DLARFT(3)

DLARFT(3) — LAPACK Library Functions Manual 💖

okay bestie, gather round: this is the LAPACK manual page for DLARFT, and it's lowkey iconic. It forms the triangular factor T of a block reflector, and every tiny constraint is staying right here. No cap. 🫶

📛 NAME

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/dlarft.f

✨ SYNOPSIS ✨

Functions/Subroutines

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

🌸 Function/Subroutine Documentation 🌸

subroutine DLARFT (character direct, character storev, integer n, integer k,
    double precision, dimension( ldv, * ) v, integer ldv, double precision,
    dimension( * ) tau, double precision, dimension( ldt, * ) t, integer
    ldt)

DLARFT forms the triangular factor T of a block reflector H = I - vtvH.

🎯 Purpose

Bestie, here's the whole vibe: DLARFT forms the triangular factor T of a real block reflector H of order n, which is defined as a product of k elementary reflectors. ✨

• 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**T

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 * T * V

🧸 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)
• STOREV — STOREV is CHARACTER*1. Specifies how the vectors which define the elementary reflectors are stored (see also Further Details):
  • = 'C': columnwise
  • = 'R': rowwise
• N — N is INTEGER. The order of the block reflector H. N >= 0.
• K — K is INTEGER. The order of the triangular factor T (= the number of elementary reflectors). K >= 1.
• V — V is DOUBLE PRECISION array, dimension

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

The matrix V. See further details.

• LDV — LDV is INTEGER. The leading dimension of the array V. If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.
• TAU — TAU is DOUBLE PRECISION array, dimension (K). TAU(i) must contain the scalar factor of the elementary reflector H(i).
• T — T is DOUBLE PRECISION 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.
• LDT — LDT is INTEGER. The leading dimension of the array T. LDT >= K.

👑 Author

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

🧚 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.

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 )

Definition at line 162 of file dlarft.f.

🤖 Author

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Source/version sign-off, bestie: LAPACK Version 3.12.0, from /home/abui...RC/dlarft.f(3). 💅✨