PCGEBD2(l)

📦 version 1.5, in ubuntu jammy

💖 NAME 💖

PCGEBD2 — bestie, this one takes a complex general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) and reduces it to upper or lower bidiagonal form B by a unitary transformation. That's the whole flex. ✨

🎀 SYNOPSIS 🎀

SUBROUTINE PCGEBD2( 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 PCGEBD2 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 a unitary transformation, and the whole relationship is:

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

Here's the tea on which one you get: if M >= N, B is upper bidiagonal; if M < N, B is lower bidiagonal. No in-between, it picks for you. 💅

📓 Notes 📓

Every global data object has a little buddy called a description vector. That vector holds all the info you need to map an element of the object to the process and memory location it actually lives in. Slay.

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 how many elements of K a process would receive if K were distributed over the p processes of its process column.

Same energy for LOCc( K ): it's how many elements of K a process would receive if K were distributed over the q processes of its process row.

You can figure out 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 ).

And if you want an upper bound for these quantities, you can compute it like this:

        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. 🌈
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.
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. 🫶
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. 🫶
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.
LWORK — (local or global input) INTEGER. The dimension of the array WORK. LWORK is local input and must be at least

LWORK >= MAX( MpA0, NqA0 )

where

NB = MB_A = NB_A,
IROFFA = MOD( IA-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+IROFFA, 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.

Heads up bestie: 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. 💀 (no yelling, just numbers)

INFO — (local 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. It's giving math but make it fashion. ✨

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). Cute, right? 🧚

🎯 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 )

No cap, if that's not true, you're not aligned and it's a whole situation. Check it before you send it. 🙈

And that's the page, bestie 💜 This was LAPACK auxiliary routine version 1.5, dated 12 May 1997 — yes she's vintage, and yes she still ate. 👑 🎀 ✨

📦 where this page comes from

ubuntu jammy: scalapack-doc 1.5-11
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash on 2026-10-14: a language model's rewrite of the original, under its terms