PSGETRF(l)

📦 version 1.5, in ubuntu jammy

✨ PSGETRF ✨

okay bestie, welcome to PSGETRF from the LAPACK fam 💖 this one lives in the routine section of the manual (that's the (l) vibe, iykyk) and it's the Parallel Level 3 BLAS, right-looking girly who does LU factorization. no cap.

🎀 NAME 🎀

PSGETRF — compute an LU factorization of a general M-by-N distributed matrix sub( A ) = (IA:IA+M-1,JA:JA+N-1) using partial pivoting with row interchanges. she's doing the most and we love her for it 💅

🌸 SYNOPSIS 🌸

SUBROUTINE PSGETRF( M, N, A, IA, JA, DESCA, IPIV, INFO )

    INTEGER         IA, INFO, JA, M, N

    INTEGER         DESCA( * ), IPIV( * )

    REAL            A( * )

🦋 PURPOSE 🦋

So PSGETRF computes an LU factorization of a general M-by-N distributed matrix sub( A ) = (IA:IA+M-1,JA:JA+N-1) using partial pivoting with row interchanges. literally her whole thing 🫶

The factorization has the form sub( A ) = P * L * U, where P is a permutation matrix, L is lower triangular with unit diagonal elements (lower trapezoidal if m > n), and U is upper triangular (upper trapezoidal if m < n). L and U are stored in sub( A ). iconic behavior honestly.

This is the right-looking Parallel Level 3 BLAS version of the algorithm. ⭐

💫 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.

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".

NOTATION        STORED IN      EXPLANATION
--------------- -------------- --------------------------------------
DTYPE_A(global) DESCA( DTYPE_ )The descriptor type.  In this case,
                               DTYPE_A = 1.
CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating
                               the BLACS process grid A is distribu-
                               ted over. The context itself is glo-
                               bal, but the handle (the integer
                               value) may vary.
M_A    (global) DESCA( M_ )    The number of rows in the global
                               array A.
N_A    (global) DESCA( N_ )    The number of columns in the global
                               array A.
MB_A   (global) DESCA( MB_ )   The blocking factor used to distribute
                               the rows of the array.
NB_A   (global) DESCA( NB_ )   The blocking factor used to distribute
                               the columns of the array.
RSRC_A (global) DESCA( RSRC_ ) The process row over which the first
                               row of the array A is distributed.
CSRC_A (global) DESCA( CSRC_ ) The process column over which the
                               first column of the array A is
                               distributed.
LLD_A  (local)  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
bestie this next bit is a whole requirement, not a suggestion: this routine requires square block decomposition ( MB_A = NB_A ). 💀

👑 ARGUMENTS 👑

  • 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. 🍓
  • 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. 🍓
  • A — (local input/local output) REAL 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 M-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. so don't go looking for them, they're not there 🙈
  • 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. 🔥
  • INFO — (global output) INTEGER. the vibe check on how it all went:
    • = 0: successful exit. slay 🎉
    • < 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, and division by zero will occur if it is used to solve a system of equations. 😭 not her being singular fr fr

and that's the whole page bestie, no notes 💜 this was the LAPACK version 1.5 tea, dated 12 May 1997 — yeah she's an elder icon and she still ate 👑✨ go forth and factorize responsibly 🦋🌈

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