# p?dbtrsv

Computes an LU factorization of a general triangular matrix with no pivoting. The routine is called by p?dbtrs.

## Syntax

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

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

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

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

## Description

This routines solves a banded triangular system of linear equations

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

A(1 :n, `ja:ja+n-1)`T * X = B(ib:ib+n-1, 1 :nrhs) (for real flavors); A(1 :n, `ja:ja+n-1)`H * X = B(ib:ib+n-1, 1 :nrhs) (for complex flavors),

where A(1 :n, `ja:ja+n-1)` is a banded triangular matrix factor produced by the Gaussian elimination code `PD@(dom_pre)BTRF` and is stored in A(1 :n, `ja:ja+n-1)` and af. The matrix stored in A(1 :n, `ja:ja+n-1)` is either upper or lower triangular according to uplo, and the choice of solving A(1 :n, `ja:ja+n-1)` or A(1 :n, `ja:ja+n-1)`T is dictated by the user by the parameter trans.

Routine p?dbtrf must be called first.

## Input Parameters

uplo

  (global) CHARACTER.

  If `uplo='U'`, the upper triangle of A(1:n, `ja:ja+n-1)` is stored,

  if uplo = 'L', the lower triangle of A(1:n, `ja:ja+n-1)` is stored.

trans

  (global) CHARACTER.

  If trans = 'N', solve with A(1:n, `ja:ja+n-1)`,

  if trans = 'C', solve with conjugate transpose A(1:n, `ja:ja+n-1)`.

n

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

bwl

  (global) INTEGER. Number of subdiagonals. 0 ≤ `bwl ≤ n`-1.

bwu

  (global) INTEGER. Number of subdiagonals. 0 ≤ `bwu ≤ n`-1.

nrhs

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

a

  (local).

  REAL for psdbtrsv

  DOUBLE PRECISION for pddbtrsv

  COMPLEX for pcdbtrsv

  COMPLEX*16 for pzdbtrsv.

  Pointer into the local memory to an array with first DIMENSION lld_a≥(bwl+bwu+1)(stored in desca). On entry, this array contains the local pieces of the n-by-n unsymmetric banded distributed Cholesky factor L or `LT*A(1 :n, `ja:ja+n-1)``. This local portion is stored in the packed banded format used in LAPACK. See the Application Notes below and the ScaLAPACK manual for more detail on the format of distributed matrices.

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 of DIMENSION (dlen_).

  if 1d type (dtype_a = 501 or 502), dlen≥ 7;

  if 2d type (dtype_a = 1), dlen≥ 9. The array descriptor for the distributed matrix A. Contains information of mapping of A to memory.

b

  (local)

  REAL for psdbtrsv

  DOUBLE PRECISION for pddbtrsv

  COMPLEX for pcdbtrsv

  COMPLEX*16 for pzdbtrsv.

  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 B to memory.

laf

  (local)

  INTEGER. Size of user-input Auxiliary Filling space af.

  laf must be `≥nb*(bwl+bwu)+6*max(bwl, bwu)*max(bwl, bwu)`. If laf is not large enough, an error code is returned and the minimum acceptable size will be returned in `af(1)`.

work

  (local).

  REAL for psdbtrsv

  DOUBLE PRECISION for pddbtrsv

  COMPLEX for pcdbtrsv

  COMPLEX*16 for pzdbtrsv.

  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≥ max(bwl, bwu)*nrhs`.

## Output Parameters

a

  (local).

  This local portion is stored in the packed banded format used in LAPACK. Please see the ScaLAPACK manual for more detail on the format of distributed matrices.

b

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

af

  (local).

  REAL for psdbtrsv

  DOUBLE PRECISION for pddbtrsv

  COMPLEX for pcdbtrsv

  COMPLEX*16 for pzdbtrsv.

  Auxiliary Filling Space. Filling is created during the factorization routine p?dbtrf and this is stored in af. If a linear system is to be solved using p?dbtrf after the factorization routine, af must not be altered after the factorization.

work

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

info

  (local).

  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`.

Parent topic: Auxiliary Routines