SRC/cgebd2.f(3)

okay so this is SRC/cgebd2.f(3) from the Library Functions Manual — yes, the manual name is folded in right here because the header was doing double duty, and honestly? iconic. 💅

✨ NAME ✨

SRC/cgebd2.f — that's the file, bestie. The whole page is about this one. 🎀

🌸 SYNOPSIS 🌸

🧩 Functions/Subroutines 🧩

subroutine cgebd2 (m, n, a, lda, d, e, tauq, taup, work, info)

CGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm. No cap, that's the summary. ✨

🫶 Function/Subroutine Documentation 🫶

subroutine cgebd2 (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 info)

CGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.

🎯 Purpose 🎯

Okay so the purpose is a whole glow-up: CGEBD2 reduces a complex general m by n matrix A to upper or lower real 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 it, that's the tea. 🍵

🧚 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(M,N)).
• 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 🔥

Bestie, here's the lore. 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, 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 189 of file cgebd2.f.

💀 Author (auto-generated) 💀

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