# p?pbsv

Solves a symmetric/Hermitian positive definite banded system of linear equations.

## Syntax

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

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

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

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

## Description

This routine solves a system of linear equations

`A(1:n, ja:ja+n-1)*X = B(ib:ib+n-1, 1:nrhs)`,

where `A(1:n, ja:ja+n-1)` is an n-by-n real/complex banded symmetric positive definite distributed matrix with bandwidth bw.

Cholesky factorization is used to factor a reordering of the matrix into L*L'.

## Input Parameters

uplo

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

  Indicates whether the upper or lower triangular of A is stored.

  If `uplo = 'U'`, the upper triangular A is stored

  If `uplo = 'L'`, the lower triangular of A is stored.

n

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

bw

  (global) INTEGER. The number of subdiagonals in L or U. `0 ≤ bw ≤ n-1`.

nrhs

  (global) INTEGER. The number of right-hand sides; the number of columns in B `(nrhs ≥ 0)`.

a

  (local). REAL for pspbsv

  DOUBLE PRECISON for pdpbsv

  COMPLEX for pcpbsv

  DOUBLE COMPLEX for pzpbsv.

  Pointer into the local memory to an array with first dimension `lld_a ≥ (bw+1)` (stored in desca). On entry, this array contains the local pieces of the distributed matrix `sub(A)` to be factored.

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.

b

  (local)

  REAL for pspbsv

  DOUBLE PRECISON for pdpbsv

  COMPLEX for pcpbsv

  DOUBLE COMPLEX for pzpbsv.

  Pointer into the local memory to an array of local lead dimension lld_b ≥ nb. On entry, this array contains the local pieces of the right hand sides `B(ib:ib+n-1, 1:nrhs)`.

ib

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

descb

  (global and local) INTEGER array of dimension dlen.

  If `1D type (dtype_b =502)`, `dlen ≥ 7`;

  If `2D type (dtype_b =1)`, `dlen ≥ 9`.

  The array descriptor for the distributed matrix B.

  Contains information of mapping of B to memory.

work

  (local).

  REAL for pspbsv

  DOUBLE PRECISON for pdpbsv

  COMPLEX for pcpbsv

  DOUBLE COMPLEX for pzpbsv.

  Temporary workspace. This space may be overwritten in between calls to routines. work must be the size given in lwork.

lwork

  (local or global) INTEGER. Size of user-input workspace work. If lwork is too small, the minimal acceptable size will be returned in `work(1)`and an error code is returned. `lwork ≥ (nb+2*bw)*bw +max((bw*nrhs), bw*bw)`

## Output Parameters

a

  On exit, this array contains information containing details of the factorization. Note that permutations are performed on the matrix, so that the factors returned are different from those returned by LAPACK.

b

  On exit, contains the local piece of the solutions distributed matrix X.

work

  On exit, `work(1)` contains the minimal lwork.

info

  (global). INTEGER. If info=0, the execution is successful.

  `< 0`: If the i-th argument is an array and the j-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`.

  `> 0`: If `info = k ≤ NPROCS`, the submatrix stored on processor info and factored locally was not positive definite, and the factorization was not completed.

  If `info = k > NPROCS`, the submatrix stored on processor `info-NPROCS` representing interactions with other processors was not positive definite, and the factorization was not completed.

Parent topic: Driver Routines