PCHEGST(l)

📦 version 1.5, in ubuntu jammy

✨ NAME ✨

bestie, this is PCHEGST — the LAPACK routine (manual section l) that reduces a complex Hermitian-definite generalized eigenproblem to standard form. it's giving ✨linear algebra glow-up✨ 💜

🌸 SYNOPSIS 🌸

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

    CHARACTER       UPLO

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

    REAL            SCALE

    INTEGER         DESCA( * ), DESCB( * )

    COMPLEX         A( * ), B( * )

💅 PURPOSE 💅

PCHEGST 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 ). write that down bestie, it matters.

  • 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.
🚨 ok so sub( B ) must have been previously factorized as U**H*U or L*L**H by PCPOTRF. like, this is not optional. do the factorization first or it's over for you 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. iconic behavior honestly 👑

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".

so here's the whole rundown of what lives where:

  • 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.
  • 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 ). 💫

  • 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. yes, the PCPOTRF from the warning up there. bestie we told you. 🌸
  • 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) REAL — 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. it's giving placeholder energy but ok, we respect the foresight 🎀
  • 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.

and that's the whole vibe check, bestie 🦋 go forth and reduce those generalized eigenproblems, you absolute main character 💖

PCHEGST(l) comes to you from LAPACK version 1.5, dated 12 May 1997 — vintage, but still slaying ✨👑

📦 where this page comes from