SLABRD 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 SGEBRD

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 REAL 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.

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 REAL array dimension (NB)


          The scalar factors of the elementary reflectors which


          represent the orthogonal matrix Q. See Further Details.

TAUP

          TAUP is REAL array, dimension (NB)


          The scalar factors of the elementary reflectors which


          represent the orthogonal matrix P. See Further Details.

X

          X is REAL 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 REAL 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**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).