Loading...
Searching...
No Matches

## dlabrd()

subroutine dlabrd | ( | integer | m,
|  | integer | n,
|  | integer | nb,
|  | 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( ldx, * ) | x,
|  | integer | ldx,
|  | double precision, dimension( ldy, * ) | y,
|  | integer | ldy )

DLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.

Download DLABRD + dependencies [TGZ] [ZIP] [TXT]

Purpose:

      !>
      !> DLABRD reduces the first NB rows and columns of a real general
      !> m by n matrix A to upper or lower bidiagonal form by an orthogonal
      !> transformation Q**T * A * P, and returns the matrices X and Y which
      !> are needed to apply the transformation to the unreduced part of A.
      !>
      !> If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
      !> bidiagonal form.
      !>
      !> This is an auxiliary routine called by DGEBRD
      !>

Parameters

  [in]

  M

      !>          M is INTEGER
      !>          The number of rows in the matrix A.
      !>

  [in]

  N

      !>          N is INTEGER
      !>          The number of columns in the matrix A.
      !>

  [in]

  NB

      !>          NB is INTEGER
      !>          The number of leading rows and columns of A to be reduced.
      !>

  [in,out]

  A

      !>          A is DOUBLE PRECISION array, dimension (LDA,N)
      !>          On entry, the m by n general matrix to be reduced.
      !>          On exit, the first NB rows and columns of the matrix are
      !>          overwritten; the rest of the array is unchanged.
      !>          If m >= n, elements on and below the diagonal in the first NB
      !>            columns, with the array TAUQ, represent the orthogonal
      !>            matrix Q as a product of elementary reflectors; and
      !>            elements above the diagonal in the first NB rows, with the
      !>            array TAUP, represent the orthogonal matrix P as a product
      !>            of elementary reflectors.
      !>          If m < n, elements below the diagonal in the first NB
      !>            columns, with the array TAUQ, represent the orthogonal
      !>            matrix Q as a product of elementary reflectors, and
      !>            elements on and above the diagonal in the first NB rows,
      !>            with the array TAUP, represent the orthogonal matrix P as
      !>            a product of elementary reflectors.
      !>          See Further Details.
      !>

  [in]

  LDA

      !>          LDA is INTEGER
      !>          The leading dimension of the array A.  LDA >= max(1,M).
      !>

  [out]

  D

      !>          D is DOUBLE PRECISION array, dimension (NB)
      !>          The diagonal elements of the first NB rows and columns of
      !>          the reduced matrix.  D(i) = A(i,i).
      !>

  [out]

  E

      !>          E is DOUBLE PRECISION array, dimension (NB)
      !>          The off-diagonal elements of the first NB rows and columns of
      !>          the reduced matrix.
      !>

  [out]

  TAUQ

      !>          TAUQ is DOUBLE PRECISION array, dimension (NB)
      !>          The scalar factors of the elementary reflectors which
      !>          represent the orthogonal matrix Q. See Further Details.
      !>

  [out]

  TAUP

      !>          TAUP is DOUBLE PRECISION array, dimension (NB)
      !>          The scalar factors of the elementary reflectors which
      !>          represent the orthogonal matrix P. See Further Details.
      !>

  [out]

  X

      !>          X is DOUBLE PRECISION array, dimension (LDX,NB)
      !>          The m-by-nb matrix X required to update the unreduced part
      !>          of A.
      !>

  [in]

  LDX

      !>          LDX is INTEGER
      !>          The leading dimension of the array X. LDX >= max(1,M).
      !>

  [out]

  Y

      !>          Y is DOUBLE PRECISION array, dimension (LDY,NB)
      !>          The n-by-nb matrix Y required to update the unreduced part
      !>          of A.
      !>

  [in]

  LDY

      !>          LDY is INTEGER
      !>          The leading dimension of the array Y. LDY >= max(1,N).
      !>

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:
      !>
      !>     Q = H(1) H(2) . . . H(nb)  and  P = G(1) G(2) . . . G(nb)
      !>
      !>  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.
      !>
      !>  If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
      !>  A(i:m,i); u(1:i) = 0, u(i+1) = 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).
      !>
      !>  If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
      !>  A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
      !>  A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
      !>
      !>  The elements of the vectors v and u together form the m-by-nb matrix
      !>  V and the nb-by-n matrix U**T which are needed, with X and Y, to apply
      !>  the transformation to the unreduced part of the matrix, using a block
      !>  update of the form:  A := A - V*Y**T - X*U**T.
      !>
      !>  The contents of A on exit are illustrated by the following examples
      !>  with nb = 2:
      !>
      !>  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):
      !>
      !>    (  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )
      !>    (  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )
      !>    (  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )
      !>    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
      !>    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
      !>    (  v1  v2  a   a   a  )
      !>
      !>  where a denotes an element of the original matrix which is unchanged,
      !>  vi denotes an element of the vector defining H(i), and ui an element
      !>  of the vector defining G(i).
      !>