PZHEGS2(l)

📦 version 1.5, in ubuntu jammy

💖 NAME 💖

okay so bestie, meet PZHEGS2(l) — a LAPACK routine (version 1.5). Her whole thing? Reduce a complex Hermitian-definite generalized eigenproblem to standard form. periodt. ✨

✨ SYNOPSIS ✨

SUBROUTINE PZHEGS2( IBTYPE, UPLO, N, A, IA, JA, DESCA, B, IB, JB, DESCB,
                    INFO )

    CHARACTER       UPLO

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

    INTEGER         DESCA( * ), DESCB( * )

    COMPLEX*16      A( * ), B( * )

🦋 PURPOSE 🦋

PZHEGS2 reduces a complex Hermitian-definite generalized eigenproblem to standard form. no cap.

In the following, sub( A ) denotes A( IA:IA+N-1, JA:JA+N-1 ) and sub( B ) denotes B( IB:IB+N-1, JB:JB+N-1 ). keep that mapping in your notes, bestie.

If IBTYPE = 1, the problem is sub( A )*x = lambda*sub( B )*x, and sub( A ) is overwritten by inv(U**H)*sub( A )*inv(U) or inv(L)*sub( A )*inv(L**H). she's getting a whole new look.

If IBTYPE = 2 or 3, the problem is sub( A )*sub( B )*x = lambda*x or sub( B )*sub( A )*x = lambda*x, and sub( A ) is overwritten by U*sub( A )*U**H or L**H*sub( A )*L. different flavor, same iconic energy.

sub( B ) must have been previously factorized as U**H*U or L*L**H by PZPOTRF. do not skip this step, fr fr.

📝 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. iykyk.

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".

Here's the notation tea ☕:

  • 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

👑 ARGUMENTS 👑

  • IBTYPE (global input) INTEGER — bestie, this is the vibe selector: = 1: compute inv(U**H)*sub( A )*inv(U) or inv(L)*sub( A )*inv(L**H); = 2 or 3: compute U*sub( A )*U**H or L**H*sub( A )*L.
  • UPLO (global input) CHARACTER — which triangle are we serving? = 'U': Upper triangle of sub( A ) is stored and sub( B ) is factored as U**H*U; = 'L': Lower triangle of sub( A ) is stored and sub( B ) is factored as L*L**H.
  • N (global input) INTEGER — the order of the matrices sub( A ) and sub( B ). N >= 0. no cap.
  • A (local input/local output) COMPLEX16 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 ). 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 matrix, and its strictly upper triangular part is not referenced. On exit, if INFO = 0, the transformed matrix, stored in the same format as sub( A ). she's giving input and output at the same time, iconic.
  • IA (global input) INTEGER — A's global row index, which points to the beginning of the submatrix which is to be operated on.
  • JA (global input) INTEGER — A's global column index, which points to the beginning of the submatrix which is to be operated on.
  • DESCA (global and local input) INTEGER array of dimension DLEN_. — the array descriptor for the distributed matrix A. protect her at all costs.
  • B (local input) COMPLEX16 pointer into the local memory to an array of dimension (LLD_B, LOCc(JB+N-1)).* — on entry, this array contains the local pieces of the triangular factor from the Cholesky factorization of sub( B ), as returned by PZPOTRF.
  • IB (global input) INTEGER — B's global row index, which points to the beginning of the submatrix which is to be operated on.
  • JB (global input) INTEGER — B's global column index, which points to the beginning of the submatrix which is to be operated on.
  • 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. so if she's negative, read the code like a text message.

and that's the whole thing, bestie 💖 LAPACK version 1.5, dated 12 May 1997, PZHEGS2(l) said what she said. slay, stay Hermitian-definite, and may your descriptors always be valid ✨👑💅

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