PZHEGST(l)

📦 version 1.5, in ubuntu jammy

✨ NAME ✨

PZHEGST — bestie, this one takes a complex Hermitian-definite generalized eigenproblem and reduces it to standard form. That's the whole vibe. 💅

🌸 SYNOPSIS 🌸

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

    CHARACTER       UPLO

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

    DOUBLE          PRECISION SCALE

    INTEGER         DESCA( * ), DESCB( * )

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

💖 PURPOSE 💖

Okay so PZHEGST reduces a complex Hermitian-definite generalized eigenproblem to standard form. Iconic behavior honestly.

In everything below, sub( A ) means A( IA:IA+N-1, JA:JA+N-1 ) and sub( B ) means B( IB:IB+N-1, JB:JB+N-1 ). Like, the little corner of the big array we're actually touching. 🫶

  • If IBTYPE = 1, the problem is sub( A )*x = lambda*sub( B )*x, and sub( A ) gets overwritten by inv(U**H)*sub( A )*inv(U) or inv(L)*sub( A )*inv(L**H)
  • 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 ) gets overwritten by U*sub( A )*U**H or L**H*sub( A )*L.
Heads up bestie: sub( B ) must have already been factorized as U**H*U or L*L**H by PZPOTRF. No shortcuts, no skipping ahead. 😤

📓 Notes 📓

Every global data object is described by an associated description vector, and that vector is storing all the info you need to map an object element to its corresponding process and memory location. It's giving "address book for your matrix." 💌

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

🎀 ARGUMENTS 🎀

  • IBTYPE — (global input) INTEGER — = 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 — = 'U': the upper triangle of sub( A ) is stored and sub( B ) is factored as U**H*U; = 'L': the 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.
  • 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 ). 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 ).

  • 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.
  • B — *(local input) COMPLEX*16 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.
  • SCALE — (global output) DOUBLE PRECISION — amount by which the eigenvalues should be scaled to compensate for the scaling performed in this routine. At present, SCALE is always returned as 1.0, it is returned here to allow for future enhancement. (Yes bestie, it's literally just 1.0 every single time. It's giving placeholder energy but we respect the forward planning. ✨)
  • 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.

LAPACK version 1.5 · 12 May 1997 — okay that's the whole page bestie, tysm for reading, go slay those Hermitian-definite generalized eigenproblems, and remember: factorize with PZPOTRF first. 💖✨🎀🦋🌸💅🫶

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