# p?pbtrs

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

## Syntax

call pspbtrs( uplo, n, bw, nrhs, a, ja, desca, b, ib, descb, af, laf, work, lwork, info)

call pdpbtrs( uplo, n, bw, nrhs, a, ja, desca, b, ib, descb, af, laf, work, lwork, info)

call pcpbtrs( uplo, n, bw, nrhs, a, ja, desca, b, ib, descb, af, laf, work, lwork, info)

call pzpbtrs( uplo, n, bw, nrhs, a, ja, desca, b, ib, descb, af, laf, work, lwork, info)

## Description

The routine p?pbtrs solves for X a system of distributed linear equations in the form:

`sub(A)*X = sub(B)` ,

where `sub(A) = A(1:n, ja:ja+n-1)` is an n-by-n real symmetric or complex Hermitian positive definite distributed band matrix, and sub(B) denotes the distributed matrix `B(ib:ib+n-1, 1:nrhs)`.

This routine uses Cholesky factorization

`sub(A) = P*UH*U*PT`, or `sub(A) = P*L*LH*PT`

computed by p?pbtrf.

## Input Parameters

uplo

  (global) CHARACTER*1. Must be 'U' or 'L'.

  If `uplo = 'U'`, upper triangle of sub(A) is stored;

  If `uplo = 'L'`, lower triangle of sub(A) is stored.

n

  (global) INTEGER. The order of the distributed submatrix sub(A) `(n≥0)`.

bw

  (global) INTEGER. The number of superdiagonals of the distributed matrix if `uplo = 'U'`, or the number of subdiagonals if `uplo = 'L' (bw≥0)`.

nrhs

  (global) INTEGER. The number of right hand sides; the number of columns of the distributed submatrix sub(B) (nrhs≥0).

a, b

  (local)

  REAL for pspbtrs

  DOUBLE PRECISION for pdpbtrs

  COMPLEX for pcpbtrs

  DOUBLE COMPLEX for pzpbtrs.

  Pointers into the local memory to arrays of local dimension `a(lld_a,LOCc(ja+n-1))` and `b(lld_b,LOCc(nrhs-1))`, respectively.

  The array a contains the permuted triangular factor U or L from the Cholesky factorization ``sub(A)` = P*UH*U*PT`, or `sub(A)` = P*L*LH*PT of the band matrix A, as returned by p?pbtrf.

  On entry, the array b contains the local pieces of the n-by-nrhs right hand side distributed matrix sub(B).

ja

  (global) INTEGER. The index in the global array A that points to the start of the matrix to be operated on (which may be either all of A or a submatrix of A).

desca

  (global and local) INTEGER array, dimension (dlen_). The array descriptor for the distributed matrix A.

  If `desca(dtype_) = 501`, then `dlen_ ≥ 7`;

  else if `desca(dtype_) = 1`, then `dlen_ ≥ 9`.

ib

  (global) INTEGER. The row index in the global array B indicating the first row of the submatrix sub(B).

descb

  (global and local) INTEGER array, dimension (dlen_). The array descriptor for the distributed matrix B.

  If `descb(dtype_) = 502`, then `dlen_ ≥ 7`;

  else if `descb(dtype_) = 1`, then `dlen_ ≥ 9`.

af, work

  (local) Arrays, same type as a.

  The array af is of dimension (laf). It contains auxiliary Fillin space. Fillin is created during the factorization routine p?dbtrf and this is stored in af.

  The array work is a workspace array of dimension lwork.

laf

  (local) INTEGER. The dimension of the array af.

  Must be `laf ≥ nrhs*bw`.

  If laf is not large enough, an error code will be returned and the minimum acceptable size will be returned in af(1).

lwork

  (local or global) INTEGER. The size of the array work, must be at least `lwork ≥ bw2`.

## Output Parameters

b

  On exit, if `info=0`, this array contains the local pieces of the n-by-nrhs solution distributed matrix X.

`work(1)`

  On exit, `work(1)` contains the minimum value of lwork required for optimum performance.

info

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

  `info < 0`:

  If the i-th argument is an array and the j-th entry had an illegal value, then `info = -(i*100+j)`; if the i-th argument is a scalar and had an illegal value, then info = -i.

Parent topic: Routines for Solving Systems of Linear Equations