PZPOTRF(l)

💖 PZPOTRF — the Cholesky glow-up 💖

okay bestie, sit DOWN, because we're talking about PZPOTRF from the LAPACK crew (manual section l, if you're keeping track, and yes this is the version 1.5 era of LAPACK). This routine computes the Cholesky factorization of an N-by-N complex hermitian positive definite distributed matrix sub( A ) denoting A(IA:IA+N-1, JA:JA+N-1). That's the whole vibe. She factors, she slays, she moves on. ✨

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🌸 SYNOPSIS 🌸

SUBROUTINE PZPOTRF( UPLO, N, A, IA, JA, DESCA, INFO )

    CHARACTER       UPLO

    INTEGER         IA, INFO, JA, N

    INTEGER         DESCA( * )

    COMPLEX*16      A( * )

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🎀 PURPOSE 🎀

So PZPOTRF computes the Cholesky factorization of an N-by-N complex hermitian positive definite distributed matrix sub( A ) denoting A(IA:IA+N-1, JA:JA+N-1). 💅

The factorization has the form

          sub( A ) = U' * U ,  if UPLO = 'U', or

          sub( A ) = L  * L',  if UPLO = 'L',

where U is an upper triangular matrix and L is lower triangular. Both of them are serving looks, just on opposite corners. 👑

📝 Notes 📝

🦋 the descriptor situation 🦋

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 "who am I and where do I live" but for matrix elements. 🫶

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". iykyk.

• 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)).

🌈 the LOCr / LOCc lore 🌈

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 decomposition ( MB_A = NB_A ). no rectangles allowed, periodt.

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✨ ARGUMENTS ✨

• UPLO — (global input) CHARACTER
  • = 'U': Upper triangle of sub( A ) is stored;
  • = 'L': Lower triangle of sub( A ) is stored.
• 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.
• 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 N-by-N Hermitian distributed matrix sub( A ) to be factored. If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the distributed matrix, and its strictly upper triangular part is not referenced. On exit, if UPLO = 'U', the upper triangular part of the distributed matrix contains the Cholesky factor U, if UPLO = 'L', the lower triangular part of the distributed matrix contains the Cholesky factor L. she comes in as one thing and leaves as another, main character transformation arc fr fr 🌟
• 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 whole roadmap, don't lose her 🗺️
• 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, the leading minor of order K, A(IA:IA+K-1,JA:JA+K-1) is not positive definite, and the factorization could not be completed.

😭 and that INFO > 0 case is the sad one — if INFO = K, the leading minor of order K, A(IA:IA+K-1,JA:JA+K-1), is not positive definite and the factorization could not be completed. not every matrix is built for this life and that's okay.

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mwah 💋 that's the whole page. go factor something hermitian and positive definite, you absolute icon.

sending you off with the receipts: this is from LAPACK, version 1.5, dated 12 May 1997. vintage, iconic, still relevant. bye bestie 🎀✨