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man pages->IRIX man pages -> complib/sgebrd (3)
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### Contents

```
SGEBRD(3F)							    SGEBRD(3F)

```

### NAME[Toc][Back]

```     SGEBRD - reduce a general real M-by-N matrix A to upper or	lower
bidiagonal	form B by an orthogonal	transformation
```

### SYNOPSIS[Toc][Back]

```     SUBROUTINE	SGEBRD(	M, N, A, LDA, D, E, TAUQ, TAUP,	WORK, LWORK, INFO )

INTEGER	INFO, LDA, LWORK, M, N

REAL		A( LDA,	* ), D(	* ), E(	* ), TAUP( * ),	TAUQ( *	),
WORK( LWORK )
```

### PURPOSE[Toc][Back]

```     SGEBRD reduces a general real M-by-N matrix A to upper or lower
bidiagonal	form B by an orthogonal	transformation:	Q**T * A * P = B.

If	m >= n,	B is upper bidiagonal; if m < n, B is lower bidiagonal.

```

### ARGUMENTS[Toc][Back]

```     M	     (input) INTEGER
The number	of rows	in the matrix A.  M >= 0.

N	     (input) INTEGER
The number	of columns in the matrix A.  N >= 0.

A	     (input/output) REAL array,	dimension (LDA,N)
On	entry, the M-by-N general matrix to be reduced.	 On exit, if m
>=	n, the diagonal	and the	first superdiagonal are	overwritten
with the upper bidiagonal matrix B; the elements below the
diagonal, with the	array TAUQ, represent the orthogonal matrix Q
as	a product of elementary	reflectors, and	the elements above the
first superdiagonal, with the array TAUP, represent the
orthogonal	matrix P as a product of elementary reflectors;	if m <
n,	the diagonal and the first subdiagonal are overwritten with
the lower bidiagonal matrix B; the	elements below the first
subdiagonal, with the array TAUQ, represent the orthogonal	matrix
Q as a product of elementary reflectors, and the elements above
the diagonal, 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 (min(M,N))
The diagonal elements of the bidiagonal matrix B:	D(i) = A(i,i).

E	     (output) REAL array, dimension (min(M,N)-1)
The off-diagonal elements of the bidiagonal matrix	B:  if m >= n,
E(i) = A(i,i+1) for i = 1,2,...,n-1; if m < n, E(i) = A(i+1,i)
for i = 1,2,...,m-1.

Page 1

SGEBRD(3F)							    SGEBRD(3F)

TAUQ    (output) REAL array dimension (min(M,N))
The scalar	factors	of the elementary reflectors which represent
the orthogonal matrix Q. See Further Details.  TAUP    (output)
REAL array, dimension (min(M,N)) The scalar factors of the
elementary	reflectors which represent the orthogonal matrix P.
See Further Details.  WORK	   (workspace/output) REAL array,
dimension (LWORK) On exit,	if INFO	= 0, WORK(1) returns the
optimal LWORK.

LWORK   (input) INTEGER
The length	of the array WORK.  LWORK >= max(1,M,N).  For optimum
performance LWORK >= (M+N)*NB, where NB is	the optimal blocksize.

INFO    (output) INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value.

FURTHER	DETAILS
The matrices Q and	P are represented as products of elementary
reflectors:

If	m >= n,

Q = H(1) H(2) .	. . H(n)  and  P = G(1)	G(2) . . . G(n-1)

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;
v(1:i-1) =	0, v(i)	= 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
u(1:i) = 0, u(i+1)	= 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
tauq is stored in TAUQ(i) and taup	in TAUP(i).

If	m < n,

Q = H(1) H(2) .	. . H(m-1)  and	 P = G(1) G(2) . . . G(m)

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;
v(1:i) = 0, v(i+1)	= 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
u(1:i-1) =	0, u(i)	= 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).

The contents of A on exit are illustrated by the following	examples:

m = 6 and n = 5 (m	> n):	       m = 5 and n = 6 (m < n):

(  d   e	  u1  u1  u1 )		 (  d	u1  u1	u1  u1	u1 )

Page 2

SGEBRD(3F)							    SGEBRD(3F)

(  v1  d	  e   u2  u2 )		 (  e	d   u2	u2  u2	u2 )
(  v1  v2  d   e	  u3 )		 (  v1	e   d	u3  u3	u3 )
(  v1  v2  v3  d	  e  )		 (  v1	v2  e	d   u4	u4 )
(  v1  v2  v3  v4  d  )		 (  v1	v2  v3	e   d	u5 )
(  v1  v2  v3  v4  v5 )

where d and e denote diagonal and off-diagonal elements of	B, vi denotes
an	element	of the vector defining H(i), and ui an element of the vector
defining G(i).
SGEBRD(3F)							    SGEBRD(3F)

