PDSYGS2(l)

📦 version 1.5, in ubuntu jammy

💖 PDSYGS2 — the LAPACK routine that turns your generalized eigenproblem into a standard one 💖

okay bestie, gather round, because today we're doing PDSYGS2 from LAPACK version 1.5, and she is giving linear algebra realness. 🦋

✨ NAME ✨

PDSYGS2 — reduce a real symmetric-definite generalized eigenproblem to standard form. no cap, that's the whole job. 👑

🌸 SYNOPSIS 🌸

SUBROUTINE PDSYGS2( 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( * )

    DOUBLE PRECISION A( * ), B( * )

like, that's the whole cast list. twelve arguments and a dream. 💅

🎀 PURPOSE 🎀

PDSYGS2 reduces a real symmetric-definite generalized eigenproblem to standard form. literally her entire personality. ✨

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 ). (so sub( A ) is just a cute nickname for that slice of A, iykyk 🫶)

  • If IBTYPE = 1, the problem is sub( A )*x = lambda*sub( B )*x, and sub( A ) is overwritten by inv(U**T)*sub( A )*inv(U) or inv(L)*sub( A )*inv(L**T)
  • 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**T or L**T*sub( A )*L.
⚠️ heads up bestie: sub( B ) must have been previously factorized as U**T*U or L*L**T by PDPOTRF. no PDPOTRF first, no vibes. that's a hard requirement, 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.

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". (iconic little underscore, doing so much work 🌟)

Here's the whole notation cheat sheet, and yes, we're keeping every single line because the details are sacred:

  • 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 👑

one bullet per argument, no skips, no shortcuts, we eat every single one of these. 🍓

  • IBTYPE — (global input) INTEGER — = 1: compute inv(U**T)*sub( A )*inv(U) or inv(L)*sub( A )*inv(L**T); = 2 or 3: compute U*sub( A )*U**T or L**T*sub( A )*L.
  • UPLO — (global input) CHARACTER — = 'U': Upper triangle of sub( A ) is stored and sub( B ) is factored as U**T*U; = 'L': Lower triangle of sub( A ) is stored and sub( B ) is factored as L*L**T.
  • N — (global input) INTEGER — The order of the matrices sub( A ) and sub( B ). N >= 0.
  • A — (local input/local output) DOUBLE PRECISION 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 ). 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 ). (glow up ✨)

  • 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) DOUBLE PRECISION 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 PDPOTRF.
  • 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.
🫶 reminder: negative INFO is never a vibe, it's the routine telling you which argument you fumbled. check the math, bestie.

and that's the whole page, you understood it, you ate, periodt. 💖

this was LAPACK version 1.5, dated 12 May 1997 — yes she's vintage, yes she still serves. 🎀✨

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