PDGEHRD(l)

📦 version 1.5, in ubuntu jammy

💖 PDGEHRD 💖

okay bestie, so this is a LAPACK routine, and it's giving distributed linear algebra in the most iconic way possible. It's a SUBROUTINE that lives in the PDGEHRD(l) page, and we're about to make it make sense. no cap. ✨

🎀 NAME 🎀

PDGEHRD — reduce a real general distributed matrix sub( A ) to upper Hessenberg form H by an orthogonal similarity transformation 🦋

✨ SYNOPSIS ✨

SUBROUTINE PDGEHRD( N, ILO, IHI, A, IA, JA, DESCA, TAU, WORK, LWORK,
                    INFO )

    INTEGER         IA, IHI, ILO, INFO, JA, LWORK, N

    INTEGER         DESCA( * )

    DOUBLE          PRECISION A( * ), TAU( * ), WORK( * )

💅 PURPOSE 💅

PDGEHRD reduces a real general distributed matrix sub( A ) to upper Hessenberg form H by an orthogonal similarity transformation, and the way that it does it is literally

Q' * sub( A ) * Q = H

where, and I need you to read this carefully because it's a whole thing,

sub( A ) = A(IA+N-1:IA+N-1,JA+N-1:JA+N-1)

(that's what the page says, character for character, and I'm not touching it 💀)

📝 Notes 📝

Every global data object is described by an associated description vector. This vector stores the info you need 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".

so here's the whole notation breakdown, one glow-up at a time 💖

  • 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 🌸

  • N (global input) INTEGER — the number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. bestie it's a size, it's giving "how big is this thing" energy 🫶
  • ILO (global input) INTEGER
  • IHI (global input) INTEGER — it is assumed that sub( A ) is already upper triangular in rows IA:IA+ILO-2 and IA+IHI:IA+N-1 and columns JA:JA+ILO-2 and JA+IHI:JA+N-1. See Further Details. If N > 0,
(yes, that trailing "If N > 0," is literally how it's written on the page and I am not fixing it, that's the original's business 💀)
  • 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 general distributed matrix sub( A ) to be reduced. On exit, the upper triangle and the first subdiagonal of sub( A ) are overwritten with the upper Hessenberg matrix H, and the elements below the first subdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details.
  • IA (global input) INTEGER — the row index in the global array A indicating the first row of sub( A ).
  • JA (global input) INTEGER — the column index in the global array A indicating the first column of sub( A ).
  • DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A.
  • TAU (local output) DOUBLE PRECISION array, dimension LOCc(JA+N-2). The scalar factors of the elementary reflectors (see Further Details). Elements JA:JA+ILO-2 and JA+IHI:JA+N-2 of TAU are set to zero. TAU is tied to the distributed matrix A. 🔗
  • WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK). On exit, WORK( 1 ) returns the minimal and optimal LWORK.
  • LWORK (local or global input) INTEGER — the dimension of the array WORK. LWORK is local input and must be at least
LWORK >= NB*NB + NB*MAX( IHIP+1, IHLP+INLQ )

where

NB = MB_A = NB_A,
IROFFA = MOD( IA-1, NB ),
ICOFFA = MOD( JA-1, NB ),
IOFF = MOD( IA+ILO-2, NB ),
IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ),
IHIP = NUMROC( IHI+IROFFA, NB, MYROW, IAROW, NPROW ),
ILROW = INDXG2P( IA+ILO-1, NB, MYROW, RSRC_A, NPROW ),
IHLP = NUMROC( IHI-ILO+IOFF+1, NB, MYROW, ILROW, NPROW ),
ILCOL = INDXG2P( JA+ILO-1, NB, MYCOL, CSRC_A, NPCOL ),
INLQ = NUMROC( N-ILO+IOFF+1, NB, MYCOL, ILCOL, NPCOL ),

INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO.

If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. 🔥

  • 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.

🦋 FURTHER DETAILS 🦋

The matrix Q is represented as a product of (ihi-ilo) elementary reflectors

Q = H(ilo) H(ilo+1) . . . H(ihi-1).

Each H(i) has the form

H(i) = I - tau * v * v'

where tau is a real scalar, and v is a real vector with v(1:I) = 0, v(I+1) = 1 and v(IHI+1:N) = 0; v(I+2:IHI) is stored on exit in A(IA+ILO+I:IA+IHI-1,JA+ILO+I-2), and tau in TAU(JA+ILO+I-2).

The contents of A(IA:IA+N-1,JA:JA+N-1) are illustrated by the following example, with N = 7, ILO = 2 and IHI = 6:

on entry

( a   a   a   a   a   a   a )
(     a   a   a   a   a   a )
(     a   a   a   a   a   a )
(     a   a   a   a   a   a )
(     a   a   a   a   a   a )
(     a   a   a   a   a   a )
(                         a )

on exit

(  a   a   h   h   h   h   a )
(      a   h   h   h   h   a )
(      h   h   h   h   h   h )
(      v2  h   h   h   h   h )
(      v2  v3  h   h   h   h )
(      v2  v3  v4  h   h   h )
(                         a )

where a denotes an element of the original matrix sub( A ), H denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(JA+ILO+I-2). ✨

🧚 Alignment requirements 🧚

The distributed submatrix sub( A ) must verify some alignment properties, namely the following expression should be true:

( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA )
fr fr, if that alignment vibe check fails, we're not doing this. 👑

that's the whole page, bestie — every flag, every formula, every little NUMROC call, untouched and thriving. now go reduce something to upper Hessenberg form, you main character. 💜

with love from LAPACK version 1.5, dated 12 May 1997, and originally signed off as PDGEHRD(l) — slay, periodt. 💅

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