PDGETRF(l)

📦 version 1.5, in ubuntu jammy

💖 PDGETRF 💖

okay bestie, so this one is a LAPACK routine (version 1.5, she's vintage but she serves) and she lives in the l manual section. Let's get into it 🫶

✨ NAME ✨

PDGETRF — she computes an LU factorization of a general M-by-N distributed matrix sub( A ) = (IA:IA+M-1,JA:JA+N-1), and she does it with partial pivoting with row interchanges. iconic behavior honestly 💅

🌸 SYNOPSIS 🌸

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

    INTEGER         IA, INFO, JA, M, N

    INTEGER         DESCA( * ), IPIV( * )

    DOUBLE          PRECISION A( * )

🎀 PURPOSE 🎀

So PDGETRF 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. no cap.

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). And where do L and U live afterwards? bestie they are stored in sub( A ). periodt. 💖

This is the right-looking Parallel Level 3 BLAS version of the algorithm. highkey the glow-up version ✨

🦋 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. it's giving "everybody needs a little paperwork" 📋

Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. And listen, in the following comments, the character _ should be read as "of the global array". iykyk 🧚

Here's the whole notation tea, laid out for you:

  • 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
🚨 real talk bestie: this routine requires square block decomposition (MB_A = NB_A). don't try to be cute and skip that part, okay? 😭

👑 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. keep her non-negative pls 🫶
  • 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) DOUBLE PRECISION 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. she comes in as A and leaves as L and U, main character energy honestly ✨
  • 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. it's the vibe check for your matrix 💅
  • 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.
    • = 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, and division by zero will occur if it is used to solve a system of equations. 🚨💀 like bestie she finished the job but the job is cursed, do not proceed.
💖 and that's the whole page! no notes, you ate that.

okay bye, this was the LAPACK version 1.5 tea, dated 12 May 1997, and she's still out here serving LU factorizations. obsessed with her fr fr 💅✨🎀🦋🌸👑💜

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