PSTRRFS(l)

📦 version 1.5, in ubuntu jammy

✨ PSTRRFS 💖 bestie's guide to the LAPACK page ✨

Okay so welcome to the PSTRRFS(l) man page, straight out of LAPACK routine version 1.5. Grab a snack, we're doing linear algebra and we're doing it together. 🦋

🌸 NAME 🌸

PSTRRFS — it provides error bounds and backward error estimates for the solution to a system of linear equations with a triangular coefficient matrix. It doesn't solve anything for you, bestie, it just tells you how good the answer you already have actually is. Vibe check for your solution. 👑

✨ SYNOPSIS ✨

SUBROUTINE PSTRRFS( UPLO, TRANS, DIAG, N, NRHS, A, IA, JA, DESCA, B, IB,
                    JB, DESCB, X, IX, JX, DESCX, FERR, BERR, WORK,
                    LWORK, IWORK, LIWORK, INFO )

    CHARACTER       DIAG, TRANS, UPLO

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

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

    REAL            A( * ), B( * ), BERR( * ), FERR( * ), WORK( * ), X(
                    * )

Like, that's a lot of arguments. She's high maintenance but she's worth it. 💅

🎀 PURPOSE 🎀

So PSTRRFS provides error bounds and backward error estimates for the solution to a system of linear equations with a triangular coefficient matrix. That's the tea. ☕

The solution matrix X must be computed by PSTRTRS or some other means before you enter this routine. Like, do the homework first, then come get it graded. ✏️

PSTRRFS does not do iterative refinement, because doing so cannot improve the backward error. Not me refusing to do extra work for zero gain — iconic behaviour honestly. 🫶

📝 Notes 📝

Every global data object is described by an associated description vector. That vector stores the info you need to map an object element to its corresponding process and memory location. It's basically the group chat contact card for your matrix. 📇

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".

  • NOTATION / STORED IN / EXPLANATION:
    • 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 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. Same energy, different axis: 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). Slicing, but make it math. 🔪✨

💜 ARGUMENTS 💜

  • UPLO (global input) CHARACTER*1
    • = 'U': sub( A ) is upper triangular;
    • = 'L': sub( A ) is lower triangular.
  • TRANS (global input) CHARACTER*1 — specifies the form of the system of equations.
    • = 'N': sub( A ) * sub( X ) = sub( B ) (No transpose)
    • = 'T': sub( A )**T * sub( X ) = sub( B ) (Transpose)
    • = 'C': sub( A )**T * sub( X ) = sub( B ) (Conjugate transpose = Transpose)
  • DIAG (global input) CHARACTER*1
    • = 'N': sub( A ) is non-unit triangular;
    • = 'U': sub( A ) is unit 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) REAL 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 original triangular 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. If DIAG = 'U', the diagonal elements of sub( A ) are also not referenced and are assumed to be 1. 🙈 (yes she ignores parts of your array, don't take it personally)
  • 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.
  • B (local input) REAL pointer into the local memory to an array of local dimension (LLD_B, LOCc(JB+NRHS-1) ). On entry, this array contains 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) REAL pointer into the local memory to an array of local dimension (LLD_X, LOCc(JX+NRHS-1) ). On entry, this array contains the local pieces of the solution vectors sub( X ).
  • 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) REAL array of local dimension LOCc(JB+NRHS-1). The estimated forward error bounds for each solution vector of sub( X ). If XTRUE is the true solution, FERR bounds the magnitude of the largest entry in (sub( X ) - XTRUE) divided by the magnitude of the largest entry 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) REAL 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. she's measuring the smallest glow-up that would make your answer perfect, obsessed 🌈
  • WORK (local workspace/local output) REAL 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. (the classic -1 sneak peek 🕵️)
  • 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.

🧚 Notes (the INFO-adjacent ones) 🧚

This routine temporarily returns when N <= 1. tiny problem, tiny nap, we love to see it 💤

The distributed submatrices sub( X ) and sub( B ) should be distributed the same way on the same processes. These conditions ensure that sub( X ) and sub( B ) are "perfectly" aligned. Besties matching outfits. 🫶

Moreover, this routine requires the distributed submatrices sub( A ), 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( IB, DESCB( MB_ ) ) = f( JB, DESCB( NB_ ) ) = 0, and f( IX, DESCX( MB_ ) ) = f( JX, DESCX( NB_ ) ) = 0.

⚠️ No cap: if your submatrices aren't aligned like this, you're not getting the behaviour you think you're getting. Read those MOD conditions twice, bestie. 💀

Okay that's the whole page, you ate it, periodt. 💅 This was PSTRRFS(l) from LAPACK version 1.5, dated 12 May 1997. Go forth and bound your errors. 🦋✨🎀

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