SRC/ztprfb.f(3)

💖 NAME

SRC/ztprfb.f(3) — this page lives in the Library Functions Manual, and it's all about ztprfb, your fave complex block-reflector bestie. ✨

✨ SYNOPSIS

🧚 Functions/Subroutines

       subroutine ztprfb (side, trans, direct, storev, m, n, k, l, v, ldv, t,
           ldt, a, lda, b, ldb, work, ldwork)
           ZTPRFB applies a complex triangular-
           pentagonal"blockreflectortoacomplexmatrix,whichiscomposedoftwoblocks.

🌸 Function/Subroutine Documentation

   subroutine ztprfb (character side, character trans, character direct,
       character storev, integer m, integer n, integer k, integer l,
       complex*16, dimension( ldv, * ) v, integer ldv, complex*16, dimension(
       ldt, * ) t, integer ldt, complex*16, dimension( lda, * ) a, integer lda,
       complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension(
       ldwork, * ) work, integer ldwork)

ZTPRFB applies a complex "triangular-pentagonal" block reflector to a complex matrix, which is composed of two blocks. That's the whole plot, bestie. 💅

🎀 Purpose

Okay so ZTPRFB takes a complex block reflector H or its conjugate transpose H**H and applies it to a complex matrix C. That C is made of two blocks: A and B. The reflector can come from the left or the right — your call. ✨

🔥 Parameters

Every single knob, no skips:

• SIDE — CHARACTER*1. = 'L': apply H or H**H from the Left. = 'R': apply H or H**H from the Right. 🫶
• TRANS — CHARACTER*1. = 'N': apply H (No transpose). = 'C': apply H**H (Conjugate transpose). 🔥
• DIRECT — CHARACTER*1. Indicates how H is formed from a product of elementary reflectors. = 'F': H = H(1) H(2) . . . H(k) (Forward). = 'B': H = H(k) . . . H(2) H(1) (Backward). 🎀
• STOREV — CHARACTER*1. Indicates how the vectors which define the elementary reflectors are stored: = 'C': Columns. = 'R': Rows. 🌈
• M — INTEGER. The number of rows of the matrix B. M >= 0.
• N — INTEGER. The number of columns of the matrix B. N >= 0.
• K — INTEGER. The order of the matrix T, i.e. the number of elementary reflectors whose product defines the block reflector. K >= 0.
• L — INTEGER. The order of the trapezoidal part of V. K >= L >= 0. See Further Details.
• V — COMPLEX*16 array, dimension (LDV,K) if STOREV = 'C'; (LDV,M) if STOREV = 'R' and SIDE = 'L'; (LDV,N) if STOREV = 'R' and SIDE = 'R'. The pentagonal matrix V, which contains the elementary reflectors H(1), H(2), ..., H(K). See Further Details.
• LDV — INTEGER. The leading dimension of the array V. If STOREV = 'C' and SIDE = 'L', LDV >= max(1,M); if STOREV = 'C' and SIDE = 'R', LDV >= max(1,N); if STOREV = 'R', LDV >= K.
• T — COMPLEX*16 array, dimension (LDT,K). The triangular K-by-K matrix T in the representation of the block reflector.
• LDT — INTEGER. The leading dimension of the array T. LDT >= K.
• A — COMPLEX*16 array, dimension (LDA,N) if SIDE = 'L' or (LDA,K) if SIDE = 'R'. On entry, the K-by-N or M-by-K matrix A. On exit, A is overwritten by the corresponding block of H*C or H**H*C or C*H or C*H**H. See Further Details.
• LDA — INTEGER. The leading dimension of the array A. If SIDE = 'L', LDA >= max(1,K); if SIDE = 'R', LDA >= max(1,M).
• B — COMPLEX*16 array, dimension (LDB,N). On entry, the M-by-N matrix B. On exit, B is overwritten by the corresponding block of H*C or H**H*C or C*H or C*H**H. See Further Details.
• LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,M).
• WORK — COMPLEX*16 array, dimension (LDWORK,N) if SIDE = 'L', (LDWORK,K) if SIDE = 'R'.
• LDWORK — INTEGER. The leading dimension of the array WORK. If SIDE = 'L', LDWORK >= K; if SIDE = 'R', LDWORK >= M.

👑 Author

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

🧚 Further Details

Okay, here's the deep lore. The matrix C is a composite matrix formed from blocks A and B. The block B is of size M-by-N; if SIDE = 'R', A is of size M-by-K, and if SIDE = 'L', A is of size K-by-N. 🫶

• If SIDE = 'R' and DIRECT = 'F', C = [A B].
• If SIDE = 'L' and DIRECT = 'F':

C = [A]
    [B].

• If SIDE = 'R' and DIRECT = 'B', C = [B A].
• If SIDE = 'L' and DIRECT = 'B':

C = [B]
    [A].

The pentagonal matrix V is composed of a rectangular block V1 and a trapezoidal block V2. The size of the trapezoidal block is determined by the parameter L, where 0<=L<=K. If L=K, the V2 block of V is triangular; if L=0, there is no trapezoidal block, thus V = V1 is rectangular. ✨

• If DIRECT = 'F' and STOREV = 'C':

V = [V1]
    [V2]

• V2 is upper trapezoidal (first L rows of K-by-K upper triangular).

• If DIRECT = 'F' and STOREV = 'R':

V = [V1 V2]

• V2 is lower trapezoidal (first L columns of K-by-K lower triangular).

• If DIRECT = 'B' and STOREV = 'C':

V = [V2]
    [V1]

• V2 is lower trapezoidal (last L rows of K-by-K lower triangular).

• If DIRECT = 'B' and STOREV = 'R':

V = [V2 V1]

• V2 is upper trapezoidal (last L columns of K-by-K upper triangular).

• If STOREV = 'C' and SIDE = 'L', V is M-by-K with V2 L-by-K.
• If STOREV = 'C' and SIDE = 'R', V is N-by-K with V2 L-by-K.
• If STOREV = 'R' and SIDE = 'L', V is K-by-M with V2 K-by-L.
• If STOREV = 'R' and SIDE = 'R', V is K-by-N with V2 K-by-L.

Definition at line 249 of file ztprfb.f.

👑 Author

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And that's the whole page, bestie — stamped LAPACK Version 3.12.0, page SRC/ztprfb.f(3). 💅✨