The IMSL_CHSOL function solves a symmetric positive definite system of real or complex linear equations Ax = b.
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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 LLH 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
row 0: 1 -3 2
row 1: -3 10 -5
row 2: 2 -5 6
RM, b, 3, 1
row 0: 27
row 1: -78
row 2: 64
x = IMSL_CHSOL(b, a)
PM, x, Title = 'Solution'
IDL prints:
Solution
1.00000
-4.00000
Chapter 4: Linear Systems 93
IDL Analyst Reference Guide IMSL_CHSOL
7.00000
PM, a # x - b, Title = 'Residual'
IDL prints:
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
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
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)
PM, x, Title = 'Solution', Format = '("(",f8.5,",",f8.5,")")'
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 #.
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 (optional)
Named variable into which an estimate of the L1 condition number is stored. The CONDITION and FACTOR keywords cannot be used together.
DOUBLE (optional)
If present and nonzero, double precision is used.
FACTOR (optional)
Named variable in which the LLH 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 (optional)
Specifies a named variable into which the inverse of the matrix A is stored. This keyword is not allowed if A is complex.
Version History
See Also
IMSL_CHFAC, IMSL_LUFAC, IMSL_LUSOL