# clabrd.f

Section: LAPACK (3)
Updated: Tue Nov 14 2017
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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 (integer M, integer N, integer NB, complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) D, real, dimension( * ) E, complex, dimension( * ) TAUQ, complex, dimension( * ) TAUP, complex, dimension( ldx, * ) X, integer LDX, complex, dimension( ldy, * ) Y, integer LDY)

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

NAG Ltd.

Date:

June 2017

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 214 of file clabrd.f.

## Author

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