PSGEQL2(l)

📦 version 1.5, in ubuntu jammy

🎀 PSGEQL2 🎀

okay bestie, listen up — this is the manual page for PSGEQL2, aka PSGEQL2(l) in the LAPACK routine library (version 1.5, if we're being precise, and we ARE). it does distributed linear algebra and it's lowkey iconic. let's get into it 💅

✨ NAME ✨

PSGEQL2 — computes a QL factorization of a real distributed M-by-N matrix

sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * L

🌸 SYNOPSIS 🌸

SUBROUTINE PSGEQL2( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO )

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

    INTEGER         DESCA( * )

    REAL            A( * ), TAU( * ), WORK( * )

💖 PURPOSE 💖

PSGEQL2 computes a QL factorization of a real distributed M-by-N matrix

sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * L

📝 Notes 📝

okay so before we go any further, you need to know about description vectors, because literally everything in this world revolves around them. every global data object has one, and it stores all the info you need to map an element of that object to the process and memory location where it actually lives. it's giving ✨address book for distributed data✨.

let's call A the generic name for any 2D block cyclicly distributed array. a global array like that has a description vector DESCA riding along with it. in the notes below, that little _ character means "of the global array". iykyk 🫶

  • 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 ) = 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 ) = 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() can be figured out 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 an upper bound for those quantities can be computed like this:

LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A
LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A

slay. now we can talk arguments 🧚

💅 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) REAL 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: if M >= N, the lower triangle of the distributed submatrix A( IA+M-N:IA+M-1, JA:JA+N-1 ) contains the N-by-N lower triangular matrix L; if M <= N, the elements on and below the (N-M)-th superdiagonal contain the M by N lower trapezoidal matrix L; the remaining elements, together with the array TAU, represent the orthogonal 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.
  • TAU — (local output) REAL, array, dimension LOCc(JA+N-1). this array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A.
  • WORK — (local workspace/local output) REAL 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 >= Mp0 + MAX( 1, Nq0 ), where
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 ),

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. so don't panic when nothing yells at you 💀
  • 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 matrix Q is represented as a product of elementary reflectors

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

each H(i) has the form

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

where tau is a real scalar, and v is a real vector with v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in A(ia:ia+m-k+i-2,ja+n-k+i-1), and tau in TAU(ja+n-k+i-1).

that's the whole vibe bestie, you ate that 🍓

💖 this page is LAPACK version 1.5, dated 12 May 1997, and lives in the manual as PSGEQL2(l) 💖

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