SGEBD2(3)

Okay so this is SGEBD2(3), from the Library Functions Manual — yes, that manual — and she’s here to reduce a general matrix to bidiagonal form using an unblocked algorithm. The source vibes are immaculate. 💅

🌸 NAME 🌸

Here’s the file path this page is about, exactly as given:

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/sgebd2.f

Periodt. No renaming, no shortcuts. 💖

✨ SYNOPSIS ✨

Under Functions/Subroutines, we have this icon:

subroutine SGEBD2 (m, n, a, lda, d, e, tauq, taup, work, info)

SGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm. Slay.

💅 Function/Subroutine Documentation 💅

Here’s the full signature, bestie:

subroutine SGEBD2 (integer m, integer n, real, dimension( lda, * ) a,
       integer lda, real, dimension( * ) d, real, dimension( * ) e, real,
       dimension( * ) tauq, real, dimension( * ) taup, real, dimension( * )
       work, integer info)

SGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.

🎀 Purpose 🎀

Okay, the glow-up: SGEBD2 reduces a real general m by n matrix A to upper or lower bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.

If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.

🧚 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 REAL 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 orthogonal matrix Q as a product of elementary reflectors, and the elements above the first superdiagonal, with the array TAUP, represent the orthogonal 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 orthogonal matrix Q as a product of elementary reflectors, and the elements above the diagonal, with the array TAUP, represent the orthogonal 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 REAL array, dimension (min(M,N)). The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. See Further Details.
• TAUP — TAUP is REAL array, dimension (min(M,N)). The scalar factors of the elementary reflectors which represent the orthogonal matrix P. See Further Details.
• WORK — WORK is REAL 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 🦋

The matrices Q and P are represented as products of elementary reflectors, and this is where it gets iconic:

💖 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**T  and G(i) = I - taup * u * u**T

where tauq and taup are real scalars, and v and u are real 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**T  and G(i) = I - taup * u * u**T

where tauq and taup are real scalars, and v and u are real 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):

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

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

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

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 188 of file sgebd2.f.

⭐ Author ⭐

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That’s the whole page, bestie — LAPACK Version 3.12.0, source page /home/abui...RC/sgebd2.f(3). Go forth and reduce something. 💖✨