SRC/dgebrd.f(3)

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version 3.12.1 · 7 versions
, in arch rolling

NAME

okay so this is the SRC/dgebrd.f(3) page from the Library Functions Manual, and it's literally all about SRC/dgebrd.f — the Fortran source file behind the dgebrd subroutine. she's the one who bidiagonalizes, and she's iconic for it. 💅

SYNOPSIS

Functions/Subroutines

       subroutine dgebrd (m, n, a, lda, d, e, tauq, taup, work, lwork, info)
           DGEBRD

Function/Subroutine Documentation

here's the full signature, bestie — look at her go:

   subroutine dgebrd (integer m, integer n, double precision, dimension( lda, *
       ) a, integer lda, double precision, dimension( * ) d, double precision,
       dimension( * ) e, double precision, dimension( * ) tauq, double
       precision, dimension( * ) taup, double precision, dimension( * ) work,
       integer lwork, integer info)
       DGEBRD

Purpose:

okay so the tea is: DGEBRD 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. slay. 💖

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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (min(M,N)). the diagonal elements of the bidiagonal matrix B: D(i) = A(i,i).

E

E is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (min(M,N)). the scalar factors of the elementary reflectors which represent the orthogonal matrix Q. see Further Details.

TAUP

TAUP is DOUBLE PRECISION array, dimension (min(M,N)). the scalar factors of the elementary reflectors which represent the orthogonal matrix P. see Further Details.

WORK

WORK is DOUBLE PRECISION 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.

bestie, the LWORK = -1 thing is the "just tell me the workspace size, don't actually compute" vibe. no error message related to LWORK from XERBLA. periodt.

INFO

INFO is INTEGER

  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value.

Author

the absolute legends behind this:

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

Further Details:

okay this is the part where the math girlies shine. 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).

Definition at line 204 of file dgebrd.f.

Author

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