PCGEBRD(l)

📦 version 1.5, in ubuntu jammy

✨ PCGEBRD ✨ (that's PCGEBRD(l), bestie, and yes the (l) is doing main character energy)

Okay so real talk — this is a LAPACK routine, specifically a ScaLAPACK one (the P stands for parallel, iykyk), and it does something genuinely iconic: it takes a complex general distributed matrix and glow-ups it into bidiagonal form using a unitary transformation. It's giving linear algebra, it's giving distributed computing, it's giving "why is my grid 4x3 and my brain 0x0". Let's get into it. 💖

🌸 NAME 🌸

PCGEBRD — reduce a complex general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper or lower bidiagonal form B by an unitary transformation. Periodt.

🦋 SYNOPSIS 🦋

SUBROUTINE PCGEBRD( M, N, A, IA, JA, DESCA, D, E, TAUQ, TAUP, WORK,
                    LWORK, INFO )

    INTEGER         IA, INFO, JA, LWORK, M, N

    INTEGER         DESCA( * )

    REAL            D( * ), E( * )

    COMPLEX         A( * ), TAUP( * ), TAUQ( * ), WORK( * )

👑 PURPOSE 👑

So here's the tea, bestie. PCGEBRD reduces a complex general M-by-N distributed matrix

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

to upper or lower bidiagonal form B by an unitary transformation:

Q' * sub( A ) * P = B.

If M >= N, B is upper bidiagonal; if M < N, B is lower bidiagonal. That's it, that's the whole vibe. 🎀

📝 Notes (the lore you actually need) 📝

Every global data object is described by an associated description vector. This vector stores the info required to establish the mapping between an object element and its corresponding process and memory location. It's basically the group chat roster for your matrix. 💅

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".

Here's the whole notation breakdown, no cap:

  • 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 💖

  • M (global input) INTEGER — The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. Yes bestie, zero is allowed, we love an inclusive range. 🌈
  • N (global input) INTEGER — The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). 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 general distributed matrix sub( A ). On exit, if M >= N, the diagonal and the first superdiagonal of sub( A ) are overwritten with the upper bidiagonal matrix B; the elements below the diagonal, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors, and the elements above the first superdiagonal, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. If M < N, the diagonal and the first subdiagonal are overwritten with the lower bidiagonal matrix B; the elements below the first subdiagonal, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors, and the elements above the diagonal, with the array TAUP, represent the orthogonal matrix P 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. This is the ID card, don't lose it. 🪪
  • D (local output) REAL array, dimension LOCc(JA+MIN(M,N)-1) if M >= N; LOCr(IA+MIN(M,N)-1) otherwise. — The distributed diagonal elements of the bidiagonal matrix B: D(i) = A(i,i). D is tied to the distributed matrix A. Besties forever, they move together. 🫶
  • E (local output) REAL array, dimension LOCr(IA+MIN(M,N)-1) if M >= N; LOCc(JA+MIN(M,N)-2) otherwise. — The distributed off-diagonal elements of the bidiagonal distributed matrix B: if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1; if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1. E is tied to the distributed matrix A. Also attached, also unbothered. 💜
  • TAUQ (local output) COMPLEX array dimension LOCc(JA+MIN(M,N)-1). — The scalar factors of the elementary reflectors which represent the unitary matrix Q. TAUQ is tied to the distributed matrix A. See Further Details.
  • TAUP (local output) COMPLEX array, dimension LOCr(IA+MIN(M,N)-1). — The scalar factors of the elementary reflectors which represent the unitary matrix P. TAUP is tied to the distributed matrix A. See Further Details.
  • WORK (local workspace/local output) COMPLEX array, dimension (LWORK). — On exit, WORK( 1 ) returns the minimal and optimal LWORK. It's the little messenger that comes back with the numbers. 🧚
  • LWORK (local or global input) INTEGER — The dimension of the array WORK. LWORK is local input and must be at least
LWORK >= NB*( MpA0 + NqA0 + 1 ) + NqA0

where

NB = MB_A = NB_A, IROFFA = MOD( IA-1, NB ), ICOFFA = MOD( JA-1, NB ),
IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ),
IACOL = INDXG2P( JA, NB, MYCOL, CSRC_A, NPCOL ),
MpA0 = NUMROC( M+IROFFA, NB, MYROW, IAROW, NPROW ),
NqA0 = NUMROC( N+ICOFFA, NB, MYCOL, IACOL, 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. So don't come for it when it's quiet, it's supposed to be quiet. 🤫

  • 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 matrices Q and P are represented as products of elementary reflectors:

If m >= n,

Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)

Each H(i) and G(i) has the form:

H(i) = I - tauq * v * v'  and G(i) = I - taup * u * u'

where tauq and taup are complex scalars, and v and u are complex vectors;

v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in
A(ia+i:ia+m-1,ja+i-1);
u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in
A(ia+i-1,ja+i+1:ja+n-1);
tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1).

If m < n,

Q = H(1) H(2) . . . H(m-1)  and  P = G(1) G(2) . . . G(m)

Each H(i) and G(i) has the form:

H(i) = I - tauq * v * v'  and G(i) = I - taup * u * u'

where tauq and taup are complex scalars, and v and u are complex vectors;

v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in
A(ia+i+1:ia+m-1,ja+i-1);
u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in
A(ia+i-1,ja+i:ja+n-1);
tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1).

The contents of sub( A ) on exit are illustrated by the following examples:

m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):

  (  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )
  (  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )
  (  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )
  (  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )
  (  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )
  (  v1  v2  v3  v4  v5 )

where d and e denote diagonal and off-diagonal elements of B, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i). Ate and left no crumbs. 💅

🎯 Alignment requirements 🎯

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

( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA )
Bestie, lowkey do NOT skip this one. If your alignment is off, sub( A ) is not valid input and everything downstream is going to be a mess. Check it first, thank me later. 💀

And that's the whole page, bestie! ✨ This was LAPACK version 1.5, dated 12 May 1997 — yes, this routine is older than most of the internet and it still slays. Stay bidiagonal, stay distributed, stay aligned. 🎀💖🦋🌸👑

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