# p?porfs

Improves the computed solution to a system of linear equations with symmetric/Hermitian positive definite distributed matrix and provides error bounds and backward error estimates for the solution.

## Syntax

call psporfs(uplo, n, nrhs, a, ia, ja, desca, af, iaf, jaf, descaf, b, ib, jb, descb, x, ix, jx, descx, ferr, berr, work, lwork, iwork, liwork, info)

call pdporfs(uplo, n, nrhs, a, ia, ja, desca, af, iaf, jaf, descaf, b, ib, jb, descb, x, ix, jx, descx, ferr, berr, work, lwork, iwork, liwork, info)

call pcporfs(uplo, n, nrhs, a, ia, ja, desca, af, iaf, jaf, descaf, b, ib, jb, descb, x, ix, jx, descx, ferr, berr, work, lwork, rwork, lrwork, info)

call pzporfs(uplo, n, nrhs, a, ia, ja, desca, af, iaf, jaf, descaf, b, ib, jb, descb, x, ix, jx, descx, ferr, berr, work, lwork, rwork, lrwork, info)

## Description

The routine p?porfs improves the computed solution to the system of linear equations

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

where `sub(A) = A(ia:ia+n-1, ja:ja+n-1)` is a real symmetric or complex Hermitian positive definite distributed matrix and

`sub(B) = B(ib:ib+n-1, jb:jb+nrhs-1)`,

`sub(X) = X(ix:ix+n-1, jx:jx+nrhs-1)`

are right-hand side and solution submatrices, respectively. This routine also provides error bounds and backward error estimates for the solution.

## Input Parameters

uplo

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

  Specifies whether the upper or lower triangular part of the symmetric/Hermitian matrix sub(A) is stored.

  If `uplo = 'U'`, sub(A) is upper triangular. If `uplo = 'L'`, sub(A) is lower triangular.

n

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

nrhs

  (global) INTEGER. The number of right-hand sides, i.e., the number of columns of the matrices sub(B) and sub(X) `(nrhs≥0)`.

a, af, b, x

  (local)

  REAL for psporfs

  DOUBLE PRECISION for pdporfs

  COMPLEX for pcporfs

  DOUBLE COMPLEX for pzporfs.

  Pointers into the local memory to arrays of local dimension

  `a(lld_a,LOCc(ja+n-1))`, `af(lld_af,LOCc(ja+n-1))`, `b(lld_b,LOCc(jb+nrhs-1))`, and `x(lld_x,LOCc(jx+nrhs-1))`, respectively.

  The array a contains the local pieces of the n-by-n symmetric/Hermitian distributed matrix sub(A).

  If `uplo = 'U'`, the leading n-by-n upper triangular part of sub(A) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced.

  If `uplo = 'L'`, the leading n-by-n lower triangular part of sub(A) contains the lower triangular part of the distributed matrix, and its strictly upper triangular part is not referenced.

  The array af contains the factors L or U from the Cholesky factorization `sub(A) = L*LH` or `sub(A) = UH*U`, as computed by p?potrf.

  On entry, the array b contains the local pieces of the distributed matrix of right hand sides sub(B).

  On entry, the array x contains the local pieces of the solution vectors sub(X).

ia, ja

  (global) INTEGER. The row and column indices in the global array A indicating the first row and the first column of the submatrix sub(A), respectively.

desca

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

iaf, jaf

  (global) INTEGER. The row and column indices in the global array AF indicating the first row and the first column of the submatrix sub(AF), respectively.

descaf

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

ib, jb

  (global) INTEGER. The row and column indices in the global array B indicating the first row and the first column of the submatrix sub(B), respectively.

descb

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

ix, jx

  (global) INTEGER. The row and column indices in the global array X indicating the first row and the first column of the submatrix sub(X), respectively.

descx

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

work

  (local)

  REAL for psporfs

  DOUBLE PRECISION for pdporfs

  COMPLEX for pcporfs

  DOUBLE COMPLEX for pzporfs.

  The array work of dimension (lwork) is a workspace array.

lwork

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

  For real flavors:

  lwork must be at least

  `lwork ≥ 3*LOCr(n+mod(ia-1,mb_a))`

  For complex flavors:

  lwork must be at least

  `lwork ≥ 2*LOCr(n+mod(ia-1,mb_a))`

iwork

  (local) INTEGER. Workspace array, DIMENSION (liwork). Used in real flavors only.

liwork

  (local or global) INTEGER. The dimension of the array iwork; used in real flavors only. Must be at least

  `liwork ≥ LOCr(n+mod(ib-1,mb_b))`.

rwork

  (local) REAL for pcporfs

  DOUBLE PRECISION for pzporfs

  Workspace array, DIMENSION (lrwork). Used in complex flavors only.

lrwork

  (local or global) INTEGER. The dimension of the array rwork; used in complex flavors only. Must be at least `lrwork ≥ LOCr(n+mod(ib-1,mb_b)))`.

## Output Parameters

x

  On exit, contains the improved solution vectors.

ferr, berr

  REAL for single precision flavors.

  DOUBLE PRECISION for double precision flavors.

  Arrays, dimension `LOCc(jb+nrhs-1)` each.

  The array ferr contains the estimated forward error bound for each solution vector of sub(X).

  If XTRUE is the true solution corresponding to sub(X), ferr is an estimated upper bound for the magnitude of the largest element in `(sub(X) - XTRUE)`divided by the magnitude of the largest element in `sub(X)`. The estimate is as reliable as the estimate for rcond, and is almost always a slight overestimate of the true error.

  This array is tied to the distributed matrix X.

  The array berr contains the component-wise relative backward error of each solution vector (that is, the smallest relative change in any entry of sub(A) or sub(B) that makes sub(X) an exact solution). This array is tied to the distributed matrix X.

`work(1)`

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

`iwork(1)`

  On exit, `iwork(1)` contains the minimum value of liwork required for optimum performance (for real flavors).

`rwork(1)`

  On exit, `rwork(1)` contains the minimum value of lrwork required for optimum performance (for complex flavors).

info

  (global) 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: Refining the Solution and Estimating Its Error