SRC/zlarft.f(3)

🎀 SRC/zlarft.f(3) — LAPACK Library Functions Manual 🎀

okay bestie, welcome to the zlarft page, straight out of the LAPACK Library Functions Manual ✨ this one's a Fortran source file page, so it's giving "recursive subroutine" main character energy from the jump 💅

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🌸 NAME 🌸

SRC/zlarft.f — literally just the file, no cap 🫶

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✨ SYNOPSIS ✨

🦋 Functions/Subroutines

• recursive 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 💖

🌈 recursive 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:

ok so here's the whole vibe 👇

!>
!> ZLARFT forms the triangular factor T of a complex block reflector H
!> of order n, which is defined as a product of k elementary reflectors.
!>
!> If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular;
!>
!> If DIRECT = 'B', 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
!>

so like: two vibes of ordering ('F' forward → upper triangular T, 'B' backward → lower triangular T), and two vibes of storage ('C' columns, 'R' rows). Pick your fighter 💅

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🔥 Parameters

DIRECT

!>          DIRECT is 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)
!>

it's a CHARACTER*1 and it decides the multiplication order, periodt 🫶

• '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):
!>          = 'C': columnwise
!>          = 'R': rowwise
!>

also CHARACTER*1, and it's literally just "how are the vectors stored" (peep Further Details for the visual bestie) 💖

• 'C' → columnwise
• 'R' → rowwise

N

!>          N is INTEGER
!>          The order of the block reflector H. N >= 0.
!>

INTEGER, the order of the block reflector H, and the rule is N >= 0 — no negative nonsense allowed 🙈

K

!>          K is INTEGER
!>          The order of the triangular factor T (= the number of
!>          elementary reflectors). K >= 1.
!>

INTEGER, the order of the triangular factor T — which is also the number of elementary reflectors, iykyk — and K >= 1 fr fr ✨

V

!>          V is COMPLEX*16 array, dimension
!>                               (LDV,K) if STOREV = 'C'
!>                               (LDV,N) if STOREV = 'R'
!>          The matrix V. See further details.
!>

COMPLEX*16 array, dimension (LDV,K) if STOREV = 'C', or (LDV,N) if STOREV = 'R'. it's the matrix V, see further details below 🦋

LDV

!>          LDV is INTEGER
!>          The leading dimension of the array V.
!>          If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.
!>

INTEGER, the leading dimension of V. check the vibe: if STOREV = 'C', then LDV >= max(1,N); if STOREV = 'R', then LDV >= K 👑

TAU

!>          TAU is COMPLEX*16 array, dimension (K)
!>          TAU(i) must contain the scalar factor of the elementary
!>          reflector H(i).
!>

COMPLEX*16 array, dimension (K). TAU(i) must contain the scalar factor of the elementary reflector H(i) — not optional, bestie 🎀

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.
!>

COMPLEX*16 array, dimension (LDT,K) — the k by k triangular factor T of the block reflector. if DIRECT = 'F' then T is upper triangular; if DIRECT = 'B' then T is lower triangular. the rest of the array? not used, ignored, irrelevant, we don't know her 💀

LDT

!>          LDT is INTEGER
!>          The leading dimension of the array T. LDT >= K.
!>

INTEGER, the leading dimension of T, and LDT >= K ⭐

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🌈 Author

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

the whole squad showed up 🫶👑

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🌸 Further Details:

okay this part actually slays, the ascii art is iconic 🎨

!>
!>  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.
!>
!>  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 )
!>

so the shape of V and the storage of the vectors defining H(i) is best illustrated with n = 5 and k = 3 — and the elements equal to 1 are not stored 🙈 that's the whole trick bestie.

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⭐ Definition

Definition at line 162 of file zlarft.f.

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💜 Author

Generated automatically by Doxygen for LAPACK from the source code. yes as in the tool made this page, not a person, we love an automated icon ✨🤖

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and that's the page!! this one came to you courtesy of LAPACK, Version 3.12.0 🫶 mwah, go forth and build your block reflectors, bestie 💅🎀