PSGERFS(l)

📦 version 1.5, in ubuntu jammy

💖 PSGERFS 💖

okay so bestie, welcome to the PSGERFS(l) page — it's a LAPACK routine, and it's lowkey one of the most useful glow-ups in the whole linear algebra group chat. it takes your kinda-sloppy computed solution and fixes it up, then hands you receipts (error bounds, backward error estimates) proving how good the fix was. iconic behavior, honestly. 🎀

✨ NAME ✨

PSGERFS</purg> — improves the computed solution to a system of linear equations, and provides error bounds and backward error estimates for the solutions. 💅

✨ SYNOPSIS ✨

SUBROUTINE PSGERFS( TRANS, N, NRHS, A, IA, JA, DESCA, AF, IAF, JAF,
                    DESCAF, IPIV, B, IB, JB, DESCB, X, IX, JX, DESCX,
                    FERR, BERR, WORK, LWORK, IWORK, LIWORK, INFO )

    CHARACTER       TRANS

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

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

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

✨ PURPOSE ✨

okay so here's the tea: PSGERFS improves the computed solution to a system of linear equations, and provides error bounds and backward error estimates for the solutions. 🫶

Notes

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. (fr fr, the descriptor is the passport that tells every process where its little slice of the matrix lives.)

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".

  • 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 ✨

  • 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)
  • 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 distributed matrix sub( A ).
  • 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) REAL pointer into the local memory to an array of local dimension (LLD_AF,LOCc(JA+N-1)). This array contains the local pieces of the distributed factors of the matrix sub( A ) = P * L * U as computed by PSGETRF.
  • 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.
  • IPIV — (local input) INTEGER array of dimension LOCr(M_AF)+MB_AF. This array contains the pivoting information as computed by PSGETRF. IPIV(i) -> The global row local row i was swapped with. This array is tied to 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)). This array contains the local pieces of the distributed matrix of 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 and output) 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 distributed matrix solution sub( X ). On exit, the improved solution vectors. (glow-up arc complete ✨)
  • 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 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) 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.
  • 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.

  • 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.

✨ PARAMETERS ✨

ITMAX is the maximum number of steps of iterative refinement. (yes bestie, it loops, it polishes, it serves 💅)

Notes

This routine temporarily returns when N <= 1.

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 ensure that sub( A ) and sub( AF ) (resp. sub( X ) and sub( B )) are "perfectly" aligned.

Moreover, 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.

and that's the whole page, bestie!! this one is LAPACK version 1.5, dated 12 May 1997, and yes the manual page itself is literally named PSGERFS(l). go forth, refine your solutions, and stay aligned on a block boundary 🦋💖✨👑

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