## Developer Reference

• 0.9
• 09/09/2020
• Public Content
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

Intel's compilers may or may not optimize to the same degree for non-Intel microprocessors for optimizations that are not unique to Intel microprocessors. These optimizations include SSE2, SSE3, and SSSE3 instruction sets and other optimizations. Intel does not guarantee the availability, functionality, or effectiveness of any optimization on microprocessors not manufactured by Intel. Microprocessor-dependent optimizations in this product are intended for use with Intel microprocessors. Certain optimizations not specific to Intel microarchitecture are reserved for Intel microprocessors. Please refer to the applicable product User and Reference Guides for more information regarding the specific instruction sets covered by this notice.

Notice revision #20110804