PZGEQPF(l)

💖 PZGEQPF 💖

okay bestie, gather round, because we're about to talk about a whole LAPACK routine from version 1.5 and honestly? she's iconic. this is the manual page for PZGEQPF, and yes it's a lot of letters, but you're gonna eat this up, i promise 🫶

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✨ NAME ✨

PZGEQPF — compute a QR factorization with column pivoting of a M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) 🌸

like, that's the whole job. QR + column pivoting. on a distributed matrix. no cap. 💅

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✨ SYNOPSIS ✨

SUBROUTINE PZGEQPF( M, N, A, IA, JA, DESCA, IPIV, TAU, WORK, LWORK,
                    RWORK, LRWORK, INFO )

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

    INTEGER         DESCA( * ), IPIV( * )

    DOUBLE          PRECISION RWORK( * )

    COMPLEX*16      A( * ), TAU( * ), WORK( * )

look at all those arguments. she's got range. 👑

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✨ PURPOSE ✨

so PZGEQPF computes a QR factorization with column pivoting of a M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1):

                       sub( A ) * P = Q * R.

that's it, that's the whole vibe. QR, but pivoted. slay. 🦋

🌈 Notes 🌈

okay this is the part where it gets a lil technical but stay with me, it's giving lore.

every global data object is described by an associated description vector. this vector stores the info you need to figure out the mapping between an object element and its corresponding process and memory location. fr fr, it's the little passport that tells you where stuff lives. 🛂

let A be a generic term for any 2D block cyclicly distributed array. such a global array has an associated description vector DESCA. and in the following comments, the character _ should be read as "of the global array". iykyk. ✨

here's the whole legend, and yes i'm keeping every single line because every single line matters:

• 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)).

now 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 ) 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

math girlies rise 🙈✨

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✨ ARGUMENTS ✨

okay bestie this is the main character section, let's go one by one, no skipping, we do not skip in this house 💅

• 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*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). on entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. on exit, the elements on and above the diagonal of sub( A ) contain the min(M,N) by N upper trapezoidal matrix R (R is upper triangular if M >= N); the elements below the diagonal, 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.
• IPIV (local output) INTEGER array, dimension LOCc(JA+N-1). on exit, if IPIV(I) = K, the local i-th column of sub( A )*P was the global K-th column of sub( A ). IPIV is tied to the distributed matrix A. 🔗
• TAU (local output) COMPLEX*16, array, dimension LOCc(JA+MIN(M,N)-1). this array contains the scalar factors TAU of the elementary reflectors. TAU is tied to the distributed matrix A. 🔗
• WORK (local workspace/local output) COMPLEX*16 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(3,Mp0 + Nq0).

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. 👀

• RWORK (local workspace/local output) DOUBLE PRECISION array, dimension (LRWORK). on exit, RWORK(1) returns the minimal and optimal LRWORK.
• LRWORK (local or global input) INTEGER — the dimension of the array RWORK. LRWORK is local input and must be at least LRWORK >= LOCc(JA+N-1)+Nq0.

and here come the little helper quantities, get ready:

        IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW =
        INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA,
        NB_A, MYCOL, CSRC_A, NPCOL ), Mp0   = NUMROC( M+IROFF, MB_A,
        MYROW, IAROW, NPROW ), Nq0   = NUMROC( N+ICOFF, NB_A, MYCOL,
        IACOL, NPCOL ), LOCc(JA+N-1) = NUMROC( JA+N-1, NB_A, MYCOL,
        CSRC_A, NPCOL )

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

if LRWORK = -1, then LRWORK 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.

• 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.

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✨ FURTHER DETAILS ✨

the matrix Q is represented as a product of elementary reflectors 💜

   Q = H(1) H(2) . . . H(n)

each H(i) has the form

   H = I - tau * v * v'

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

the matrix P is represented in jpvt as follows: if

   jpvt(j) = i

then the jth column of P is the ith canonical unit vector. ⭐

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heads up bestie: this is a LAPACK routine from version 1.5, dated 12 May 1997, so she's an elder icon but she still serves 💅 treat the arguments with respect and double-check your workspace sizes, because PXERBLA will absolutely not be texting you about it during a workspace query.

and that's the whole page, mwah 🎀 lots of love from the LAPACK crew — LAPACK version 1.5, 12 May 1997 — bye bestie, go pivot some columns 💖✨👑