PZPORFS(l)

📦 version 1.5, in ubuntu jammy

💖 PZPORFS — LAPACK routine (version 1.5) 💖

okay bestie, sit DOWN, because this one is a whole journey. welcome to PZPORFS from LAPACK, version 1.5 — she's a parallel girlie, she runs on a whole process grid, and she is literally here to make your Hermitian positive definite linear systems better. 💅

✨ NAME ✨

PZPORFS — improve the computed solution to a system of linear equations when the coefficient matrix is Hermitian positive definite, and provides error bounds and backward error estimates for the solutions. 🎀

🦋 SYNOPSIS 🦋

SUBROUTINE 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 )

    CHARACTER       UPLO

    INTEGER         IA, IAF, IB, INFO, IX, JA, JAF, JB, JX, LRWORK,
                    LWORK, N, NRHS

    INTEGER         DESCA( * ), DESCAF( * ), DESCB( * ), DESCX( * )

    COMPLEX*16      A( * ), AF( * ), B( * ), BERR( * ), FERR( * ), WORK(
                    * ), X( * )

    DOUBLE          PRECISION RWORK( * )

🌸 PURPOSE 🌸

PZPORFS improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian positive definite, and provides error bounds and backward error estimates for the solutions. no cap, that's the whole assignment. 💖

💫 Notes 💫

Every global data object out here has its own description vector, and that vector is basically the group chat where the object keeps all the info needed to establish the mapping between an object element and its corresponding process and memory location. 📱

Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array".

okay so here's the lore, one bullet at a time 👑:

DTYPE_A (global) — stored in DESCA( DTYPE_ ). The descriptor type. In this case, DTYPE_A = 1.
CTXT_A (global) — stored in DESCA( CTXT_ ). The BLACS context handle, indicating the BLACS process grid A is distributed over. The context itself is global, but the handle (the integer value) may vary.
M_A (global) — stored in DESCA( M_ ). The number of rows in the global array A.
N_A (global) — stored in DESCA( N_ ). The number of columns in the global array A.
MB_A (global) — stored in DESCA( MB_ ). The blocking factor used to distribute the rows of the array.
NB_A (global) — stored in DESCA( NB_ ). The blocking factor used to distribute the columns of the array.
RSRC_A (global) — stored in DESCA( RSRC_ ). The process row over which the first row of the array A is distributed.
CSRC_A (global) — stored in DESCA( CSRC_ ). The process column over which the first column of the array A is distributed.
LLD_A (local) — stored in DESCA( LLD_ ). The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)).

Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q.

LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column.

Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row.

The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC:

LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ),
LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ).

An upper bound for these quantities may be computed by:

LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A
LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A

In the following comments, sub( A ), sub( X ) and sub( B ) denote respectively A(IA:IA+N-1,JA:JA+N-1), X(IX:IX+N-1,JX:JX+NRHS-1) and B(IB:IB+N-1,JB:JB+NRHS-1). 📝

