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 Advanced Math and Stats license. For more information, contact your ITT Visual Information Solutions sales or technical support representative.

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) contains the 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 L^H. 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.

Discussion

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

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 
   7.00000 
PM, a # x - b, Title = 'Residual' 
Residual 
   0.00000 
   0.00000 
   0.00000

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)

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 #.

Version History


6.4	Introduced