# slabrd.f - Man Page

SRC/slabrd.f

## Synopsis

### Functions/Subroutines

subroutine slabrd (m, n, nb, a, lda, d, e, tauq, taup, x, ldx, y, ldy)
SLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.

## Function/Subroutine Documentation

### subroutine slabrd (integer m, integer n, integer nb, real, dimension( lda, * ) a, integer lda, real, dimension( * ) d, real, dimension( * ) e, real, dimension( * ) tauq, real, dimension( * ) taup, real, dimension( ldx, * ) x, integer ldx, real, dimension( ldy, * ) y, integer ldy)

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

Purpose:

``` 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```
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 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).```
Author

Univ. of Tennessee

Univ. of California Berkeley

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).```

Definition at line 208 of file slabrd.f.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man page slabrd(3) is an alias of slabrd.f(3).

Tue Nov 28 2023 12:08:42 Version 3.12.0 LAPACK