🎀 ARGUMENTS 🎀

UPLO (global input) CHARACTER*1 — Specifies whether the upper or lower triangular part of the Hermitian matrix sub( A ) is stored. = 'U': Upper triangular. = 'L': Lower triangular. 💅
N (global input) INTEGER — The order of the matrix sub( A ). N >= 0.
NRHS (global input) INTEGER — The number of right hand sides, i.e., the number of columns of the matrices sub( B ) and sub( X ). NRHS >= 0.
A (local input) COMPLEX*16 pointer into the local memory to an array of local dimension (LLD_A,LOCc(JA+N-1) ). This array contains the local pieces of the N-by-N Hermitian distributed matrix sub( A ) to be factored. 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.
IA (global input) INTEGER — The row index in the global array A indicating the first row of sub( A ).
JA (global input) INTEGER — The column index in the global array A indicating the first column of sub( A ).
DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. 🗂️
AF (local input) COMPLEX*16 pointer into the local memory to an array of local dimension (LLD_AF,LOCc(JA+N-1)). On entry, this array contains the factors L or U from the Cholesky factorization sub( A ) = L*L**H or U**H*U, as computed by PZPOTRF. 🔥
IAF (global input) INTEGER — The row index in the global array AF indicating the first row of sub( AF ).
JAF (global input) INTEGER — The column index in the global array AF indicating the first column of sub( AF ).
DESCAF (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix AF.
B (local input) COMPLEX*16 pointer into the local memory to an array of local dimension (LLD_B, LOCc(JB+NRHS-1) ). On entry, this array contains the the local pieces of the right hand sides sub( B ). 🫶
IB (global input) INTEGER — The row index in the global array B indicating the first row of sub( B ).
JB (global input) INTEGER — The column index in the global array B indicating the first column of sub( B ).
DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B.
X (local input) COMPLEX*16 pointer into the local memory to an array of local dimension (LLD_X, LOCc(JX+NRHS-1) ). On entry, this array contains the the local pieces of the solution vectors sub( X ). On exit, it contains the improved solution vectors. ✨ (glow up arc, honestly)
IX (global input) INTEGER — The row index in the global array X indicating the first row of sub( X ).
JX (global input) INTEGER — The column index in the global array X indicating the first column of sub( X ).
DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X.
FERR (local output) DOUBLE PRECISION array of local dimension LOCc(JB+NRHS-1). 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. 😭 (she's doing her best, and her best is a slight overestimate)
BERR (local output) DOUBLE PRECISION array of local dimension LOCc(JB+NRHS-1). The componentwise relative backward error of each solution vector (i.e., 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 (local workspace/local output) COMPLEX*16 array, dimension (LWORK). On exit, WORK(1) returns the minimal and optimal LWORK.
LWORK (local or global input) INTEGER — The dimension of the array WORK. LWORK is local input and must be at least

LWORK >= 2*LOCr( N + MOD( IA-1, MB_A ) )

If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. 🧚

RWORK (local workspace/local output) DOUBLE PRECISION array, dimension (LRWORK). On exit, RWORK(1) returns the minimal and optimal LRWORK.
LRWORK (local or global input) INTEGER — The dimension of the array RWORK. LRWORK is local input and must be at least

LRWORK >= LOCr( N + MOD( IB-1, MB_B ) ).

If LRWORK = -1, then LRWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. 🌈

INFO (global output) INTEGER —
- = 0: successful exit 🎉
- < 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. 💀

okay that's the setup, the lore, and every single argument, bestie — and we are SO not done yet, there's more of this page coming so don't you dare close the tab 🌸💜

✨ PARAMETERS ✨

ITMAX — the maximum number of steps of iterative refinement, bestie. That's the cap. It goes, it refines, and when it's hit ITMAX steps it's done. 💅

📝 Notes 📝

Okay so here's the tea you actually need before you run this, because the vibes will not save you if the data layout is wrong. 🙈

Not me 👀 — this routine temporarily returns when N <= 1. Yes, it just dips. So if your N is tiny, don't come crying that it bailed early, that's literally the design.
The distributed submatrices op( A ) and op( AF ) (respectively sub( X ) and sub( B )) should be distributed the same way on the same processes. These conditions are what ensure that sub( A ) and sub( AF ) (resp. sub( X ) and sub( B )) are "perfectly" aligned. It's giving matching outfits on purpose, fr fr — same layout, same processes, or it simply doesn't work. 🫶
On top of that, this routine requires the distributed submatrices sub( A ), sub( AF ), sub( X ), and sub( B ) to be aligned on a block boundary, i.e.:

if f(x,y) = MOD( x-1, y ):
    f( IA, DESCA( MB_ ) ) = f( JA, DESCA( NB_ ) ) = 0,
    f( IAF, DESCAF( MB_ ) ) = f( JAF, DESCAF( NB_ ) ) = 0,
    f( IB, DESCB( MB_ ) ) = f( JB, DESCB( NB_ ) ) = 0, and
    f( IX, DESCX( MB_ ) ) = f( JX, DESCX( NB_ ) ) = 0.

Lowkey the most important thing on this whole page: if your submatrices aren't aligned like that, you're not "close enough" — you're just wrong. Block boundary. Periodt. 🎀

And that's the whole page, babe. 💜 This icon of a routine is living in LAPACK version 1.5, and it was blessed upon the world on 12 May 1997 — a vintage queen, honestly, and still serving. Slay. ✨👑

📦 where this page comes from

ubuntu jammy: scalapack-doc 1.5-11
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash on 2026-10-14: a language model's rewrite of the original, under its terms