PCHEGS2(l)

📦 version 1.5, in ubuntu jammy

💖 PCHEGS2 💖

okay bestie, sit down, hydrate, because we're about to talk about PCHEGS2 from LAPACK (yes that LAPACK, the one and only) — it reduces a complex Hermitian-definite generalized eigenproblem to standard form. it's giving linear algebra royalty, fr fr 🫶

✨ NAME ✨

PCHEGS2 — reduce a complex Hermitian-definite generalized eigenproblem to standard form. no cap, that's literally the whole job description 💅

🌸 SYNOPSIS 🌸

SUBROUTINE PCHEGS2( 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         A( * ), B( * )

👑 PURPOSE 👑

PCHEGS2 reduces a complex Hermitian-definite generalized eigenproblem to standard form. that's the mission, and it's eating, honestly.

so 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 ). keeping it cute and abbreviated, iykyk.

  • 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)
  • 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.
⚠️ important bestie: sub( B ) must have been previously factorized as U**H*U or L*L**H by PCPOTRF. do NOT come in here with an unfactorized B, it's not the vibe.

🦋 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. it's basically the group chat admin for your array 📱

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". adorable, honestly.

  • 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': 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. (yes, zero is allowed, we love a boundary case 🎀)
  • A — (local input/local output) COMPLEX 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 ). glow up complete ✨
  • 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 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 PCPOTRF.
  • 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.
🫶 quick reminder bestie: that INFO math is not a typo, -(i*100+j) is real and it will find you. respect the indexing.

that's the whole page, you ate it, you understood it, main character energy confirmed 🌈⭐

this was PCHEGS2(l), a LAPACK routine, version 1.5, dated 12 May 1997. mwah, bye bestie 💖🦋🌸

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