IMSL_CHSOL

The IMSL_CHSOL function solves a symmetric positive definite system of real or complex linear equations Ax = b.

Note: This routine requires an IDL Analyst license. For more information, contact your Exelis VIS sales or technical support representative.

The IMSL_CHSOL function solves a system of linear algebraic equations having a symmetric positive definite coefficient matrix A. The function first computes the Cholesky factorization LL^H of A. The solution of the linear system is then found by solving the two simpler systems, y = L^–1b and x = L^–Hy. An estimate of the L₁ condition number of A is computed using the same algorithm as in Dongarra et al. (1979). If the estimated condition number is greater than 1/ε (where ε 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 IMSL_CHSOL function fails if L, the lower-triangular matrix in the factorization, has a zero diagonal element.

Examples

For additional information on using IMSL_CHSOL, see Additional Examples.

Example 1

RM, a, 3, 3

; Define the coefficient matrix.

row 0: 1 -3 2

row 1: -3 10 -5

row 2: 2 -5 6

RM, b, 3, 1

; Define the right-hand side.

row 0: 27

row 1: -78

row 2: 64

x = IMSL_CHSOL(b, a)

; Call IMSL_CHSOL to compute the solution.

PM, x, Title = 'Solution'

Solution

1.00000

-4.00000

Chapter 4: Linear Systems 93

IDL Analyst Reference Guide IMSL_CHSOL

7.00000

PM, a # x - b, Title = 'Residual'

Residual

0.00000

0.00000

0.00000

Syntax

Result =IMSL_CHSOL( b [, a] [, CONDITION=variable] [, /DOUBLE] [, FACTOR=variable] [, INVERSE=variable] )

Return Value

The solution of the linear system Ax = b.

Arguments

b

One-dimensional matrix containing the right-hand side.

a

Two-dimensional matrix containing the coefficient matrix. Matrix A (i, j) containsthe j-th coefficient of the i-th equation.

Keywords

CONDITION

Named variable into which an estimate of the L₁ condition number is stored. The CONDITION and FACTOR keywords cannot be used together.

DOUBLE

If present and nonzero, double precision is used.

FACTOR

Named variable in which the LL^H factorization of A is stored. The lower-triangular part of this matrix contains L, and the upper-triangular part contains ^LH. The CONDITION and FACTOR keywords cannot be used together.

INVERSE

Specifies a named variable into which the inverse of the matrix A is stored. This keyword is not allowed if A is complex.

Errors

Warning Errors

MATH_ILL_CONDITIONED - Input matrix is too ill-conditioned. An estimate of the reciprocal of its L₁ condition number is #. The solution might not be accurate.

Fatal Errors

MATH_NONPOSITIVE_MATRIX - Leading # by # submatrix of the input matrix is not positive definite.

MATH_SINGULAR_MATRIX - Input matrix is singular

MATH_SINGULAR_TRI_MATRIX - Input triangular matrix is singular. The index of the first zero diagonal element is #.

Additional Examples

Example 2

This example solves a system of five linear equations with Hermitian positive definite coefficient matrix. The equations are as follows:

2x₀ + (–1 + i ) x₁ = 1 + 5i

(–1 –i ) x₀ + 4x₁ + (1 + 2i ) x₂ = 12 – 6i

(–1 –2i ) x₁ + 10x₂ + 4ix₃ = 1 + (–16i )

(–4ix₂₎ + 6x₃ + (i + 1)x₄ = –3 –3i

(1 – i ) x₃ + 9x₄ = 25 + 16i

RM, a, 5, 5, /Complex

; Input the complex matrix A.

row 0: 2 (-1,1) 0 0 0

row 1: (-1,-1) 4 (1,2) 0 0

row 2: 0 (1,-2) 10 (0,4) 0

row 3: 0 0 (0,-4) 6 (1,1)

row 4: 0 0 0 (1,-1) 9

RM, b, 5, 1, /Complex

; Input the right-hand side.

row 0: (1, 5)

row 1: (12, -6)

row 2: (1, -16)

row 3: (-3, -3)

row 4: (25, 16)

x = IMSL_CHSOL(b, a)

; Compute the solution.

PM, x, Title = 'Solution', Format = '("(",f8.5,",",f8.5,")")'

; Output the results.

Solution

( 2.00000, 1.00000)

( 3.00000,-0.00000)

(-1.00000,-1.00000)

( 0.00000,-2.00000)

( 3.00000, 2.00000)

PM, a # x-b, Title = 'Residual', Format='("(",f8.5,",",f8.5,")")'

Residual

( 0.00000, 0.00000)

( 0.00000,-0.00000)

( 0.00000, 0.00000)

( 0.00000, 0.00000)

( 0.00000, 0.00000)

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