ZLARFT(3)

💅 NAME 💅

okay so we're deep in the LAPACK Library Functions Manual, section 3, page zlarft.f(3) — and the star of the show is the source file living at /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zlarft.f. That's the whole name situation, bestie. ✨

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🌸 SYNOPSIS 🌸

Functions/Subroutines

• subroutine ZLARFT (direct, storev, n, k, v, ldv, tau, t, ldt) — ZLARFT forms the triangular factor T of a block reflector H = I - vtvH

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🎀 Function/Subroutine Documentation 🎀

subroutine ZLARFT (character direct, character storev, integer n, integer k, complex*16, dimension( ldv, * ) v, integer ldv, complex*16, dimension( * ) tau, complex*16, dimension( ldt, * ) t, integer ldt)

ZLARFT forms the triangular factor T of a block reflector H = I - vtvH 💖

🦋 Purpose

ngl this is the whole vibe of the routine: ZLARFT forms the triangular factor T of a complex block reflector H of order n, and that H is defined as a product of k elementary reflectors. literally that's it, that's the job.

• 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 array V, and:

   H  =  I - V * T * V**H

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**H * T * V

periodt. 💅

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✨ Parameters ✨

💜 DIRECT

DIRECT is CHARACTER*1 — this one specifies the order in which the elementary reflectors are multiplied to form the block reflector, iykyk:

• = 'F': H = H(1) H(2) . . . H(k) (Forward)
• = 'B': H = H(k) . . . H(2) H(1) (Backward)

💜 STOREV

STOREV is CHARACTER*1 — specifies how the vectors which define the elementary reflectors are stored (see also Further Details, it's cute down there):

• = 'C': columnwise
• = 'R': rowwise

💜 N

N is INTEGER — the order of the block reflector H. Constraint check: N >= 0. 👑

💜 K

K is INTEGER — the order of the triangular factor T (which, fun fact, equals the number of elementary reflectors). Constraint check: K >= 1. 🔥

💜 V

V is COMPLEX*16 array, dimension:

                     (LDV,K) if STOREV = 'C'
                     (LDV,N) if STOREV = 'R'

The matrix V. See further details, bestie (spoiler: there's a whole diagram).

💜 LDV

LDV is INTEGER — the leading dimension of the array V. Respectively: if STOREV = 'C', then LDV >= max(1,N); if STOREV = 'R', then LDV >= K. don't skip the vibe check on these bounds fr fr.

💜 TAU

TAU is COMPLEX*16 array, dimension (K) — TAU(i) must contain the scalar factor of the elementary reflector H(i). must, not "should", not "maybe", must. 🙈

💜 T

T is COMPLEX*16 array, dimension (LDT,K) — the k by k triangular factor T of the block reflector. If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is lower triangular. The rest of the array is not used — the leftovers are just… there, minding their business. 💀

💜 LDT

LDT is INTEGER — the leading dimension of the array T. Constraint check: LDT >= K. slay. ⭐

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👑 Author 👑

• Univ. of Tennessee
• Univ. of California Berkeley
• Univ. of Colorado Denver
• NAG Ltd.

all icons, no notes. 🫶

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🌈 Further Details: 🌈

okay this part is genuinely so satisfying, 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 (they're implied, main character energy, they don't need to be written down):

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

               V = (  1       )                 V = (  1 v1 v1 v1 v1 )
                   ( v1  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 = ( v1 v2 v3 )                 V = ( v1 v1  1       )
                   ( v1 v2 v3 )                     ( v2 v2 v2  1    )
                   (  1 v2 v3 )                     ( v3 v3 v3 v3  1 )
                   (     1 v3 )
                   (        1 )

it's giving "triangular but make it fashion." 🎀

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💖 Official-ish note: this routine's definition is at line 162 of file zlarft.f.

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🧚 Author 🧚

Generated automatically by Doxygen for LAPACK from the source code. (not me writing it by hand, the robots ate and left no crumbs ✨)

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and that's the page!! 🦋 this lil doc is brought to you by LAPACK, Version 3.12.0, for zlarft.f — go off and form some triangular factors, bestie. 🍓💜