PCPOSV(l)

📦 version 1.5, in ubuntu jammy

💖 PCPOSV 💖 (a LAPACK routine, version 1.5)

okay bestie, sit down, hydrate, because we're about to solve a whole distributed complex linear system and it's gonna be iconic. This is PCPOSV from the LAPACK crew — lowercase (l) energy, big brain energy. 🧠✨

✨ NAME ✨

PCPOSV — computes the solution to a complex system of linear equations

sub( A ) * X = sub( B ),

that's it, that's the job. slay 💅

🌸 SYNOPSIS 🌸

SUBROUTINE PCPOSV( UPLO, N, NRHS, A, IA, JA, DESCA, B, IB, JB, DESCB,
                   INFO )

    CHARACTER      UPLO

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

    INTEGER        DESCA( * ), DESCB( * )

    COMPLEX        A( * ), B( * )

🎀 PURPOSE 🎀

PCPOSV computes the solution to a complex system of linear equations — the whole thing, no cap:

  • sub( A ) denotes A(IA:IA+N-1,JA:JA+N-1) and it's an N-by-N hermitian distributed positive definite matrix 👑
  • X and sub( B ) denote B(IB:IB+N-1,JB:JB+NRHS-1), and they're N-by-NRHS distributed matrices

The Cholesky decomposition is used to factor sub( A ) as

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

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

where U is an upper triangular matrix and L is a lower triangular matrix. Then the factored form of sub( A ) gets used to solve the system of equations. chef's kiss 😘

📝 Notes

bestie this part is the ✨lore✨ — read it, it's giving essential context

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".

  • 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
🚨 This routine requires square block decomposition ( MB_A = NB_A ). Not optional. No exceptions. Periodt.

💅 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.
  • 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/local output) COMPLEX 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 symmetric 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 INFO = 0, this array contains the local pieces of the factor U or L from the Cholesky factorization sub( A ) = U**H*U or L*L**H. 🦋
  • 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.
  • B — (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_B,LOC(JB+NRHS-1)). On entry, the local pieces of the right hand sides distributed matrix sub( B ). On exit, if INFO = 0, sub( B ) is overwritten with the solution distributed matrix X. glow up ✨
  • 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
    • = 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 the solution has not been computed. 💀 yeah that one hurts.
lowkey the whole vibe: hermitian positive definite, Cholesky factorized, distributed across a process grid, and it ate. iykyk. 🫶

mwah 💜 that's the whole page, bestie. This was LAPACK version 1.5, dated 12 May 1997 — yes she's vintage, yes she still runs, and yes she's still that girl. 🎀👑✨

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