Contents

# ?sytrf_rook

Computes the bounded 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 bounded Bunch-Kaufman ("rook") 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
.
Input Parameters
matrix_layout
Specifies whether matrix storage layout for array
b
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
lda
*
n
. The array
a
contains either the upper or the lower triangular part of the matrix
A
(see
uplo
).
lda
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
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
) < 0
and
ipiv
(
k
- 1) < 0
, then rows and columns
k
and -
ipiv
(
k
) were interchanged, rows and columns
k
- 1 and -
ipiv
(
k
- 1) were interchanged, and
D
k
-1:
k
,
k
-1:
k
is a 2-by-2 diagonal block.
If
uplo
=
'L'
and
ipiv
(
k
) < 0
and
ipiv
(
k
+ 1) < 0
, then rows and columns
k
and
-ipiv
(
k
) were interchanged, rows and columns
k
+ 1 and
-ipiv
(
k
+ 1) were interchanged, and
D
k
:
k
+1,
k
:
k
+1
is a 2-by-2 diagonal block.
Return Values
This function returns a value
info
.
If
info
= 0
, the execution is successful.
If
info
=
-i
, the
i-
th parameter 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 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
?sycon_rook
(Fortran only)
to estimate the condition number of
A
?sytri_rook
(Fortran only)
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.