SRC/slarzt.f(3)

💅 SRC/slarzt.f(3)

Ok so bestie, buckle up — this is a LAPACK manpage from the SRC/slarzt.f file, and it's genuinely serving linear-algebra main character energy. It's giving "Library Functions Manual" realness, so you already know it's a fortran thing. Let's get into it ✨

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🦋 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. 🎀

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✨ 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

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. 🙈 (yes bestie, that's a real limitation, don't come for me)

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🎀 Parameters

• 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, not supported yet 💀)
  • = 'B': H = H(k) . . . H(2) H(1) (Backward) 👑
• STOREV — is 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 — is INTEGER. The order of the block reflector H. N >= 0. ✨
• K — is INTEGER. The order of the triangular factor T (= the number of elementary reflectors). K >= 1. 🫶
• V — is REAL array, dimension

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

The matrix V. See further details. 💜

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

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👑 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

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💖 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':

        ( v1 v2 v3 )
        ( v1 v2 v3 )
    V = ( v1 v2 v3 )
        ( v1 v2 v3 )
        ( v1 v2 v3 )
           .  .  .
           .  .  .
           1  .  .
              1  .
                 1

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':

           1
           .  1
           .  .  1
           .  .  .
           .  .  .
        ( v1 v2 v3 )
        ( v1 v2 v3 )
    V = ( v1 v2 v3 )
        ( v1 v2 v3 )
        ( v1 v2 v3 )

DIRECT = 'B' and STOREV = 'R':

                                               ______V_____
                                           /            \
                  ( 1 . . . . v1 v1 v1 v1 v1 )
                  ( . 1 . . . v2 v2 v2 v2 v2 )
                  ( . . 1 . . v3 v3 v3 v3 v3 )
           1
           .  1
           .  .  1
           .  .  .
           .  .  .
        ( v1 v2 v3 )
        ( v1 v2 v3 )
    V = ( v1 v2 v3 )
        ( v1 v2 v3 )
        ( v1 v2 v3 )

ngl those diagrams are a whole vibe check on their own — stare at them long enough and the row/column storage situation lowkey clicks. 🌸

Definition at line 184 of file slarzt.f.

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🌟 Author

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