SLARZT(3)

SLARZT(3) 🎀

from the Library Functions Manual — yes bestie, this is the whole slay.

🦋 NAME 🦋

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/slarzt.f — that's the file, path and all. No cap.

✨ SYNOPSIS ✨

Functions/Subroutines

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

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

💅 Function/Subroutine Documentation 💅

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

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

🔥 Purpose 🔥

Okay so SLARZT 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', 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**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. That's the vibe check, bestie: the other combos are not supported yet.

🧸 Parameters 🧸

Okay parameter time — every single one, no skipping.

• DIRECT — 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, not supported yet)
  • = 'B': H = H(k) . . . H(2) H(1) (Backward)
• STOREV — CHARACTER*1. Specifies how the vectors which define the elementary reflectors are stored (see also Further Details):
  • = 'C': columnwise (not supported yet)
  • = 'R': rowwise
• N — INTEGER. The order of the block reflector H. N >= 0.
• K — INTEGER. The order of the triangular factor T (= the number of elementary reflectors). K >= 1.
• V — REAL array, dimension (LDV,K) if STOREV = 'C'; (LDV,N) if STOREV = 'R'. The matrix V. See further details.
• LDV — INTEGER. The leading dimension of the array V. If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.
• TAU — REAL array, dimension (K). TAU(i) must contain the scalar factor of the elementary reflector H(i).
• T — REAL 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 — 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.

Here's the literal layout from the source, comment markers and all, because precision is the whole point:

           !>
           !>  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 slarzt.f.

🤖 Author 🤖

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