SRC/sgebrd.f(3)

📦
version 3.12.1 · 7 versions
, in arch rolling

💅 NAME

SRC/sgebrd.f — okay so this is the manpage for one single function living in that file, bestie, and it's literally called sgebrd. That's it, that's the name section. Short and iconic. 🎀

✨ SYNOPSIS

🦋 Functions/Subroutines

  • subroutine sgebrd (m, n, a, lda, d, e, tauq, taup, work, lwork, info) — the whole squad, right there. This is SGEBRD, say hi 👋

🌸 Function/Subroutine Documentation

💖 The full signature (no cap, every single arg)

subroutine sgebrd (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 lwork, integer info)

Bestie. Look at that. SGEBRD. Eleven arguments and not one of them is optional. Obsessed. 💜

🎀 Purpose

SGEBRD reduces a general real 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. That's the whole vibe check — the shape of A decides which flavor of bidiagonal you're getting. ✨

🌈 Parameters

🍓 M

M is INTEGER. It's the number of rows in the matrix A. Constraint: M >= 0. (Zero rows? Legal. Weird, but legal. 🙈)

🍓 N

N is INTEGER. Number of columns in the matrix A. Constraint: N >= 0.

🍓 A

A is a REAL array, dimension (LDA,N).

On entry: the M-by-N general matrix that's about to get reduced. On exit, it gets rewritten, and which parts get overwritten depends entirely on the shape, so pay attention:

  • if m >= n, the diagonal and the first superdiagonal are overwritten with the upper bidiagonal matrix B; the elements below the diagonal, together with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors, and the elements above the first superdiagonal, together 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, together with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors, and the elements above the diagonal, together with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors.

See Further Details. 👀 (spoiler: there's a whole diagram down there and it ate.)

🍓 LDA

LDA is INTEGER. It's the leading dimension of the array A. Constraint: LDA >= max(1,M). Respect it. 🫶

🍓 D

D is a REAL array, dimension (min(M,N)). These are the diagonal elements of the bidiagonal matrix B: D(i) = A(i,i). Main character energy, literally the diagonal. 👑

🍓 E

E is a REAL array, dimension (min(M,N)-1). These are 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 a 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 a 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 a REAL array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. Cute little return value, we love a self-reporting workspace. ✨

🍓 LWORK

LWORK is INTEGER. It's the length of the array WORK.

Constraint time, bestie: 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.

🚨 And here's the spicy one: 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. So don't be surprised when it stays quiet. That's on purpose. 💀

🍓 INFO

INFO is INTEGER.

  • = 0: successful exit. slay. 🎉
  • < 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.

A whole committee of legends, honestly. 🫶

🔥 Further Details:

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**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):          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).

Like, that diagram? art. Frame it. 🖼️✨

🦋 Definition at line 204 of file sgebrd.f.

Yes bestie, line 204. Go look, it's there. 💅

💜 Author (the meta one)

Generated automatically by Doxygen for LAPACK from the source code. Not a human, just a robot doing its lil thing. 🤖✨

🎀 ok that's the whole page!! this was LAPACK, Version 3.12.0, and the file is SRC/sgebrd.f — mwah, go reduce something bidiagonal, ily, bye!! 💖🦋🌸
📦 where this page comes from