IDL Analyst Reference Guide: Linear Systems

IMSL_SP_BDPDSOL

The IMSL_SP_BDPDSOL function solves a symmetric positive definite system of linear equations Ax = b in band symmetric storage mode. Using keywords, any of several related computations can be performed.

Syntax

Result = IMSL_SP_BDPDSOL(b, ncoda[, a] [, CONDITION=variable]
[, /DOUBLE] [, FACTOR=array])

Return Value

A one-dimensional array containing the solution of the linear system Ax = b.

Arguments

b

One-dimensional matrix containing the right-hand side.

ncoda

Number of upper codiagonals in a.

a

(Optional) Array of size (ncoda + 1) x n containing the n x n banded coefficient matrix in band symmetric storage mode A(i, j). See Band Storage Format for a description of band symmetric storage mode.

Keywords

CONDITION

Named variable into which an estimate of the L₁ condition number is stored. This keyword cannot be used if a previously computed factorization is specified with the keyword FACTOR.

DOUBLE

If present and nonzero, double precision is used.

FACTOR

An array of size (ncoda + 1) x N_ELEMENTS(b) containing the R^TR factorization of A in band symmetric storage mode, as returned from IMSL_SP_BDPDFAC.

Discussion

The IMSL_SP_BDPDSOL function solves a system of linear algebraic equations with a symmetric positive definite band coefficient matrix A. It computes the R^TR Cholesky factorization of A. R is an upper triangular band matrix.

The L₁ condition number of A is computed using Higham's modifications to Hager's method, as given in Higham (1988). If the estimated condition number is greater than 1/e (where e 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_SP_BDPDSOL function fails if any submatrix of R is not positive definite or if R has a zero diagonal element. These errors occur only if A is very close to a singular matrix or to a matrix which is not positive definite.

The IMSL_SP_BDPDSOL function is partially based on the LINPACK subroutines CPBFA and SPBSL; see Dongarra et al. (1979).

Example

Solve a system of linear equations Ax = b, where

:

n = 4L  
ncoda = 2L  
a = DBLARR((ncoda+1)*n)  
a(0:n-1) = [0, 0, -1, 1]  
a(n:2L*n-1) = [0, 0, 2, -1]  
a(2L*n:*) = [2, 4, 7, 3]  
; Define A in band symmetric storage mode.  
b = [6, -11, -11, 19]  
x = IMSL_SP_BDPDSOL(b, ncoda, a)  
; Compute the solution  
PM, x  
   4.0000000  
   -6.0000000  
   2.0000000  
   9.0000000