?potri
?potri
Computes the inverse of a symmetric (Hermitian) positive-definite matrix using the Cholesky factorization.
Syntax
lapack_int
LAPACKE_spotri
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
float
*
a
,
lapack_int
lda
);
lapack_int
LAPACKE_dpotri
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
double
*
a
,
lapack_int
lda
);
lapack_int
LAPACKE_cpotri
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
lapack_complex_float
*
a
,
lapack_int
lda
);
lapack_int
LAPACKE_zpotri
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
lapack_complex_double
*
a
,
lapack_int
lda
);
Include Files
- mkl.h
Description
The routine computes the inverse
inv(
of a symmetric positive definite or, for complex flavors, Hermitian positive-definite matrix A
)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 matrixAhas been factored:If, the upper triangle ofuplo='U'Ais stored.If, the lower triangle ofuplo='L'Ais stored.
- n
- The order of the matrixA;.n≥0
- a
- lda
- The leading dimension ofa.lda≥max(1,n).
Output Parameters
- a
- Overwritten by the upper or lower triangle of the inverse ofA.
Return Values
This function returns a value
info
.If , the execution is successful.
info
= 0If , parameter
info
= -i
i
had an illegal value.If , the
info
= i
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 κ2(A), ||AX - I||2≤ c(n)ε κ2(A),
where is a modest linear function of
c
(n
)n
, and ε
is the machine precision; I
denotes the identity matrix.The 2-norm is defined by .
||
of a matrix A
||2
A
is defined by ||(
, and the condition number A
||2
= max
x
·
x
=1A
x
·
A
x
)1/2
κ
2
(A
)κ
2
(A
) = ||A
||2
||A
-1
||2
The total number of floating-point operations is approximately
(2/3)
for real flavors and n
3
(8/3)
for complex flavors.n
3