PDPORFS(l)

📦 version 1.5, in ubuntu jammy

PDPORFS 💖 (that's PDPORFS(l) bestie — a LAPACK routine, version 1.5)

✨ NAME ✨

PDPORFS — ok so what she DOES is: she improves the computed solution to a system of linear equations when the coefficient matrix is symmetric positive definite, and she also hands you error bounds and backward error estimates for the solutions. queen behavior, honestly. 👑

🌸 SYNOPSIS 🌸

SUBROUTINE 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 )

    CHARACTER       UPLO

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

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

    DOUBLE          PRECISION A( * ), AF( * ), B( * ), BERR( * ), FERR(
                    * ), WORK( * ), X( * )

look at her. so many arguments. so much main character energy. 💅

🦋 PURPOSE 🦋

PDPORFS improves the computed solution to a system of linear equations when the coefficient matrix is symmetric positive definite, and provides error bounds and backward error estimates for the solutions.

Notes (the lore, bestie — read it, iykyk)

Each global data object is described by an associated description vector. This vector stores the information required 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".

ok so here's the whole decoder ring, one bullet at a time because we don't do tables in this house ✨:

  • 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 👑

every single one of them. no skips. periodt. 💅

  • UPLO — (global input) CHARACTER*1. Specifies whether the upper or lower triangular part of the symmetric 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) DOUBLE PRECISION 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 symmetric 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) DOUBLE PRECISION 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**T or U**T*U, as computed by PDPOTRF.
  • 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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 ✨
  • 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.
  • 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) DOUBLE PRECISION 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 >= 3*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.

  • IWORK — (local workspace/local output) INTEGER array, dimension (LIWORK). On exit, IWORK(1) returns the minimal and optimal LIWORK.
  • LIWORK — (local or global input) INTEGER. The dimension of the array IWORK. LIWORK is local input and must be at least
LIWORK >= LOCr( N + MOD( IB-1, MB_B ) )

If LIWORK = -1, then LIWORK 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.
heads up bestie: those WORK / LWORK and IWORK / LIWORK minimums are hard requirements, not suggestions. and the LWORK = -1 / LIWORK = -1 thing flips them into global input workspace-query mode — and when that happens, PXERBLA stays quiet, no error message, she's just quietly handing you the sizes. 🫶

✨ PARAMETERS ✨

  • ITMAX — the maximum number of steps of iterative refinement. That's the whole parameter, bestie. Iconic. 💅

💭 Notes 💭

Okay so first thing's first: this routine temporarily returns when N <= 1. Like, it just peaces out for a sec. Not me being dramatic, it's literally in the code. 👋

The distributed submatrices op( A ) and op( AF ) (respectively sub( X ) and sub( B ) ) should be distributed the same way on the same processes. Why does this matter? Because these conditions ensure that sub( A ) and sub( AF ) (resp. sub( X ) and sub( B ) ) are "perfectly" aligned. No cap. ✨

Also, real talk: 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.
So yeah, those all need to be zero, or it's giving misaligned and the routine is not gonna have a good time. 😭

That's the tea, bestie. This page is PDPORFS(l) from LAPACK version 1.5, dated 12 May 1997 — vintage but still slaying. 👑💖

📦 where this page comes from