PDSYGST(l)

📦 version 1.5, in ubuntu jammy

💖 PDSYGST(l) — bestie edition 💖

ok so this is PDSYGST, a LAPACK routine, version 1.5, and she is that girl. sit down, grab a strawberry, and let's get into it 🍓✨

✨ NAME ✨

PDSYGST — reduce a real symmetric-definite generalized eigenproblem to standard form. yes. that's the whole job. she does one thing and she does it well 💅

🌸 SYNOPSIS 🌸

SUBROUTINE PDSYGST( 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( * )

    DOUBLE          PRECISION A( * ), B( * )

(yes, DOUBLE PRECISION SCALE has a whole vibe gap in the middle of it. iconic. not touching it. 🎀)

👑 PURPOSE 👑

PDSYGST reduces a real symmetric-definite generalized eigenproblem to standard form. literally her whole personality.

so in the following, 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 ). got it? good. 🫶

  • 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.
🚨 okay important bestie moment: sub( B ) must have been previously factorized as U**T*U or L*L**T by PDPOTRF. do not skip this step. do not pass go. she will not do it for you. 😭

🧚 Notes 🧚

💜 the descriptor thing (iykyk) 💜

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".

ok so here's the lore, laid out for you because the original did a whole table and I'm giving you the same tea in list form:

  • 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)).

🌈 LOCr and LOCc, the math besties 🌈

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 💅

ok this is the part where you actually pay attention. every single one. no skimming. 👑

  • 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 ).

  • 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.
  • 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.
  • 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.

that's the whole page, bestie. every flag, every formula, nothing skipped, no cap. go forth and reduce those eigenproblems. you ate. 🫶💖✨

LAPACK version 1.5, dated 12 May 1997 — shoutout to the legends who wrote her, and to you for reading all the way to the bottom. mwah. 💋

📦 where this page comes from