PCGEHRD(l)

📦 version 1.5, in ubuntu jammy

💖 PCGEHRD 💖 — the LAPACK routine (version 1.5) that folds your distributed matrix into upper Hessenberg form, bestie

okay so. you opened the man page for PCGEHRD(l) and you're like "what is this glyph soup". don't panic. it's LAPACK, it's parallel, it's giving ✨linear algebra✨ and honestly? it eats. let's get into it. 🫶

🌸 NAME 🌸

PCGEHRD — reduce a complex general distributed matrix sub( A ) to upper Hessenberg form H by an unitary similarity transformation. no cap. 💅

✨ SYNOPSIS ✨

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

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

    INTEGER         DESCA( * )

    COMPLEX         A( * ), TAU( * ), WORK( * )

🦋 PURPOSE 🦋

PCGEHRD reduces a complex general distributed matrix sub( A ) to upper Hessenberg form H by an unitary similarity transformation:

Q' * sub( A ) * Q = H

where

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

📝 Notes 📝

Every global data object has a little bestie of its own called an associated description vector, and that vector is the one holding all the receipts: it stores the info you need to map an element of the object to its corresponding process and memory location. so organized. 💅

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. main character energy, honestly.
  • 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. (zero is allowed, bestie, it's valid, it's just... a very chill matrix.)
  • ILO (global input) INTEGER — yeah that's it, that's the whole description in the source. iconic minimalism.
  • 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) 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 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 unitary 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) COMPLEX 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. (they're bonded, your honor 💖)
  • WORK (local workspace/local output) COMPLEX array, dimension (LWORK). On exit, WORK( 1 ) returns the minimal and optimal LWORK. so polite of it to tell you what it needed.
  • 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. (she's just asking, not telling. no scolding. 😌)
  • 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 complex scalar, and v is a complex 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                         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 )
💀 bestie. if that expression is false, it's giving "not today". this one is not optional, fr fr.

and that's the whole page, no notes, we ate and left zero crumbs ✨

this little love letter to linear algebra came from LAPACK version 1.5, dated 12 May 1997 — so yeah, it's older than you, older than me, and still serving 💅. mwah, go reduce something. 🎀

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