```

### NAME[Toc][Back]

```     SGEBRD - reduce a general real M-by-N matrix A to upper or	lower
bidiagonal	form B by an orthogonal	transformation
```

### SYNOPSIS[Toc][Back]

```     SUBROUTINE	SGEBRD(	M, N, A, LDA, D, E, TAUQ, TAUP,	WORK, LWORK, INFO )

INTEGER	INFO, LDA, LWORK, M, N

REAL		A( LDA,	* ), D(	* ), E(	* ), TAUP( * ),	TAUQ( *	),
WORK( LWORK )
```

### PURPOSE[Toc][Back]

```     SGEBRD reduces a general real M-by-N matrix A to upper or lower
bidiagonal	form B by an orthogonal	transformation:	Q**T * A * P = B.

If	m >= n,	B is upper bidiagonal; if m < n, B is lower bidiagonal.

```

### ARGUMENTS[Toc][Back]

```     M	     (input) INTEGER
The number	of rows	in the matrix A.  M >= 0.

N	     (input) INTEGER
The number	of columns in the matrix A.  N >= 0.

A	     (input/output) REAL array,	dimension (LDA,N)
On	entry, the M-by-N general matrix to be reduced.	 On exit, if m
>=	n, the diagonal	and the	first superdiagonal are	overwritten
with the upper bidiagonal matrix B; the elements below the
diagonal, with the	array TAUQ, represent the orthogonal matrix Q
as	a product of elementary	reflectors, and	the elements above the
first superdiagonal, with the array TAUP, represent the
orthogonal	matrix P as a product of elementary reflectors;	if m <
n,	the diagonal and the first subdiagonal are overwritten with
the lower bidiagonal matrix B; the	elements below the first
subdiagonal, with the array TAUQ, represent the orthogonal	matrix
Q as a product of elementary reflectors, and the elements above
the diagonal, 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 (min(M,N))
The diagonal elements of the bidiagonal matrix B:	D(i) = A(i,i).

E	     (output) REAL array, dimension (min(M,N)-1)
The off-diagonal elements of the bidiagonal matrix	B:  if m >= n,
E(i) = A(i,i+1) for i = 1,2,...,n-1; if m < n, E(i) = A(i+1,i)
for i = 1,2,...,m-1.

Page 1

SGEBRD(3F)							    SGEBRD(3F)

TAUQ    (output) REAL array dimension (min(M,N))
The scalar	factors	of the elementary reflectors which represent
the orthogonal matrix Q. See Further Details.  TAUP    (output)
REAL array, dimension (min(M,N)) The scalar factors of the
elementary	reflectors which represent the orthogonal matrix P.
See Further Details.  WORK	   (workspace/output) REAL array,
dimension (LWORK) On exit,	if INFO	= 0, WORK(1) returns the
optimal LWORK.

LWORK   (input) INTEGER
The length	of the array WORK.  LWORK >= max(1,M,N).  For optimum
performance LWORK >= (M+N)*NB, where NB is	the optimal blocksize.

INFO    (output) INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value.

FURTHER	DETAILS
The matrices Q and	P are represented as products of elementary
reflectors:

If	m >= n,

Q = H(1) H(2) .	. . H(n)  and  P = G(1)	G(2) . . . G(n-1)

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;
v(1:i-1) =	0, v(i)	= 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
u(1:i) = 0, u(i+1)	= 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
tauq is stored in TAUQ(i) and taup	in TAUP(i).

If	m < n,

Q = H(1) H(2) .	. . H(m-1)  and	 P = G(1) G(2) . . . G(m)

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;
v(1:i) = 0, v(i+1)	= 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
u(1:i-1) =	0, u(i)	= 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).

The contents of A on exit are illustrated by the following	examples:

m = 6 and n = 5 (m	> n):	       m = 5 and n = 6 (m < n):

(  d   e	  u1  u1  u1 )		 (  d	u1  u1	u1  u1	u1 )

Page 2

SGEBRD(3F)							    SGEBRD(3F)

(  v1  d	  e   u2  u2 )		 (  e	d   u2	u2  u2	u2 )
(  v1  v2  d   e	  u3 )		 (  v1	e   d	u3  u3	u3 )
(  v1  v2  v3  d	  e  )		 (  v1	v2  e	d   u4	u4 )
(  v1  v2  v3  v4  d  )		 (  v1	v2  v3	e   d	u5 )
(  v1  v2  v3  v4  v5 )

where d and e denote diagonal and off-diagonal elements of	B, vi denotes
an	element	of the vector defining H(i), and ui an element of the vector
defining G(i).

PPPPaaaaggggeeee 3333```
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