# p?dbtrs

Solves a system of linear equations with a diagonally dominant-like banded distributed matrix using the factorization computed by p?dbtrf.

## Syntax

call psdbtrs(trans, n, bwl, bwu, nrhs, a, ja, desca, b, ib, descb, af, laf, work, lwork, info)

call pddbtrs(trans, n, bwl, bwu, nrhs, a, ja, desca, b, ib, descb, af, laf, work, lwork, info)

call pcdbtrs(trans, n, bwl, bwu, nrhs, a, ja, desca, b, ib, descb, af, laf, work, lwork, info)

call pzdbtrs(trans, n, bwl, bwu, nrhs, a, ja, desca, b, ib, descb, af, laf, work, lwork, info)

## Description

The routine p?dbtrs solves for X one of the systems of equations:

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

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

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

where `sub(A) = A(1:n, ja:ja+n-1)` is a diagonally dominant-like banded distributed matrix, and sub(B) denotes the distributed matrix `B(ib:ib+n-1, 1:nrhs)`.

This routine uses the LU factorization computed by p?dbtrf.

## Input Parameters

trans

  (global) CHARACTER*1. Must be 'N' or 'T' or 'C'.

  Indicates the form of the equations:

  If `trans = 'N'`, then `sub(A)*X = sub(B)` is solved for X.

  If `trans = 'T'`, then `(sub(A))T*X = sub(B)` is solved for X.

  If `trans = 'C'`, then `(sub(A))H*X = sub(B)` is solved for X.

n

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

bwl

  (global) INTEGER. The number of subdiagonals within the band of A

  `( 0 ≤ bwl ≤ n-1 )`.

bwu

  (global) INTEGER. The number of superdiagonals within the band of A

  `( 0 ≤ bwu ≤ n-1 )`.

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 psdbtrs

  DOUBLE PRECISON for pddbtrs

  COMPLEX for pcdbtrs

  DOUBLE COMPLEX for pzdbtrs.

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

  On entry, the array a contains details of the LU factorization of the band matrix A, as computed by p?dbtrf.

  On entry, the array b contains the local pieces of the 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 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, 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)

  REAL for psdbtrs

  DOUBLE PRECISION for pddbtrs

  COMPLEX for pcdbtrs

  DOUBLE COMPLEX for pzdbtrs.

  Arrays of dimension (laf) and (lwork), respectively The array af 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.

laf

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

  Must be laf ≥ `NB*(bwl+bwu)+6*(max(bwl,bwu))2` .

  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 ≥ (max(bwl,bwu))2`.

## Output Parameters

b

  On exit, this array contains the local pieces of the 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 ith 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