PZLARFG(l)

📦 version 1.5, in ubuntu jammy

🎀 PZLARFG 💖 (LAPACK auxiliary routine, bestie edition)

okay so welcome to the PZLARFG(l) glow-up page ✨ this is a LAPACK auxiliary routine and honestly? she's giving linear algebra royalty 👑 stick with me and you'll get it, no cap.

🌸 NAME 🌸

  • PZLARFG — generates a complex elementary reflector H of order n, such that H * sub( X ) = H * ( x(iax,jax) ) = ( alpha ), H' * H = I 💅

✨ SYNOPSIS ✨

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

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

    COMPLEX*16      ALPHA

    INTEGER         DESCX( * )

    COMPLEX*16      TAU( * ), X( * )

🦋 PURPOSE 🦋

PZLARFG generates a complex elementary reflector H of order n, such that

                      (      x     )   (   0   )

where alpha is a real scalar, and sub( X ) is an (N-1)-element complex 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 complex scalar and v is a complex (N-1)-element vector. Note that H is not Hermitian. yeah, she plays by her own rules 💅

If the elements of sub( X ) are all zero and X(IAX,JAX) is real, then tau = 0 and H is taken to be the unit matrix.

Otherwise 1 <= real(tau) <= 2 and abs(tau-1) <= 1.

💫 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".

okay so here's the notation rundown, keep up bestie 🫶

  • 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. she's a matrix, she's a vector, she contains multitudes 💜

💅 ARGUMENTS 💅

  • N — (global input) INTEGER. The global order of the elementary reflector. N >= 0.
  • ALPHA — (local output) COMPLEX*16. 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) COMPLEX*16, 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. so real, she changes in place 💖
  • 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, don't try it 🙈
  • TAU — (local output) COMPLEX*16, 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. 🎀

and that's the whole page bestie, you ate that 🌈 go forth and build your reflectors 💅

signing off with all the love: this was LAPACK version 1.5, dated 12 May 1997 — iconic, vintage, timeless 👑✨🦋🍓

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