# ?pbtrs

Solves a system of linear equations with a Cholesky-factored symmetric (Hermitian) positive-definite band matrix.

## Syntax

FORTRAN 77:

call spbtrs( uplo, n, kd, nrhs, ab, ldab, b, ldb, info )

call dpbtrs( uplo, n, kd, nrhs, ab, ldab, b, ldb, info )

call cpbtrs( uplo, n, kd, nrhs, ab, ldab, b, ldb, info )

call zpbtrs( uplo, n, kd, nrhs, ab, ldab, b, ldb, info )

Fortran 95:

call pbtrs( a, b [,uplo] [,info] )

## Description

This routine solves for real data a system of linear equations `A*X = B` with a symmetric positive-definite or, for complex data, Hermitian positive-definite band matrix A, given the Cholesky factorization of A:

`A = UT*U` for real data, `A = UH*U` for complex data

  If `uplo='U'`

`A = L*LT` for real data, `A = L*LH` for complex data

  If `uplo='L'`,

where L is a lower triangular matrix and U is upper triangular. The system is solved with multiple right-hand sides stored in the columns of the matrix B.

Before calling this routine, you must call ?pbtrf to compute the Cholesky factorization of A in the band storage form.

## Input Parameters

uplo

  CHARACTER*1. Must be 'U' or 'L'.

  Indicates how the input matrix A has been factored:

  If `uplo = 'U'`, the upper triangular factor is stored in ab.

  If `uplo = 'L'`, the lower triangular factor is stored in ab.

n

  INTEGER. The order of matrix A; n ≥ 0.

kd

  INTEGER. The number of superdiagonals or subdiagonals in the matrix A; kd ≥ 0.

nrhs

  INTEGER. The number of right-hand sides; nrhs ≥ 0.

ab, b

  REAL for spbtrs

  DOUBLE PRECISION for dpbtrs

  COMPLEX for cpbtrs

  DOUBLE COMPLEX for zpbtrs.

  Arrays: `ab(ldab,*)`,`b(ldb,*)`.

  The array ab contains the Cholesky factor, as returned by the factorization routine, in band storage form.

  The array b contains the matrix B whose columns are the right-hand sides for the systems of equations.

  The second dimension of ab must be at least `max(1, n)`, and the second dimension of b at least `max(1,nrhs)`.

ldab

  INTEGER. The first dimension of the array ab; ldab ≥ kd +1.

ldb

  INTEGER. The first dimension of b; `ldb ≥ max(1, n)`.

## Output Parameters

b

  Overwritten by the solution matrix X.

info

  INTEGER. If `info=0`, the execution is successful.

  If `info = -i`, the i-th parameter had an illegal value.

## Fortran 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their Fortran 77 counterparts. For general conventions applied to skip redundant or reconstructible arguments, see Fortran 95 Interface Conventions.

Specific details for the routine pbtrs interface are as follows:

a

  Stands for argument ab in Fortan 77 interface. Holds the array A of size `(kd+1,n)`.

b

  Holds the matrix B of size (n, nrhs).

uplo

  Must be 'U' or 'L'. The default value is 'U'.

## Application Notes

For each right-hand side b, the computed solution is the exact solution of a perturbed system of equations `(A + E)x = b`, where
`|E| ≤ c(kd + 1)ε P|UH||U| or |E| ≤ c(kd + 1)ε P|LH||L|`

`c(k)` is a modest linear function of k, and ε is the machine precision.

If `x0` is the true solution, the computed solution x satisfies this error bound:

where `cond(A,x)`= || |A-1||A| |x| ||∞ / ||x||∞ ≤ ||A-1||∞ ||A||∞ = κ∞(A).

Note that `cond(A,x)` can be much smaller than `κ∞(A)`.

The approximate number of floating-point operations for one right-hand side vector is 4n*kd for real flavors and 16n*kd for complex flavors.

To estimate the condition number `κ∞(A)`, call ?pbcon.

To refine the solution and estimate the error, call ?pbrfs.

Parent topic: Routines for Solving Systems of Linear Equations