PSGEHRD(l)

📦 version 1.5, in ubuntu jammy

🌸 PSGEHRD 🌸

okay bestie, gather round, because today we're talking about PSGEHRD, a routine from the LAPACK crew, version 1.5, and she is so much less scary than her name makes her look 💅

✨ NAME ✨

PSGEHRD — reduce a real general distributed matrix sub( A ) to upper Hessenberg form H by an orthogonal similarity transformation. that's it, that's the girl 🎀

🦋 SYNOPSIS 🦋

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

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

    INTEGER         DESCA( * )

    REAL            A( * ), TAU( * ), WORK( * )

💖 PURPOSE 💖

So PSGEHRD takes a real general distributed matrix sub( A ) and reduces it to upper Hessenberg form H by an orthogonal similarity transformation:

Q' * sub( A ) * Q = H

where

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

(no cap, that's literally what the page says, we reproduce it faithfully 💀)

Notes

okay so this is the part where it gets a lil bookkeepy, but stay with me, it's important ✨

Every global data object has an associated description vector that stores all the info needed to map an object element to its 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". 📚

  • 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 its process grid has dimension p x q.

LOCr( K ) is 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 ) is the number of elements of K that a process would receive if K were distributed over the q processes of its process row. 🌸

You can get LOCr() and LOCc() by calling 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 can be computed like:

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. 🍓
  • 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,
  • A (local input/local output) REAL 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) REAL 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) REAL 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 (fair warning, bestie, the source page renders this one a little crunchy — we're giving it to you exactly as it comes 💀):

on entry                         on exit

( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a ) (     a
a   a   a   a   a )    (      a   h   h   h   h   a ) (     a   a   a
a   a   a )    (      h   h   h   h   h   h ) (     a   a   a   a   a
a )    (      v2  h   h   h   h   h ) (     a   a   a   a   a   a )    (
v2  v3  h   h   h   h ) (     a   a   a   a   a   a )    (      v2  v3
v4  h   h   h ) (                         a )    (
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 )
⭐ real talk: that alignment check up there? it's not optional. if it's false, things are not gonna go how you want them to go. check it before you commit, periodt. 💅

and that's the whole page, bestie!! you understood all of it, I know you did 🌈💜 this was LAPACK routine, version 1.5, dated 12 May 1997 — iconic behavior from 1997 honestly, she ate 👑✨

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