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

```
DSYTRF(3F)							    DSYTRF(3F)

```

### NAME[Toc][Back]

```     DSYTRF - compute the factorization	of a real symmetric matrix A using the
Bunch-Kaufman diagonal pivoting method
```

### SYNOPSIS[Toc][Back]

```     SUBROUTINE	DSYTRF(	UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )

CHARACTER	UPLO

INTEGER	INFO, LDA, LWORK, N

INTEGER	IPIV( *	)

DOUBLE		PRECISION A( LDA, * ), WORK( LWORK )
```

### PURPOSE[Toc][Back]

```     DSYTRF computes the factorization of a real symmetric matrix A using the
Bunch-Kaufman diagonal pivoting method.  The form of the factorization is

A = U*D*U**T  or  A = L*D*L**T

where U (or L) is a product of permutation	and unit upper (lower)
triangular	matrices, and D	is symmetric and block diagonal	with 1-by-1
and 2-by-2	diagonal blocks.

This is the blocked version of the	algorithm, calling Level 3 BLAS.

```

### ARGUMENTS[Toc][Back]

```     UPLO    (input) CHARACTER*1
= 'U':  Upper triangle of A is stored;
= 'L':  Lower triangle of A is stored.

N	     (input) INTEGER
The order of the matrix A.	 N >= 0.

A	     (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On	entry, the symmetric matrix A.	If UPLO	= 'U', the leading Nby-N
upper	triangular part	of A contains the upper	triangular
part of the matrix	A, and the strictly lower triangular part of A
is	not referenced.	 If UPLO = 'L',	the leading N-by-N lower
triangular	part of	A contains the lower triangular	part of	the
matrix A, and the strictly	upper triangular part of A is not
referenced.

On	exit, the block	diagonal matrix	D and the multipliers used to
obtain the	factor U or L (see below for further details).

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

Page 1

DSYTRF(3F)							    DSYTRF(3F)

IPIV    (output) INTEGER array, dimension (N)
Details of	the interchanges and the block structure of D.	If
IPIV(k) > 0, then rows and	columns	k and IPIV(k) were
interchanged and D(k,k) is	a 1-by-1 diagonal block.  If UPLO =
'U' and IPIV(k) = IPIV(k-1) < 0, then rows	and columns k-1	and
-IPIV(k) were interchanged	and D(k-1:k,k-1:k) is a	2-by-2
diagonal block.  If UPLO =	'L' and	IPIV(k)	= IPIV(k+1) < 0, then
rows and columns k+1 and -IPIV(k) were interchanged and
D(k:k+1,k:k+1) is a 2-by-2	diagonal block.

WORK    (workspace/output)	DOUBLE PRECISION array,	dimension (LWORK)
On	exit, if INFO =	0, WORK(1) returns the optimal LWORK.

LWORK   (input) INTEGER
The length	of WORK.  LWORK	>=1.  For best performance LWORK >=
N*NB, where NB is the block size returned by ILAENV.

INFO    (output) INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value
> 0:  if INFO = i,	D(i,i) is exactly zero.	 The factorization has
been completed, but the block diagonal matrix D is	exactly
singular, and division by zero will occur if it is	used to	solve
a system of equations.

FURTHER	DETAILS
If	UPLO = 'U', then A = U*D*U', where
U = P(n)*U(n)* ... *P(k)U(k)* ...,
i.e., U is	a product of terms P(k)*U(k), where k decreases	from n to 1 in
steps of 1	or 2, and D is a block diagonal	matrix with 1-by-1 and 2-by-2
diagonal blocks D(k).  P(k) is a permutation matrix as defined by
IPIV(k), and U(k) is a unit upper triangular matrix, such that if the
diagonal block D(k) is of order s (s = 1 or 2), then

(   I	 v    0	  )   k-s
U(k) =	(   0	 I    0	  )   s
(   0	 0    I	  )   n-k
k-s	 s   n-k

If	s = 1, D(k) overwrites A(k,k), and v overwrites	A(1:k-1,k).  If	s = 2,
the upper triangle	of D(k)	overwrites A(k-1,k-1), A(k-1,k), and A(k,k),
and v overwrites A(1:k-2,k-1:k).

If	UPLO = 'L', then A = L*D*L', where
L = P(1)*L(1)* ... *P(k)*L(k)* ...,
i.e., L is	a product of terms P(k)*L(k), where k increases	from 1 to n in
steps of 1	or 2, and D is a block diagonal	matrix with 1-by-1 and 2-by-2
diagonal blocks D(k).  P(k) is a permutation matrix as defined by
IPIV(k), and L(k) is a unit lower triangular matrix, such that if the
diagonal block D(k) is of order s (s = 1 or 2), then

(   I	 0     0   )  k-1

Page 2

DSYTRF(3F)							    DSYTRF(3F)

L(k) =	(   0	 I     0   )  s
(   0	 v     I   )  n-k-s+1
k-1	 s  n-k-s+1

If	s = 1, D(k) overwrites A(k,k), and v overwrites	A(k+1:n,k).  If	s = 2,
the lower triangle	of D(k)	overwrites A(k,k), A(k+1,k), and A(k+1,k+1),
and v overwrites A(k+2:n,k:k+1).
DSYTRF(3F)							    DSYTRF(3F)

