PDLARFG(l)

📦 version 1.5, in ubuntu jammy

PDLARFG 💖✨

okay bestie, gather round, because this is a LAPACK auxiliary routine — specifically from version 1.5 — and it's giving serious linear algebra main character energy. it lives in the l (double precision) family, and it's one of those ScaLAPACK joints where every array is described by a descriptor vector and every index is global. iykyk. 🦋

NAME 🎀

PDLARFG — generate a real elementary reflector H of order n, such that

H * sub( X ) = H * ( x(iax,jax) ) = ( alpha ), H' * H = I

like. it makes a Householder reflector. that's the whole vibe. no cap. 💅

SYNOPSIS ✨

SUBROUTINE PDLARFG( N, ALPHA, IAX, JAX, X, IX, JX, DESCX, INCX, TAU )

    INTEGER         IAX, INCX, IX, JAX, JX, N

    DOUBLE          PRECISION ALPHA

    INTEGER         DESCX( * )

    DOUBLE          PRECISION TAU( * ), X( * )

PURPOSE 🌈

PDLARFG generates a real elementary reflector H of order n, such that

                          (      x     )   (   0   )

where alpha is a scalar, and sub( X ) is an (N-1)-element real distributed vector X(IX:IX+N-2,JX) if INCX = 1 and X(IX,JX:JX+N-2) if INCX = DESCX(M_). H is represented in the form

      H = I - tau * ( 1 ) * ( 1 v' ) ,
                    ( v )

where tau is a real scalar and v is a real (N-1)-element vector.

if the elements of sub( X ) are all zero, then tau = 0 and H is taken to be the unit matrix. slay, no work to do. 💅

otherwise 1 <= tau <= 2.

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".

  • NOTATION — STORED IN — EXPLANATION
  • DTYPE_A(global) — DESCA( DTYPE_ ) — The descriptor type. In this case, DTYPE_A = 1.
  • CTXT_A (global) — 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) — DESCA( M_ ) — The number of rows in the global array A.
  • N_A (global) — DESCA( N_ ) — The number of columns in the global array A.
  • MB_A (global) — DESCA( MB_ ) — The blocking factor used to distribute the rows of the array.
  • NB_A (global) — DESCA( NB_ ) — The blocking factor used to distribute the columns of the array.
  • RSRC_A (global) — DESCA( RSRC_ ) — The process row over which the first row of the array A is distributed.
  • CSRC_A (global) — DESCA( CSRC_ ) — The process column over which the first column of the array A is distributed.
  • LLD_A (local) — 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

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

because vectors may be viewed as a subclass of matrices, a distributed vector is considered to be a distributed matrix. periodt. 💀

ARGUMENTS 🫶

N — (global input) INTEGER

The global order of the elementary reflector. N >= 0.

ALPHA — (local output) DOUBLE PRECISION

On exit, alpha is computed in the process scope having the vector sub( X ).

IAX — (global input) INTEGER

The global row index in X of X(IAX,JAX).

JAX — (global input) INTEGER

The global column index in X of X(IAX,JAX).

X — (local input/local output) DOUBLE PRECISION, pointer into the local memory to an array of dimension (LLD_X,*). This array contains the local pieces of the distributed vector sub( X ). Before entry, the incremented array sub( X ) must contain the vector x. On exit, it is overwritten with the vector v.

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.

INCX — (global input) INTEGER

The global increment for the elements of X. Only two values of INCX are supported in this version, namely 1 and M_X. INCX must not be zero.

bestie. INCX must not be zero. i'm not playing. 🚨

and that's the whole page, fr fr 💜 this was LAPACK version 1.5, dated 12 May 1997 — yes, an icon from the 90s that's still out here computing reflectors. we love a legend. 🎀✨👑

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