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# slabrd

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NAME
SLABRD - reduce the first NB rows and columns of a real gen-
eral m by n matrix A to upper or lower bidiagonal form by an
orthogonal transformation Q' * A * P, and returns the
matrices X and Y which are needed to apply the transforma-
tion to the unreduced part of A

SYNOPSIS
SUBROUTINE SLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X,
LDX, Y, LDY )

INTEGER        LDA, LDX, LDY, M, N, NB

REAL           A( LDA, * ), D( * ), E( * ), TAUP( * ),
TAUQ( * ), X( LDX, * ), Y( LDY, * )

PURPOSE
SLABRD reduces the first NB rows and columns of a real gen-
eral m by n matrix A to upper or lower bidiagonal form by an
orthogonal transformation Q' * A * P, and returns the
matrices X and Y which are needed to apply the transforma-
tion 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

ARGUMENTS
M       (input) INTEGER
The number of rows in the matrix A.

N       (input) INTEGER
The number of columns in the matrix A.

NB      (input) INTEGER
The number of leading rows and columns of A to be
reduced.

A       (input/output) 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
(input) INTEGER The leading dimension of the array
A.  LDA >= max(1,M).

D       (output) REAL array, dimension (NB)
The diagonal elements of the first NB rows and
columns of the reduced matrix.  D(i) = A(i,i).

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

TAUQ    (output) REAL array dimension (NB)
The scalar factors of the elementary reflectors
which represent the orthogonal matrix Q. See Further
Details.  TAUP    (output) REAL array, dimension
(NB) The scalar factors of the elementary reflectors
which represent the orthogonal matrix P. See Further
Details.  X       (output) REAL array, dimension
(LDX,NB) The m-by-nb matrix X required to update the
unreduced part of A.

LDX     (input) INTEGER
The leading dimension of the array X. LDX >= M.

Y       (output) REAL array, dimension (LDY,NB)
The n-by-nb matrix Y required to update the unre-
duced part of A.

LDY     (output) INTEGER
The leading dimension of the array Y. LDY >= N.

FURTHER DETAILS
The matrices Q and P are represented as products of elemen-
tary 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'  and G(i) = I - taup * u * u'

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' 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' - X*U'.

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).
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