```

### NAME[Toc][Back]

```     DSYTRF - compute the factorization	of a real symmetric matrix A using the
Bunch-Kaufman diagonal pivoting method
```

### SYNOPSIS[Toc][Back]

```     SUBROUTINE	DSYTRF(	UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )

CHARACTER	UPLO

INTEGER	INFO, LDA, LWORK, N

INTEGER	IPIV( *	)

DOUBLE		PRECISION A( LDA, * ), WORK( LWORK )
```

### PURPOSE[Toc][Back]

```     DSYTRF computes the factorization of a real symmetric matrix A using the
Bunch-Kaufman diagonal pivoting method.  The form of the factorization is

A = U*D*U**T  or  A = L*D*L**T

where U (or L) is a product of permutation	and unit upper (lower)
triangular	matrices, and D	is symmetric and block diagonal	with 1-by-1
and 2-by-2	diagonal blocks.

This is the blocked version of the	algorithm, calling Level 3 BLAS.

```

### ARGUMENTS[Toc][Back]

```     UPLO    (input) CHARACTER*1
= 'U':  Upper triangle of A is stored;
= 'L':  Lower triangle of A is stored.

N	     (input) INTEGER
The order of the matrix A.	 N >= 0.

A	     (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On	entry, the symmetric matrix A.	If UPLO	= 'U', the leading Nby-N
upper	triangular part	of A contains the upper	triangular
part of the matrix	A, and the strictly lower triangular part of A
is	not referenced.	 If UPLO = 'L',	the leading N-by-N lower
triangular	part of	A contains the lower triangular	part of	the
matrix A, and the strictly	upper triangular part of A is not
referenced.

On	exit, the block	diagonal matrix	D and the multipliers used to
obtain the	factor U or L (see below for further details).

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

Page 1

DSYTRF(3F)							    DSYTRF(3F)

IPIV    (output) INTEGER array, dimension (N)
Details of	the interchanges and the block structure of D.	If
IPIV(k) > 0, then rows and	columns	k and IPIV(k) were
interchanged and D(k,k) is	a 1-by-1 diagonal block.  If UPLO =
'U' and IPIV(k) = IPIV(k-1) < 0, then rows	and columns k-1	and
-IPIV(k) were interchanged	and D(k-1:k,k-1:k) is a	2-by-2
diagonal block.  If UPLO =	'L' and	IPIV(k)	= IPIV(k+1) < 0, then
rows and columns k+1 and -IPIV(k) were interchanged and
D(k:k+1,k:k+1) is a 2-by-2	diagonal block.

WORK    (workspace/output)	DOUBLE PRECISION array,	dimension (LWORK)
On	exit, if INFO =	0, WORK(1) returns the optimal LWORK.

LWORK   (input) INTEGER
The length	of WORK.  LWORK	>=1.  For best performance LWORK >=
N*NB, where NB is the block size returned by ILAENV.

INFO    (output) INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value
> 0:  if INFO = i,	D(i,i) is exactly zero.	 The factorization has
been completed, but the block diagonal matrix D is	exactly
singular, and division by zero will occur if it is	used to	solve
a system of equations.

FURTHER	DETAILS
If	UPLO = 'U', then A = U*D*U', where
U = P(n)*U(n)* ... *P(k)U(k)* ...,
i.e., U is	a product of terms P(k)*U(k), where k decreases	from n to 1 in
steps of 1	or 2, and D is a block diagonal	matrix with 1-by-1 and 2-by-2
diagonal blocks D(k).  P(k) is a permutation matrix as defined by
IPIV(k), and U(k) is a unit upper triangular matrix, such that if the
diagonal block D(k) is of order s (s = 1 or 2), then

(   I	 v    0	  )   k-s
U(k) =	(   0	 I    0	  )   s
(   0	 0    I	  )   n-k
k-s	 s   n-k

If	s = 1, D(k) overwrites A(k,k), and v overwrites	A(1:k-1,k).  If	s = 2,
the upper triangle	of D(k)	overwrites A(k-1,k-1), A(k-1,k), and A(k,k),
and v overwrites A(1:k-2,k-1:k).

If	UPLO = 'L', then A = L*D*L', where
L = P(1)*L(1)* ... *P(k)*L(k)* ...,
i.e., L is	a product of terms P(k)*L(k), where k increases	from 1 to n in
steps of 1	or 2, and D is a block diagonal	matrix with 1-by-1 and 2-by-2
diagonal blocks D(k).  P(k) is a permutation matrix as defined by
IPIV(k), and L(k) is a unit lower triangular matrix, such that if the
diagonal block D(k) is of order s (s = 1 or 2), then

(   I	 0     0   )  k-1

Page 2

DSYTRF(3F)							    DSYTRF(3F)

L(k) =	(   0	 I     0   )  s
(   0	 v     I   )  n-k-s+1
k-1	 s  n-k-s+1

If	s = 1, D(k) overwrites A(k,k), and v overwrites	A(k+1:n,k).  If	s = 2,
the lower triangle	of D(k)	overwrites A(k,k), A(k+1,k), and A(k+1,k+1),
and v overwrites A(k+2:n,k:k+1).

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