PZGESV(l)

📦 version 1.5, in ubuntu jammy

💅 NAME 💅

okay so bestie, welcome to PZGESV(l) — that's the LAPACK routine, version 1.5, and she's here to compute the solution to a complex system of linear equations sub( A ) * X = sub( B ) ✨

that's it. that's the whole job. she solves. she slays. 🫶

🦋 SYNOPSIS 🦋

SUBROUTINE PZGESV( N, NRHS, A, IA, JA, DESCA, IPIV, B, IB, JB, DESCB,
                   INFO )

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

    INTEGER        DESCA( * ), DESCB( * ), IPIV( * )

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

look at her signature. iconic. no cap. 🎀

💖 PURPOSE 💖

PZGESV computes the solution to a complex system of linear equations

sub( A ) * X = sub( B )

where sub( A ) = A(IA:IA+N-1,JA:JA+N-1) is an N-by-N distributed matrix, and X and sub( B ) = B(IB:IB+N-1,JB:JB+NRHS-1) are N-by-NRHS distributed matrices. 💫

the LU decomposition with partial pivoting and row interchanges is used to factor sub( A ) as sub( A ) = P * L * U, where P is a permutation matrix, L is unit lower triangular, and U is upper triangular. L and U are stored in sub( A ). the factored form of sub( A ) is then used to solve the system of equations sub( A ) * X = sub( B ). 🔥

📝 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. (yes queen, it's giving ✨bookkeeping✨)

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
🚨 this routine requires square block decomposition ( MB_A = NB_A ). she said SQUARE. no rectangle energy allowed. 💀

🎀 ARGUMENTS 🎀

  • N — (global input) INTEGER. the number of rows and columns to be operated on, i.e. the order 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 submatrix sub( A ). NRHS >= 0. (yes the original really says sub( A ) here, we don't question the ancestors 🫶)
  • A — (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). on entry, the local pieces of the N-by-N distributed matrix sub( A ) to be factored. on exit, this array contains the local pieces of the factors L and U from the factorization sub( A ) = P*L*U; the unit diagonal elements of L are not stored. 💜
  • 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.
  • IPIV — (local output) INTEGER array, dimension ( LOCr(M_A)+MB_A ). this array contains the pivoting information. IPIV(i) -> the global row local row i was swapped with. this array is tied to the distributed matrix A.
  • B — (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_B,LOCc(JB+NRHS-1)). on entry, the right hand side distributed matrix sub( B ). on exit, if INFO = 0, sub( B ) is overwritten by the solution distributed matrix X. ⭐
  • 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.
  • INFO — (global output) INTEGER. the vibe check, and here's how she reports back:
    • = 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.
    • > 0: if INFO = K, U(IA+K-1,JA+K-1) is exactly zero. the factorization has been completed, but the factor U is exactly singular, so the solution could not be computed. 😭💀

and that's the whole page, bestie. PZGESV(l) from the LAPACK routine, version 1.5, dated 12 May 1997 — she's been serving linear algebra realness since before you were born and she has never missed. 👑💖 go solve something. periodt. 🦋✨

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