SRC/clarft.f(3)

Bestie, welcome to SRC/clarft.f from the Library Functions Manual — this page is all about clarft, and it’s lowkey iconic. 💜

🌸 NAME 🌸

SRC/clarft.f — that’s the file, periodt. ✨

✨ SYNOPSIS ✨

🧚 Functions/Subroutines

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

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

💅 Function/Subroutine Documentation 💅

🫶 recursive subroutine clarft

Here’s the whole signature, bestie — no cap, copied exactly:

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

And the one-liner vibe is literally:

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

🎯 Purpose

Okay so CLARFT 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

📝 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 COMPLEX 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 COMPLEX array, dimension (K). TAU(i) must contain the scalar factor of the elementary reflector H(i).
• T — T is COMPLEX 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

Definition at line 162 of file clarft.f.

👑 Author

Generated automatically by Doxygen for LAPACK from the source code.

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That’s the whole page, bestie. Final sign-off vibe: LAPACK Version 3.12.0, page SRC/clarft.f(3) — slay, bye! 💖