PCGGQRF(l)

📦 version 1.5, in ubuntu jammy

💖 PCGGQRF — the generalized QR factorization bestie 💖

okay so. you know how the LAPACK routine PCGGQRF is a whole vibe? it lives in the l manual (LAPACK routine, version 1.5, if we're being precise 💅). no cap, this one is a main character.

✨ NAME ✨

PCGGQRF computes a generalized QR factorization of an N-by-M matrix sub( A ) = A(IA:IA+N-1,JA:JA+M-1) and an N-by-P matrix sub( B ) = B(IB:IB+N-1,JB:JB+P-1). iconic behavior, honestly.

✨ SYNOPSIS ✨

SUBROUTINE PCGGQRF( N, M, P, A, IA, JA, DESCA, TAUA, B, IB, JB, DESCB,
                    TAUB, WORK, LWORK, INFO )

    INTEGER         IA, IB, INFO, JA, JB, LWORK, M, N, P

    INTEGER         DESCA( * ), DESCB( * )

    COMPLEX         A( * ), B( * ), TAUA( * ), TAUB( * ), WORK( * )

✨ PURPOSE ✨

so PCGGQRF computes a generalized QR factorization of an N-by-M matrix sub( A ) = A(IA:IA+N-1,JA:JA+M-1) and an N-by-P matrix sub( B ) = B(IB:IB+N-1,JB:JB+P-1), like:

            sub( A ) = Q*R,        sub( B ) = Q*T*Z,

where Q is an N-by-N unitary matrix, Z is a P-by-P unitary matrix, and R and T assume one of these forms — and bestie, R and T are so flexible:

if N >= M, R looks like:

R = ( R11 ) M  ,   or if N < M,  R = ( R11  R12 ) N,
    (  0  ) N-M                         N   M-N
       M

where R11 is upper triangular, and

if N <= P, T looks like:

T = ( 0  T12 ) N,   or if N > P,  T = ( T11 ) N-P,
     P-N  N                           ( T21 ) P
                                         P

where T12 or T21 is upper triangular. periodt.

in particular, if sub( B ) is square and nonsingular, the GQR factorization of sub( A ) and sub( B ) implicitly gives the QR factorization of inv( sub( B ) )* sub( A ):

             inv( sub( B ) )*sub( A )= Z'*(inv(T)*R)

where inv( sub( B ) ) denotes the inverse of the matrix sub( B ), and Z' denotes the conjugate transpose of matrix Z. no big deal, just math things 🔥

🦋 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".

here's the whole notation tea, bestie:

  • 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. lowkey wild.
  • 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 to be operated on i.e the number of rows of the distributed submatrices sub( A ) and sub( B ). N >= 0.
  • M (global input) INTEGER — the number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). M >= 0.
  • P (global input) INTEGER — the number of columns to be operated on i.e the number of columns of the distributed submatrix sub( B ). P >= 0.
  • A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+M-1)). on entry, the local pieces of the N-by-M distributed matrix sub( A ) which is to be factored. on exit, the elements on and above the diagonal of sub( A ) contain the min(N,M) by M upper trapezoidal matrix R (R is upper triangular if N >= M); the elements below the diagonal, with the array TAUA, represent the unitary matrix Q as a product of min(N,M) 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. bestie behavior.
  • TAUA (local output) COMPLEX, array, dimension LOCc(JA+MIN(N,M)-1). this array contains the scalar factors TAUA of the elementary reflectors which represent the unitary matrix Q. TAUA is tied to the distributed matrix A. (see Further Details).
  • B (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_B, LOCc(JB+P-1)). on entry, the local pieces of the N-by-P distributed matrix sub( B ) which is to be factored. on exit, if N <= P, the upper triangle of B(IB:IB+N-1,JB+P-N:JB+P-1) contains the N by N upper triangular matrix T; if N > P, the elements on and above the (N-P)-th subdiagonal contain the N by P upper trapezoidal matrix T; the remaining elements, with the array TAUB, represent the unitary matrix Z as a product of elementary reflectors (see Further Details).
  • IB (global input) INTEGER — the row index in the global array B indicating the first row of sub( B ).
  • JB (global input) INTEGER — the column index in the global array B indicating the first column of sub( B ).
  • DESCB (global and local input) INTEGER array of dimension DLEN_. the array descriptor for the distributed matrix B.
  • TAUB (local output) COMPLEX, array, dimension LOCr(IB+N-1). this array contains the scalar factors of the elementary reflectors which represent the unitary matrix Z. TAUB is tied to the distributed matrix B (see Further Details).
  • WORK (local workspace/local output) COMPLEX array, dimension (LWORK). on exit, WORK(1) returns the minimal and optimal LWORK. she's giving workspace AND output, multitasking queen 💅
  • LWORK (local or global input) INTEGER — the dimension of the array WORK. LWORK is local input and must be at least
LWORK >= MAX( NB_A * ( NpA0 + MqA0 + NB_A ), MAX( (NB_A*(NB_A-1))/2,
(PqB0 + NpB0)*NB_A ) + NB_A * NB_A, MB_B * ( NpB0 + PqB0 + MB_B ) ),

