PZLARZT(l)

📦 version 1.5, in ubuntu jammy

💖 PZLARZT 💖

okay bestie, gather round, because today we're talking about PZLARZT(l) — a LAPACK auxiliary routine, version 1.5, and she is doing the most in the most elegant way possible. 🎀

✨ NAME ✨

PZLARZT — she forms the triangular factor T of a complex block reflector H of order > n, which is defined as a product of k elementary reflectors as returned by PZTZRZF. that's the whole job description, no cap. 💅

🦋 SYNOPSIS 🦋

SUBROUTINE PZLARZT( 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( * )

like. look at her. so many arguments and she handles all of them. main character energy fr fr. 👑

🌸 PURPOSE 🌸

PZLARZT forms the triangular factor T of a complex block reflector H of order > n, which is defined as a product of k elementary reflectors as returned by PZTZRZF.

so here's the vibe check on how the reflectors get multiplied:

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.

and depending on where the vectors live:

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

H  =  I - V' * T * V

🚨 important bestie alert 🚨 Currently, only STOREV = 'R' and DIRECT = 'B' are supported. the other combos are giving "coming soon" and not "works right now". don't come for me.

📝 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".

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 — Specifies the order in which the elementary reflectors are multiplied to form the block reflector:
- = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet 😭)
- = 'B': H = H(k) . . . H(2) H(1) (Backward) ✅
STOREV (global input) CHARACTER — Specifies how the vectors which define the elementary reflectors are stored (see also Further Details):
- = 'R': rowwise
N (global input) INTEGER — The number of meaningful entries of the block reflector H. N >= 0. keep it non-negative, bestie.
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+K-1),LOCc(JV+N-1)). 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 (MB_V,MB_V). It contains the k-by-k triangular factor of the block reflector associated with V. T is lower triangular.
WORK (local workspace) COMPLEX*16 array, dimension (K*(K-1)/2). that's it. that's the workspace. give her the room she needs. 🌈

🧚 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.

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

                                           ______V_____
      ( v1 v2 v3 )                        /                   ( v1 v2
v3 )                      ( v1 v1 v1 v1 v1 . . . . 1 )
  V = ( v1 v2 v3 )                      ( v2 v2 v2 v2 v2 . . . 1   )
      ( v1 v2 v3 )                      ( v3 v3 v3 v3 v3 . . 1     )
      ( v1 v2 v3 )
         .  .  .
         .  .  .
         1  .  .
            1  .
               1

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

                                                     ______V_____
         1                                          /
.  1                           ( 1 . . . . v1 v1 v1 v1 v1 )
         .  .  1                        ( . 1 . . . v2 v2 v2 v2 v2 )
         .  .  .                        ( . . 1 . . v3 v3 v3 v3 v3 )
         .  .  .
      ( v1 v2 v3 )
      ( v1 v2 v3 )
  V = ( v1 v2 v3 )
      ( v1 v2 v3 )
      ( v1 v2 v3 )

those little 1s that "are not stored" but get modified and restored on exit? iconic behaviour honestly. they leave, they come back, no drama. 💀

and that's the whole page bestie — PZLARZT from LAPACK, version 1.5, dated 12 May 1997. she's been serving since '97 and she has not missed a day. 💖✨ go forth and form those triangular factors. mwah. 🎀🦋

📦 where this page comes from

ubuntu jammy: scalapack-doc 1.5-11
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash on 2026-10-14: a language model's rewrite of the original, under its terms