sgebrd.f(3)

NAME
sgebrd.f -

SYNOPSIS

  Functions/Subroutines
subroutine sgebrd (M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, INFO)
SGEBRD

Function/Subroutine Documentation

  subroutine sgebrd (integerM, integerN, real, dimension( lda, * )A, integerLDA, real, dimension( * )D, real, dimension( * )E, real, dimension( * )TAUQ, real, dimension( * )TAUP, real, dimension( * )WORK, integerLWORK, integerINFO)

SGEBRD

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.

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(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The length of the array WORK.  LWORK >= max(1,M,N).
              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.

Date:
    November 2011

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).

Definition at line 205 of file sgebrd.f.

Author

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