where

IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ),
IAROW  = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ),
IACOL  = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ),
NpA0   = NUMROC( N+IROFFA, MB_A, MYROW, IAROW, NPROW ),
MqA0   = NUMROC( M+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ),

IROFFB = MOD( IB-1, MB_B ), ICOFFB = MOD( JB-1, NB_B ),
IBROW  = INDXG2P( IB, MB_B, MYROW, RSRC_B, NPROW ),
IBCOL  = INDXG2P( JB, NB_B, MYCOL, CSRC_B, NPCOL ),
NpB0   = NUMROC( N+IROFFB, MB_B, MYROW, IBROW, NPROW ),
PqB0   = NUMROC( P+ICOFFB, NB_B, MYCOL, IBCOL, NPCOL ),

and NUMROC, INDXG2P 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. cute little shortcut, ngl.
  • 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 elementary reflectors

Q = H(ja) H(ja+1) . . . H(ja+k-1), where k = min(n,m).

each H(i) has the form

H(i) = I - taua * v * v'

where taua is a complex scalar, and v is a complex vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(ia+i:ia+n-1,ja+i-1), and taua in TAUA(ja+i-1).

to form Q explicitly, use ScaLAPACK subroutine PCUNGQR.

to use Q to update another matrix, use ScaLAPACK subroutine PCUNMQR.

the matrix Z is represented as a product of elementary reflectors

Z = H(ib)' H(ib+1)' . . . H(ib+k-1)', where k = min(n,p).

each H(i) has the form

H(i) = I - taub * v * v'

where taub is a complex scalar, and v is a complex vector with v(p-k+i+1:p) = 0 and v(p-k+i) = 1; conjg(v(1:p-k+i-1)) is stored on exit in B(ib+n-k+i-1,jb:jb+p-k+i-2), and taub in TAUB(ib+n-k+i-1).

to form Z explicitly, use ScaLAPACK subroutine PCUNGRQ.

to use Z to update another matrix, use ScaLAPACK subroutine PCUNMRQ.

🌈 Alignment requirements 🌈

the distributed submatrices sub( A ) and sub( B ) must verify some alignment properties, namely the following expression should be true — do not skip this bestie, it's the vibe check:

( MB_A.EQ.MB_B .AND. IROFFA.EQ.IROFFB .AND. IAROW.EQ.IBROW )

and that's the whole thing, bestie!! you just ate that manual page fr fr 💖✨🦋

this little love letter came to you from LAPACK version 1.5, dated 12 May 1997 (yes she's vintage, she's still iconic, we stan a classic 👑). mwah, go factor something. 🎀

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