PZLARFT(l)

📦 version 1.5, in ubuntu jammy

💖 PZLARFT 💖

okay so bestie, welcome to the PZLARFT(l) page — it's a LAPACK auxiliary routine, version 1.5, and it's genuinely so much fun once you clock what it's doing. sit down, hydrate, let's go 🫶

✨ NAME ✨

PZLARFT — forms the triangular factor T of a complex block reflector H of order n, which is defined as a product of k elementary reflectors. that's the whole job. she does one thing and she ATE 💅

🌸 SYNOPSIS 🌸

SUBROUTINE PZLARFT( DIRECT, STOREV, N, K, V, IV, JV, DESCV, TAU, T, WORK
                    )

    CHARACTER       DIRECT, STOREV

    INTEGER         IV, JV, K, N

    INTEGER         DESCV( * )

    COMPLEX*16      TAU( * ), T( * ), V( * ), WORK( * )

👑 PURPOSE 👑

so, PZLARFT forms the triangular factor T of a complex block reflector H of order n, which is defined as a product of k elementary reflectors. iconic behaviour, honestly.

  • if DIRECT = 'F', then H = H(1) H(2) . . . H(k) and T is upper triangular 🔺
  • if DIRECT = 'B', then H = H(k) . . . H(2) H(1) and T is lower triangular 🔻

if STOREV = 'C', the vector which defines the elementary reflector H(i) is stored in the i-th column of the distributed matrix V, and

H  =  I - V * T * V'

if STOREV = 'R', the vector which defines the elementary reflector H(i) is stored in the i-th row of the distributed matrix V, and

H  =  I - V' * T * V

📝 Notes 📝

every global data object is described by an associated description vector, fr fr. this vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. it's giving ✨receipts✨.

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

🎀 ARGUMENTS 🎀

  • DIRECT (global input) CHARACTER*1 — specifies the order in which the elementary reflectors are multiplied to form the block reflector:
    • = 'F': H = H(1) H(2) . . . H(k) (Forward)
    • = 'B': H = H(k) . . . H(2) H(1) (Backward)
  • STOREV (global input) CHARACTER*1 — specifies how the vectors which define the elementary reflectors are stored (see also Further Details):
    • = 'R': rowwise
  • N (global input) INTEGER — the order of the block reflector H. N >= 0.
  • K (global input) INTEGER — the order of the triangular factor T (= the number of elementary reflectors). 1 <= K <= MB_V (= NB_V).
  • V (input/output) COMPLEX*16 pointer into the local memory to an array of local dimension (LOCr(IV+N-1),LOCc(JV+K-1)) if STOREV = 'C', and (LOCr(IV+K-1),LOCc(JV+N-1)) if STOREV = 'R'. the distributed matrix V contains the Householder vectors. see further details.
  • IV (global input) INTEGER — the row index in the global array V indicating the first row of sub( V ).
  • JV (global input) INTEGER — the column index in the global array V indicating the first column of sub( V ).
  • DESCV (global and local input) INTEGER array of dimension DLEN_. the array descriptor for the distributed matrix V.
  • TAU (local input) COMPLEX*16, array, dimension LOCr(IV+K-1) if INCV = M_V, and LOCc(JV+K-1) otherwise. this array contains the Householder scalars related to the Householder vectors. TAU is tied to the distributed matrix V.
  • T (local output) COMPLEX*16 array, dimension (NB_V,NB_V) if STOREV = 'Col', and (MB_V,MB_V) otherwise. it contains the k-by-k triangular factor of the block reflector associated with V. if DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is lower triangular.
  • WORK (local workspace) COMPLEX*16 array, dimension (K*(K-1)/2)

🦋 FURTHER DETAILS 🦋

the shape of the matrix V and the storage of the vectors which define the H(i) is best illustrated by the following example with n = 5 and k = 3. the elements equal to 1 are not stored; the corresponding array elements are modified but restored on exit. the rest of the array is not used. periodt. 💅

DIRECT = 'F' and STOREV = 'C':   DIRECT = 'F' and STOREV = 'R':

V( IV:IV+N-1,    (  1       )    V( IV:IV+K-1,    (  1 v1 v1 v1 v1 )
   JV:JV+K-1 ) = ( v1  1    )       JV:JV+N-1 ) = (     1 v2 v2 v2 )
                 ( v1 v2  1 )                     (        1 v3 v3 )
                 ( v1 v2 v3 )
                 ( v1 v2 v3 )

DIRECT = 'B' and STOREV = 'C':   DIRECT = 'B' and STOREV = 'R':

V( IV:IV+N-1,    ( v1 v2 v3 )    V( IV:IV+K-1,    ( v1 v1  1       )
   JV:JV+K-1 ) = ( v1 v2 v3 )       JV:JV+N-1 ) = ( v2 v2 v2  1    )
                 (  1 v2 v3 )                     ( v3 v3 v3 v3  1 )
                 (     1 v3 )
                 (        1 )
and that's the whole page bestie — every flag, every descriptor, every little diagram, all of it. go forth and build your block reflectors, you absolute main character 👑

kisses, this was LAPACK version 1.5, dated 12 May 1997, and the page itself is PZLARFT(l) — catch you on the next one 🎀💖✨

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