PSLARFG(l)

📦 version 1.5, in ubuntu jammy

✨ PSLARFG ✨

ok so bestie, welcome to PSLARFG from the LAPACK auxiliary routines — she's a ScaLAPACK girly, version 1.5 energy, and she has ONE job: build you a real elementary reflector. let's get into it 💅

🌸 NAME 🌸

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

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

periodt. that's the whole vibe. 🎀

✨ SYNOPSIS ✨

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

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

    REAL            ALPHA

    INTEGER         DESCX( * )

    REAL            TAU( * ), X( * )

💖 PURPOSE 💖

PSLARFG 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. (she's giving nothing, and that's valid ✨)

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. (it's the little address book of the distributed world, no cap)

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

Because vectors may be viewed as a subclass of matrices, a distributed vector is considered to be a distributed matrix. (a vector is just a matrix that decided to be low-maintenance, iykyk)

🎀 ARGUMENTS 🎀

  • N — (global input) INTEGER — the global order of the elementary reflector. N >= 0.
  • ALPHA — (local output) REAL — 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) REAL, 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. (glow up moment 💅)
  • 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. (fr fr, do NOT pass her a zero, she will not cope 😭)
  • TAU — (local output) REAL, array, dimension LOCc(JX) if INCX = 1, and LOCr(IX) otherwise. This array contains the Householder scalars related to the Householder vectors. TAU is tied to the distributed matrix X.

ok bestie that's the whole page, every flag accounted for, nothing skipped, we ate and left no crumbs 🍓✨

this was LAPACK version 1.5, dated 12 May 1997 — like, vintage vintage, but she still slays. mwah 💋🫶💖

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