PCLABRD(l)

📦 version 1.5, in ubuntu jammy

💖 PCLABRD 💖 — the LAPACK auxiliary routine (manual section l, bestie)

okay so you're in the <pink>LAPACK</pink> docs, section l, and you clicked on PCLABRD. slay. let's get into it.

✨ NAME ✨

PCLABRD — reduce the first NB rows and columns of 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 by an unitary transformation Q' * A * P, and return the matrices X and Y which are needed to apply the transformation to the unreduced part of sub( A ).

🌸 SYNOPSIS 🌸

SUBROUTINE PCLABRD( M, N, NB, A, IA, JA, DESCA, D, E, TAUQ, TAUP, X, IX,
                    JX, DESCX, Y, IY, JY, DESCY, WORK )

    INTEGER         IA, IX, IY, JA, JX, JY, M, N, NB

    INTEGER         DESCA( * ), DESCX( * ), DESCY( * )

    REAL            D( * ), E( * )

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

🎀 PURPOSE 🎀

PCLABRD reduces the first NB rows and columns of 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 by an unitary transformation Q' * A * P, and it hands you back the matrices X and Y which are needed to apply the transformation to the unreduced part of sub( A ).

If M >= N, sub( A ) gets reduced to upper bidiagonal form. If M < N, it's lower bidiagonal form. periodt.

This is an auxiliary routine called by PCGEBRD. it's giving supporting cast but honestly it's doing the most 💅

Notes

Each global data object is described by an associated description vector. This vector stores the information required 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".

  • 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.
  • 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.
  • NB — (global input) INTEGER — the number of leading rows and columns of sub( A ) to be reduced.
  • 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 ) to be reduced. On exit, the first NB rows and columns of the matrix are overwritten; the rest of the distributed matrix sub( A ) is unchanged. If m >= n, elements on and below the diagonal in the first NB columns, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors; and elements above the diagonal in the first NB rows, with the array TAUP, represent the unitary matrix P as a product of elementary reflectors. If m < n, elements below the diagonal in the first NB columns, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors, and elements on and above the diagonal in the first NB rows, with the array TAUP, represent the unitary 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 LOCr(IA+MIN(M,N)-1) if M >= N; LOCc(JA+MIN(M,N)-1) otherwise. The distributed diagonal elements of the bidiagonal matrix B: D(i) = A(ia+i-1,ja+i-1). 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(ia+i-1,ja+i) for i = 1,2,...,n-1; if m < n, E(i) = A(ia+i,ja+i-1) 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.
  • X — (local output) COMPLEX pointer into the local memory to an array of dimension (LLD_X,NB). On exit, the local pieces of the distributed M-by-NB matrix X(IX:IX+M-1,JX:JX+NB-1) required to update the unreduced part of sub( A ).
  • IX — (global input) INTEGER — the row index in the global array X indicating the first row of sub( X ).
  • JX — (global input) INTEGER — the column index in the global array X indicating the first column of sub( X ).
  • DESCX — (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X.
  • Y — (local output) COMPLEX pointer into the local memory to an array of dimension (LLD_Y,NB). On exit, the local pieces of the distributed N-by-NB matrix Y(IY:IY+N-1,JY:JY+NB-1) required to update the unreduced part of sub( A ).
  • IY — (global input) INTEGER — the row index in the global array Y indicating the first row of sub( Y ).
  • JY — (global input) INTEGER — the column index in the global array Y indicating the first column of sub( Y ).
  • DESCY — (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix Y.
  • WORK — (local workspace) COMPLEX array, dimension (LWORK). LWORK >= NB_A + NQ, with
NQ = NUMROC( N+MOD( IA-1, NB_Y ), NB_Y, MYCOL, IACOL, NPCOL )
IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL )

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

👑 FURTHER DETAILS 👑

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

Q = H(1) H(2) . . . H(nb)  and  P = G(1) G(2) . . . G(nb)

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.

If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in A(ia+i-1:ia+m-1,ja+i-1); u(1:i) = 0, u(i+1) = 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).

If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1: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: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 elements of the vectors v and u together form the m-by-nb matrix V and the nb-by-n matrix U' which are needed, with X and Y, to apply the transformation to the unreduced part of the matrix, using a block update of the form:

sub( A ) := sub( A ) - V*Y' - X*U'.

The contents of sub( A ) on exit are illustrated by the following examples with nb = 2:

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

  (  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )
  (  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )
  (  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )
  (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
  (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
  (  v1  v2  a   a   a  )

where a denotes an element of the original matrix which is unchanged, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i).

no cap, that's the whole thing. go reduce something. 💜

and that's a wrap bestie!! this page is <hotpink>LAPACK version 1.5</hotpink>, dated <cyan>12 May 1997</cyan>, and it lives in manual section l as PCLABRD(l). 🎀✨🦋

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