ZLABRD 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 ZGEBRD

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*16 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (NB)


          The off-diagonal elements of the first NB rows and columns of


          the reduced matrix.

TAUQ

          TAUQ is COMPLEX*16 array dimension (NB)


          The scalar factors of the elementary reflectors which


          represent the unitary matrix Q. See Further Details.

TAUP

          TAUP is COMPLEX*16 array, dimension (NB)


          The scalar factors of the elementary reflectors which


          represent the unitary matrix P. See Further Details.

X

          X is COMPLEX*16 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*16 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).

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

September 2012

  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).