PSPOSV(l)

📦 version 1.5, in ubuntu jammy

💖 PSPOSV(l) — bestie's guide to solving sub( A ) * X = sub( B ) 💖

okay so this is a LAPACK routine, version 1.5, and it is literally here to solve a real system of linear equations for you. No cap.

✨ NAME ✨

PSPOSV — compute the solution to a real system of linear equations

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

🌸 SYNOPSIS 🌸

SUBROUTINE PSPOSV( 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( * )

    REAL           A( * ), B( * )

🦋 PURPOSE 🦋

Okay so here's the tea. PSPOSV computes the solution to a real system of linear equations

where sub( A ) denotes A(IA:IA+N-1,JA:JA+N-1) and is an N-by-N symmetric distributed positive definite matrix, and X and sub( B ) — denoting B(IB:IB+N-1,JB:JB+NRHS-1) — are N-by-NRHS distributed matrices. 💅

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

                   sub( A ) = U**T * U,  if UPLO = 'U', or

                   sub( A ) = L * L**T,  if UPLO = 'L',

where U is an upper triangular matrix and L is a lower triangular matrix. The factored form of sub( A ) is then used to solve the system of equations. slay.

📓 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".

  • 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). No exceptions, it's giving very strict. 💀

👑 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) 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 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**T*U or L*L**T.
  • 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) REAL 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 moment ✨
  • 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.

and that's the whole vibe check, bestie — you now know PSPOSV better than your fave playlist. 🎀

This was LAPACK version 1.5, dated 12 May 1997. 💖 mwah, go solve something symmetric and positive definite. periodt. 💅

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