PZGETRF(l)

📦 version 1.5, in ubuntu jammy

💖 PZGETRF 💖

okay bestie, sit down, hydrate, and let me tell you about PZGETRF from the LAPACK routine collection (this one lives in the (l) manual, version 1.5, no cap) ✨

✨ NAME ✨

PZGETRF — 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 🫶

🌸 SYNOPSIS 🌸

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

    INTEGER         IA, INFO, JA, M, N

    INTEGER         DESCA( * ), IPIV( * )

    COMPLEX*16      A( * )

👑 PURPOSE 👑

PZGETRF 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 the whole job 💅

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 storage behavior, we love to see it.

This is the right-looking Parallel Level 3 BLAS version of the algorithm. main character energy fr fr 🔥

📝 Notes 📝

Every global data object is described by an associated description vector. That vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. it's giving passport-for-your-array 🛂

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". yes bestie, read the underscore out loud, it's a whole thing.

  • 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
🚨 bestie heads up: this routine requires square block decomposition ( MB_A = NB_A ). not optional. periodt.

🎀 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) 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 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 like, 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
    • = 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.
💀 the INFO > 0 case is not a vibe. the math is done, U is singular, and if you try to solve a system with it you're getting division by zero. read the room and read your INFO.

and that's the whole page bestie, you ate 💅 go forth and factor responsibly 🦋

LAPACK version 1.5 — dated 12 May 1997 — PZGETRF(l) 🌈⭐💖

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