PZTRRFS(l)

📦 version 1.5, in ubuntu jammy

PZTRRFS 💖

okay bestie, welcome to PZTRRFS, a LAPACK routine (we're on version 1.5 here, vintage era, still iconic). She's a ScaLAPACK girlie: distributed matrices, BLACS grids, descriptors everywhere. Buckle up. 🦋

✨ NAME ✨

PZTRRFS — she provides error bounds and backward error estimates for the solution to a system of linear equations with a triangular coefficient matrix. That's the whole job. That's the assignment. 💅

🌸 SYNOPSIS 🌸

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

    CHARACTER       DIAG, TRANS, UPLO

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

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

    DOUBLE          PRECISION BERR( * ), FERR( * ), RWORK( * )

    COMPLEX*16      A( * ), B( * ), WORK( * ), X( * )

💜 PURPOSE 💜

PZTRRFS provides error bounds and backward error estimates for the solution to a system of linear equations with a triangular coefficient matrix.

Real talk: the solution matrix X has to already exist before you walk in here. You get it from PZTRTRS or some other way. PZTRRFS does not do iterative refinement — because doing that literally cannot improve the backward error. So she doesn't. Periodt. 💅

📝 Notes 📝

Every global data object gets described by its own little description vector. That vector is the one holding all the info you need to map an element of the object to its process and its memory location. Think of it as the group chat admin. 🫶

Let A be a generic term for any 2D block cyclicly distributed array. That global array has an associated description vector DESCA. In the comments below, whenever you see the character _, read it as "of the global array".

  • 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)).

Now 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 ) is 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, other direction: LOCc( K ) is the number of elements of K that a process would receive if K were distributed over the q processes of its process row.

You can get LOCr() and LOCc() by calling 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 ).

And an upper bound for these quantities can be computed like:

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

In all the comments that follow, sub( A ), sub( X ) and sub( B ) mean, 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
    • = '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 )**H * sub( X ) = sub( B ) (Conjugate 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) 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 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. (she's off the clock, don't @ her)
  • 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) 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 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 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) DOUBLE PRECISION 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) 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.

🧚 Notes 🧚

This routine temporarily returns when N <= 1. She said not today. 🙈
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. 💫
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.

and that's the whole page bestie, every flag, every descriptor, every LOCr — no cap, nothing skipped. 💖

this absolute classic is LAPACK version 1.5, dated 12 May 1997 — yes she's older than most of the internet and she still ate. 🎀👑

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