# p?gbsv

Computes the solution to the system of linear equations with a general banded distributed matrix and multiple right-hand sides.

## Syntax

call psgbsv(n, bwl, bwu, nrhs, a, ja, desca, ipiv, b, ib, descb, work, lwork, info)

call pdgbsv(n, bwl, bwu, nrhs, a, ja, desca, ipiv, b, ib, descb, work, lwork, info)

call pcgbsv(n, bwl, bwu, nrhs, a, ja, desca, ipiv, b, ib, descb, work, lwork, info)

call pzgbsv(n, bwl, bwu, nrhs, a, ja, desca, ipiv, b, ib, descb, work, lwork, info)

## Description

The routine p?gbsv computes the solution to a real or complex system of linear equations

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

where `sub(A) = A(1:n, ja:ja+n-1)` is an n-by-n real/complex general banded distributed matrix with bwl subdiagonals and bwu superdiagonals, and X and `sub(B)= B(ib:ib+n-1, 1:rhs)` are n-by-nrhs distributed matrices.

The LU decomposition with partial pivoting and row interchanges is used to factor sub(A) as `sub(A) = P*L*U*Q`, where P and Q are permutation matrices, and L and U are banded lower and upper triangular matrices, respectively. The matrix Q represents reordering of columns for the sake of parallelism, while P represents reordering of rows for numerical stability using classic partial pivoting.

## Input Parameters

n

  (global) INTEGER. The number of rows and columns to be operated on, that is, 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 psgbsv

  DOUBLE PRECISON for pdgbsv

  COMPLEX for pcgbsv

  DOUBLE COMPLEX for pzgbsv.

  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 the local pieces of the global array A.

  On entry, the array b contains 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`.

work

  (local)

  REAL for psgbsv

  DOUBLE PRECISON for pdgbsv

  COMPLEX for pcgbsv

  DOUBLE COMPLEX for pzgbsv.

  Workspace array of dimension (lwork).

lwork

  (local or global) INTEGER. The size of the array work, must be at least `lwork ≥ (NB+bwu)*(bwl+bwu)+6*(bwl+bwu)*(bwl+2*bwu) +`

  `+ max(nrhs *(NB+2*bwl+4*bwu), 1)`.

## Output Parameters

a

  On exit, contains details of the factorization. Note that the resulting factorization is not the same factorization as returned from LAPACK. Additional permutations are performed on the matrix for the sake of parallelism.

b

  On exit, this array contains the local pieces of the solution distributed matrix X.

ipiv

  (local) INTEGER array.

  The dimension of ipiv must be at least desca(NB). This array contains pivot indices for local factorizations. You should not alter the contents between factorization and solve.

`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 ith argument is a scalar and had an illegal value, then `info = -i`.

  `info > 0`:

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

Parent topic: Driver Routines