)
(      0.000,      0.000)  (      0.000,      0.000)
 
                           3                          4
(      0.000,     -0.000)  (      0.000,     -0.000)
(      0.577,      1.155)  (      0.000,     -0.000)
(      2.887,      0.000)  (      0.000,      1.386)
(      0.000,     -1.386)  (      2.020,      0.000)
(      0.000,      0.000)  (      0.495,     -0.495)
 
                           5
(      0.000,     -0.000)
(      0.000,     -0.000)
(      0.000,     -0.000)
(      0.495,      0.495)
(      2.917,      0.000)

Warning Errors¶

`IMSL_HERMITIAN_DIAG_REAL_1`	The diagonal of a Hermitian matrix must be real. Its imaginary part is set to zero.
`IMSL_HERMITIAN_DIAG_REAL_2`	The diagonal of a Hermitian matrix must be real. The imaginary part will be used as zero in the algorithm.
`IMSL_ILL_CONDITIONED`	The input matrix is too ill-conditioned. An estimate of the reciprocal of its $L_1$ condition number is “`rcond`” = #. The solution might not be accurate.

Fatal Errors¶

`IMSL_NONPOSITIVE_MATRIX`	The leading # by # minor matrix of the input matrix is not positive definite.
`IMSL_HERMITIAN_DIAG_REAL`	During the factorization the matrix has a large imaginary component on the diagonal. Thus, it cannot be positive definite.
`IMSL_SINGULAR_TRI_MATRIX`	The triangular matrix is singular. The index of the first zero diagonal term is #.