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.

Usage

result = SP_BDPDSOL(b, ncoda[, a])

Input Parameters

b—One-dimensional matrix containing the right-hand side.

ncoda—Number of upper codiagonals in a.

a—(Optional) Array of size (ncoda + 1) × n containing the n × n banded coefficient matrix in band symmetric storage mode A(i, j). See the chapter introduction for a description of band symmetric storage mode.

Returned Value

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

Input Keywords

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

Double—If present and nonzero, double precision is used.

Output Keywords

Condition—Specifies a 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 Factor.

Discussion

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

Example 1

Solve a system of linear equations Ax = b, where:

,

n = 4L

ncoda = 2L

; Define A in band symmetric storage mode.

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]

b = [6, -11, -11, 19]

; Compute the solution

x = SP_BDPDSOL(b, ncoda, a)

PM, x ; PV-WAVE prints the following:

   ; 4.0000000

   ; -6.0000000

   ; 2.0000000

   ; 9.0000000