PZGETRS(l)

📦 version 1.5, in ubuntu jammy

✨ PZGETRS ✨

okay bestie, gather round, because this is a LAPACK routine (that's the l manual section, iykyk) and it is so much less scary than it looks. Let's get into it 💅

🌸 NAME 🌸

PZGETRS — solve a system of distributed linear equations op( sub( A ) ) * X = sub( B ) with a general N-by-N distributed matrix sub( A ) using the LU factorization computed by PZGETRF.

Like. It literally just solves for X. That's the whole gig. Periodt. 💖

🎀 SYNOPSIS 🎀

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

    CHARACTER       TRANS

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

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

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

🦋 PURPOSE 🦋

PZGETRS solves a system of distributed linear equations. Here sub( A ) denotes A(IA:IA+N-1,JA:JA+N-1), and op( A ) is either A, A**T or A**H. Meanwhile sub( B ) denotes B(IB:IB+N-1,JB:JB+NRHS-1).

💜 Notes 💜

Every global data object has an associated description vector bestie. This vector stores all the info you need to figure out the mapping between an object element and its corresponding process and memory location. Slay.

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". No cap.

  • 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. It's giving "same energy, different number" honestly.
  • 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
💀 heads up bestie: this routine requires square block data decomposition (MB_A=NB_A). Not optional. Do not come for me about it.

👑 ARGUMENTS 👑

  • TRANS (global input) CHARACTER — Specifies the form of the system of equations, and this is the moment where you pick your vibe:
    • = 'N': sub( A ) * X = sub( B ) (No transpose)
    • = 'T': sub( A )**T * X = sub( B ) (Transpose)
    • = 'C': sub( A )**H * X = sub( B ) (Conjugate transpose)
  • 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. Zero is allowed bestie, we don't judge.
  • NRHS (global input) INTEGER — The number of right hand sides, i.e., the number of columns of the distributed submatrix sub( B ). NRHS >= 0.
  • A (local input) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, 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. They're implied, they're the main character, they simply do not need to be written down. 💅
  • 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 input) 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. They're a package deal, fr fr.
  • 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 sides sub( B ). On exit, sub( B ) is overwritten by the solution distributed matrix X. She transforms okurrr ✨
  • 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 result:
    • = 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.

and that's the whole page bestie!! you understood all of it, I'm so proud, literally obsessed 🫶

PZGETRS lives in LAPACK version 1.5, and this little love letter was written on 12 May 1997. mwah 💋

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