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### IMSL_SP_BDPDSOL

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.

The IMSL_SP_BDPDSOL function solves a system of linear algebraic equations with a symmetric positive definite band coefficient matrix A. It computes the RTR 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/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_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`

## 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 (optional)

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 (optional)

If present and nonzero, double precision is used.

### FACTOR (optional)

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

## Version History

 6.4 Introduced