clabrd (3) - Linux Manuals

NAME

clabrd.f -

SYNOPSIS


Functions/Subroutines


subroutine clabrd (M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY)
CLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.

Function/Subroutine Documentation

subroutine clabrd (integerM, integerN, integerNB, complex, dimension( lda, * )A, integerLDA, real, dimension( * )D, real, dimension( * )E, complex, dimension( * )TAUQ, complex, dimension( * )TAUP, complex, dimension( ldx, * )X, integerLDX, complex, dimension( ldy, * )Y, integerLDY)

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

Purpose: 

 CLABRD reduces the first NB rows and columns of a complex general
 m by n matrix A to upper or lower real bidiagonal form by a unitary
 transformation Q**H * 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 CGEBRD


 

Parameters:

M 

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


N 

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


NB 

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


A 

          A is COMPLEX 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 unitary
            matrix Q as a product of elementary reflectors; and
            elements above the diagonal in the first NB rows, with the
            array TAUP, represent the unitary 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 unitary
            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 unitary 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 (NB)
          The diagonal elements of the first NB rows and columns of
          the reduced matrix.  D(i) = A(i,i).


E 

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


TAUQ 

          TAUQ is COMPLEX array dimension (NB)
          The scalar factors of the elementary reflectors which
          represent the unitary matrix Q. See Further Details.


TAUP 

          TAUP is COMPLEX array, dimension (NB)
          The scalar factors of the elementary reflectors which
          represent the unitary matrix P. See Further Details.


X 

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


LDX 

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


Y 

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


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.

Date:

September 2012

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**H  and G(i) = I - taup * u * u**H

  where tauq and taup are complex scalars, and v and u are complex
  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**H 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**H - X*U**H.

  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).


 

Definition at line 212 of file clabrd.f.

Author

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