PCGEQPF(l)

📦 version 1.5, in ubuntu jammy

💖 PCGEQPF ✨ (NAME)

ok so bestie, gather round 🫶 the routine we're obsessing over today is PCGEQPF — she's a LAPACK routine and she 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). no cap. iconic behavior honestly 💅

🎀 SYNOPSIS (the receipts)

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

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

    INTEGER         DESCA( * ), IPIV( * )

    REAL            RWORK( * )

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

like look at her. thirteen arguments. she's got range 💜

🌸 PURPOSE (what she actually does)

PCGEQPF 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.

periodt. that's the whole vibe. 🦋

📝 Notes

Every global data object out here is described by an associated description vector. This vector stores all the info you need to establish the mapping between an object element and its corresponding process and memory location. it's giving ✨ administrative excellence ✨

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

math but make it fashion 💅

🔥 ARGUMENTS (meet the cast)

  • 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, 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, 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 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) REAL 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.
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.
ngl the INFO < 0 encoding is lowkey a little unhinged but she's been doing it since forever so we respect her 🙈

🧚 FURTHER DETAILS (the lore)

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.

like. the canonical unit vector said "I'm the moment" and honestly? she was 👑

that's the whole page bestie, you ate that reading session, go drink some water 🍓💜

✿ this has been PCGEQPF(l), from LAPACK version 1.5, dated 12 May 1997 — yeah she's vintage, she's archive-core, she's a classic and we love her ✿

📦 where this page comes from