Contents

# ?potri

Computes the inverse of a symmetric (Hermitian) positive-definite matrix using the Cholesky factorization.

## Syntax

Include Files
• mkl.h
Description
The routine computes the inverse
inv(
A
)
of a symmetric positive definite or, for complex flavors, Hermitian positive-definite matrix
A
. Before calling this routine, call
?potrf
to factorize
A
.
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 how the input matrix
A
has been factored:
If
uplo
=
'U'
, the upper triangle of
A
is stored.
If
uplo
=
'L'
, the lower triangle of
A
is stored.
n
The order of the matrix
A
;
n
0
.
a
Array
a
(size max(1,
lda
*
n
)).
Contains the factorization of the matrix
A
, as returned by
?potrf
.
lda
a
.
lda
max(1,
n
).
Output Parameters
a
Overwritten by the upper or lower triangle of the inverse of
A
.
Return Values
This function returns a value
info
.
If
info
= 0
, the execution is successful.
If
info
=
-i
, parameter
i
If
info
=
i
, the
i
-th diagonal element of the Cholesky factor (and therefore the factor itself) is zero, and the inversion could not be completed.
Application Notes
The computed inverse
X
satisfies the following error bounds:
||
XA
-
I
||
2
c
(
n
)
ε
κ
2
(
A
), ||
AX
-
I
||
2
c
(
n
)
ε
κ
2
(
A
),
where
c
(
n
)
is a modest linear function of
n
, and
ε
is the machine precision;
I
denotes the identity matrix.
The 2-norm
||
A
||
2
of a matrix
A
is defined by
||
A
||
2
=
max
x
·
x
=1
(
A
x
·
A
x
)
1/2
, and the condition number
κ
2
(
A
)
is defined by
κ
2
(
A
) = ||
A
||
2
||
A
-1
||
2
.
The total number of floating-point operations is approximately
(2/3)
n
3
for real flavors and
(8/3)
n
3
for complex flavors.

#### 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