SRC/dlarzt.f(3)

✨ SRC/dlarzt.f(3) ✨ — live from the Library Functions Manual, bestie 🫶💖

🌸 NAME 🌸

SRC/dlarzt.f — that’s the whole name, and honestly it ate 💅

✨ SYNOPSIS ✨

🌈 Functions/Subroutines 🌈

subroutine dlarzt (direct, storev, n, k, v, ldv, tau, t, ldt)

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

💖 Function/Subroutine Documentation 💖

   subroutine dlarzt (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)

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

🎀 Purpose 🎀

Okay so picture this: DLARZT builds the triangular factor T of a real block reflector H of order > n, and that H is defined as a product of k elementary reflectors. No cap, that’s the whole assignment. 🌟

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

Currently, only STOREV = 'R' and DIRECT = 'B' are supported. 🚨 So don’t come for the other combos yet — they are not supported, bestie.

🧚 Parameters 🧚

✨ DIRECT ✨

DIRECT is CHARACTER*1 — it 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 🦋

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

• = 'C': columnwise (not supported yet) 🙈
• = 'R': rowwise 🎀

⭐ N ⭐

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

💅 K 💅

K is INTEGER — the order of the triangular factor T (= the number of elementary reflectors). Constraint: K >= 1. 🔥

🌈 V 🌈

V is a DOUBLE PRECISION array, dimension:

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

This is the matrix V. See further details below, bestie. 🫶

🎯 LDV 🎯

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

🌟 TAU 🌟

TAU is a DOUBLE PRECISION array, dimension (K). TAU(i) must contain the scalar factor of the elementary reflector H(i). ✨

🍓 T 🍓

T is a 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. Constraint: LDT >= K. 🔥

🏆 Author 🏆

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

🌟 Contributors: 🌟

• A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

🔍 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 )

Definition at line 184 of file dlarzt.f. 📍

👑 Author 👑

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Okay that’s the whole page, bestie — brought to you by LAPACK version 3.12.0 and SRC/dlarzt.f(3). Stay triangular, stay iconic, and don’t forget to restore those array elements on exit 😭💖✨