CLARZT(3)

🌸 NAME 🌸

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

Okay bestie, this is clarzt.f(3) from the Library Functions Manual, and it's all about the CLARZT subroutine. ✨

✨ SYNOPSIS ✨

Functions/Subroutines

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

🦋 Function/Subroutine Documentation 🦋

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

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

💖 Purpose 💖

Okay so here's the whole point, no cap:

CLARZT 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', 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

Currently, only STOREV = 'R' and DIRECT = 'B' are supported. 💀 So don't come for me with the other ones yet.

🎀 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. 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.

🌈 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 clarzt.f.

📜 Author 📜

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