PZGELS(l)

📦 version 1.5, in ubuntu jammy

💖 PZGELS 💖 — the LAPACK routine, but explained by your bestie

okay so bestie, welcome to PZGELS(l) 🫶 she's a LAPACK routine, version 1.5, and she is genuinely out here solving linear systems like it's nothing. buckle up, we're doing math but make it cute. ✨

🎀 NAME 🎀

PZGELS solves overdetermined or underdetermined complex linear systems involving an M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1), no cap. 💅

✨ SYNOPSIS ✨

SUBROUTINE PZGELS( TRANS, M, N, NRHS, A, IA, JA, DESCA, B, IB, JB,
                   DESCB, WORK, LWORK, INFO )

    CHARACTER      TRANS

    INTEGER        IA, IB, INFO, JA, JB, LWORK, M, N, NRHS

    INTEGER        DESCA( * ), DESCB( * )

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

🌸 PURPOSE 🌸

PZGELS solves overdetermined or underdetermined complex linear systems involving an M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1), or its conjugate-transpose, using a QR or LQ factorization of sub( A ). and the assumption, the vibe check she passes every time, is that sub( A ) has full rank. 👑

here's the menu, and honestly every option ate:

  1. if TRANS = 'N' and m >= n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem
minimize || sub( B ) - sub( A )*X ||.
  1. if TRANS = 'N' and m < n: find the minimum norm solution of an underdetermined system sub( A ) * X = sub( B ).
  2. if TRANS = 'C' and m >= n: find the minimum norm solution of an undetermined system sub( A )**H * X = sub( B ).
  3. if TRANS = 'C' and m < n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem
minimize || sub( B ) - sub( A )**H * X ||.

where sub( B ) denotes B( IB:IB+M-1, JB:JB+NRHS-1 ) when TRANS = 'N' and B( IB:IB+N-1, JB:JB+NRHS-1 ) otherwise. several right hand side vectors b and solution vectors x can be handled in a single call — one call, many columns, she's efficient like that 🦋 — When TRANS = 'N', the solution vectors are stored as the columns of the N-by-NRHS right hand side matrix sub( B ) and the M-by-NRHS right hand side matrix sub( B ) otherwise.

💜 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. iykyk.

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)).

now 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

👑 ARGUMENTS 👑

  • TRANS — (global input) CHARACTER
    • = 'N': the linear system involves sub( A );
    • = 'C': the linear system involves sub( A )**H.
  • M — (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. (so no negative rows, babe 💀)
  • N — (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0.
  • NRHS — (global input) INTEGER The number of right hand sides, i.e. the number of columns of the distributed submatrices sub( B ) and X. NRHS >= 0.
  • A — (local input/local output) COMPLEX*16 pointer into the local memory to an array of local dimension ( LLD_A, LOCc(JA+N-1) ). On entry, the M-by-N matrix A. if M >= N, sub( A ) is overwritten by details of its QR factorization as returned by PZGEQRF; if M < N, sub( A ) is overwritten by details of its LQ factorization as returned by PZGELQF. (yes she takes over your array, that's the deal, she's the main character 💅)
  • 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/local output) 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 distributed matrix B of right hand side vectors, stored columnwise; sub( B ) is M-by-NRHS if TRANS='N', and N-by-NRHS otherwise. On exit, sub( B ) is overwritten by the solution vectors, stored columnwise: if TRANS = 'N' and M >= N, rows 1 to N of sub( B ) contain the least squares solution vectors; the residual sum of squares for the solution in each column is given by the sum of squares of elements N+1 to M in that column; if TRANS = 'N' and M < N, rows 1 to N of sub( B ) contain the minimum norm solution vectors; if TRANS = 'C' and M >= N, rows 1 to M of sub( B ) contain the minimum norm solution vectors; if TRANS = 'C' and M < N, rows 1 to M of sub( B ) contain the least squares solution vectors; the residual sum of squares for the solution in each column is given by the sum of squares of elements M+1 to N in that column. 🍓
  • 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.
  • 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 >= LTAU + MAX( LWF, LWS ) where If M >= N,
then LTAU = NUMROC( JA+MIN(M,N)-1, NB_A, MYCOL, CSRC_A, NPCOL ),
LWF  = NB_A * ( MpA0 + NqA0 + NB_A ) LWS  = MAX(
(NB_A*(NB_A-1))/2, (NRHSqB0 + MpB0)*NB_A ) + NB_A * NB_A Else
LTAU = NUMROC( IA+MIN(M,N)-1, MB_A, MYROW, RSRC_A, NPROW ), LWF
= MB_A * ( MpA0 + NqA0 + MB_A ) LWS  = MAX( (MB_A*(MB_A-1))/2, (
NpB0 + MAX( NqA0 + NUMROC( NUMROC( N+IROFFB, MB_A, 0, 0, NPROW
), MB_A, 0, 0, LCMP ), NRHSqB0 ) )*MB_A ) + MB_A * MB_A End if

where

LCMP = LCM / NPROW with LCM = ILCM( NPROW, NPCOL ),

IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW =
INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA,
NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A,
MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL,
IACOL, NPCOL ),

IROFFB = MOD( IB-1, MB_B ), ICOFFB = MOD( JB-1, NB_B ), IBROW =
INDXG2P( IB, MB_B, MYROW, RSRC_B, NPROW ), IBCOL = INDXG2P( JB,
NB_B, MYCOL, CSRC_B, NPCOL ), MpB0 = NUMROC( M+IROFFB, MB_B,
MYROW, IBROW, NPROW ), NpB0 = NUMROC( N+IROFFB, MB_B, MYROW,
IBROW, NPROW ), NRHSqB0 = NUMROC( NRHS+ICOFFB, NB_B, MYCOL,
IBCOL, NPCOL ),

ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO.

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 "just tell me how much room you need, queen" move 🎀)

  • 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.
⚠️ bestie, real talk: if INFO < 0 you fed her something illegal and she is NOT going to pretend otherwise. check your arguments, fr fr.

and that's the whole page, periodt. 💖 this was LAPACK version 1.5, dated 12 May 1997 — vintage, iconic, still slaying. go forth and factorize responsibly. 🌈🦋👑

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