SRC/cgebrd.f(3)

🌸 NAME 🌸

SRC/cgebrd.f — okay bestie, this page is from the Library Functions Manual, and it's all about the CGEBRD subroutine. 💅

✨ SYNOPSIS ✨

✨ Functions/Subroutines ✨

• subroutine cgebrd (m, n, a, lda, d, e, tauq, taup, work, lwork, info)
  • CGEBRD

💖 Function/Subroutine Documentation 💖

subroutine cgebrd (integer m, integer n, complex, dimension( lda, * ) a,
    integer lda, real, dimension( * ) d, real, dimension( * ) e, complex,
    dimension( * ) tauq, complex, dimension( * ) taup, complex, dimension( *
    ) work, integer lwork, integer info)

CGEBRD — no cap, this is the whole subroutine signature, copied exactly so you can trust it. 🫶

🎀 Purpose 🎀

Okay so CGEBRD reduces a general complex M-by-N matrix A to upper or lower bidiagonal form B by a unitary transformation: Q**H * A * P = B. ✨

If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal. That's the vibe check. 💅

🌈 Parameters 🌈

• M — M is INTEGER. The number of rows in the matrix A. M >= 0. 🍓
• N — N is INTEGER. The number of columns in the matrix A. N >= 0. 🍓
• A — A is COMPLEX array, dimension (LDA,N). On entry, the M-by-N general matrix to be reduced. On exit: if m >= n, the diagonal and the first superdiagonal are overwritten with the upper bidiagonal matrix B; the elements below the diagonal, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors, and the elements above the first superdiagonal, with the array TAUP, represent the unitary matrix P as a product of elementary reflectors; if m < n, the diagonal and the first subdiagonal are overwritten with the lower bidiagonal matrix B; the elements below the first subdiagonal, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors, and the elements above the diagonal, with the array TAUP, represent the unitary matrix P as a product of elementary reflectors. See Further Details. 👑
• LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M). 🎀
• D — D is REAL array, dimension (min(M,N)). The diagonal elements of the bidiagonal matrix B: D(i) = A(i,i). 💖
• E — E is REAL array, dimension (min(M,N)-1). The off-diagonal elements of the bidiagonal matrix B: if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1; if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1. 💖
• TAUQ — TAUQ is COMPLEX array, dimension (min(M,N)). The scalar factors of the elementary reflectors which represent the unitary matrix Q. See Further Details. 🦋
• TAUP — TAUP is COMPLEX array, dimension (min(M,N)). The scalar factors of the elementary reflectors which represent the unitary matrix P. See Further Details. 🦋
• WORK — WORK is COMPLEX array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 🌸
• LWORK — LWORK is INTEGER. The length of the array WORK. LWORK >= 1, if MIN(M,N) = 0, and LWORK >= MAX(M,N), otherwise. For optimum performance LWORK >= (M+N)*NB, where NB is the optimal blocksize. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. 💀
• INFO — INFO is INTEGER. = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. 🫶

👑 Author 👑

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

🦋 Further Details 🦋

Okay this is the iconic part where Q and P are represented as products of elementary reflectors. Iykyk. 💅

The matrices Q and P are represented as products of elementary
reflectors:

If m >= n,

     Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)

Each H(i) and G(i) has the form:

     H(i) = I - tauq * v * v**H  and G(i) = I - taup * u * u**H

where tauq and taup are complex scalars, and v and u are complex
vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in
A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in
A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

If m < n,

     Q = H(1) H(2) . . . H(m-1)  and  P = G(1) G(2) . . . G(m)

Each H(i) and G(i) has the form:

     H(i) = I - tauq * v * v**H  and G(i) = I - taup * u * u**H

where tauq and taup are complex scalars, and v and u are complex
vectors; v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in
A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in
A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

The contents of A on exit are illustrated by the following examples:

m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):

    (  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )
    (  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )
    (  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )
    (  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )
    (  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )
    (  v1  v2  v3  v4  v5 )

where d and e denote diagonal and off-diagonal elements of B, vi
denotes an element of the vector defining H(i), and ui an element of
the vector defining G(i).

Definition at line 205 of file cgebrd.f. 📍

👑 Author 👑

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