Contents

# ?sytrf

Computes the Bunch-Kaufman factorization of a symmetric matrix.

## Syntax

Include Files
• mkl.h
Description
The routine computes the factorization of a real/complex symmetric matrix
A
using the Bunch-Kaufman diagonal pivoting method. The form of the factorization is:
• if
uplo
=
'U'
,
A
=
U*D*U
T
• if
uplo
=
'L'
,
A
=
L*D*L
T
where
A
is the input matrix,
U
and
L
are products of permutation and triangular matrices with unit diagonal (upper triangular for
U
and lower triangular for
L
), and
D
is a symmetric block-diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks.
U
and
L
have 2-by-2 unit diagonal blocks corresponding to the 2-by-2 blocks of
D
.
This routine supports the Progress Routine feature. See Progress Routine for details.
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
uplo
Must be
'U'
or
'L'
.
Indicates whether the upper or lower triangular part of
A
is stored and how
A
is factored:
If
uplo
=
'U'
, the array
a
stores the upper triangular part of the matrix
A
, and
A
is factored as
U*D*U
T
.
If
uplo
=
'L'
, the array
a
stores the lower triangular part of the matrix
A
, and
A
is factored as
L*D*L
T
.
n
The order of matrix
A
;
n
0.
a
Array, size
max(1,
lda
*
n
)
. The array
a
contains either the upper or the lower triangular part of the matrix
A
(see
uplo
).
lda
The leading dimension of
a
; at least
max(1,
n
)
.
Output Parameters
a
The upper or lower triangular part of
a
is overwritten by details of the block-diagonal matrix
D
and the multipliers used to obtain the factor
U
(or
L
).
ipiv
Array, size at least
max(1,
n
)
. Contains details of the interchanges and the block structure of
D
. If
ipiv
[
i
-1] =
k
>0
, then
d
i
i
is a 1-by-1 block, and the
i
-th row and column of
A
was interchanged with the
k
-th row and column.
If
uplo
=
'U'
and
ipiv
[
i
] =
ipiv
[
i
-1] = -
m
< 0, then
D
has a 2-by-2 block in rows/columns
i
and
i
+1
, and
i
-th row and column of
A
was interchanged with the
m
-th row and column.
If
uplo
=
'L'
and
ipiv
[
i
] =
ipiv
[
i
-1] = -
m
< 0, then
D
has a 2-by-2 block in rows/columns
i
and
i
+1, and (
i
+1)-th row and column of
A
was interchanged with the
m
-th row and column.
Return Values
This function returns a value
info
.
If
info
= 0
, the execution is successful.
If
info
=
-i
, parameter
i
had an illegal value.
If
info
=
i
,
D
i
i
is 0. The factorization has been completed, but
D
is exactly singular. Division by 0 will occur if you use
D
for solving a system of linear equations.
Application Notes
The 2-by-2 unit diagonal blocks and the unit diagonal elements of
U
and
L
are not stored. The remaining elements of
U
and
L
are stored in the corresponding columns of the array
a
, but additional row interchanges are required to recover
U
or
L
explicitly (which is seldom necessary).
If
ipiv
[
i
-1] =
i
for all
i
=1...
n
, then all off-diagonal elements of
U
(
L
) are stored explicitly in the corresponding elements of the array
a
.
If
uplo
=
'U'
, the computed factors
U
and
D
are the exact factors of a perturbed matrix
A
+
E
, where
```|E| ≤
c(n)ε
P|U||D||U

T
|P

T

```
c
(
n
)
is a modest linear function of
n
, and
ε
is the machine precision. A similar estimate holds for the computed
L
and
D
when
uplo
=
'L'
.
The total number of floating-point operations is approximately
(1/3)
n
3
for real flavors or
(4/3)
n
3
for complex flavors.
After calling this routine, you can call the following routines:
to solve
A*X
=
B
to estimate the condition number of
A
to compute the inverse of
A
.

If
uplo
=
'U'
, then
A
=
U
*
D
*
U'
, where
```
U = P(n)*U(n)* ... *P(k)*U(k)*...,
```
that is,
U
is a product of terms
P
(
k
)*
U
(
k
), where
• k
decreases from
n
to 1 in steps of 1 and 2.
• 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
-1]
.
• 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
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)*...,
```
that is,
L
is a product of terms
P
(
k
)*
L
(
k
), where
• k
increases from 1 to
n
in steps of 1 and 2.
• 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
).
• 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
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).

#### Product and Performance Information

1

Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex.