Contents

# ?hptrf

Computes the Bunch-Kaufman factorization of a complex Hermitian matrix using packed storage.

## Syntax

Include Files
• mkl.h
Description
The routine computes the factorization of a complex Hermitian packed matrix
A
using the Bunch-Kaufman diagonal pivoting method:
• if
uplo
=
'U'
,
A
=
U*D*U
H
• if
uplo
=
'L'
,
A
=
L*D*L
H
,
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 Hermitian 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 Function 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 packed and how
A
is factored:
If
uplo
=
'U'
, the array
ap
stores the upper triangular part of the matrix
A
, and
A
is factored as
U*D*U
H
.
If
uplo
=
'L'
, the array
ap
stores the lower triangular part of the matrix
A
, and
A
is factored as
L*D*L
H
.
n
The order of matrix
A
;
n
0.
ap
Array, size at least max(1,
n
(
n
+1)/2). The array
ap
contains the upper or the lower triangular part of the matrix
A
(as specified by
uplo
) in packed storage.
Output Parameters
ap
The upper or lower triangle of
A
(as specified by
uplo
) 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
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 array
ap
, 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)εP|U||D||UT|PT`
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 (4/3)
n
3
.
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
.

#### Product and Performance Information

1

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Notice revision #20110804