Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

  • 2021.1
  • 12/04/2020
  • Public Content
Contents

?pocon

Estimates the reciprocal of the condition number of a symmetric (Hermitian) positive-definite matrix.

Syntax

lapack_int LAPACKE_spocon
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
const float*
a
,
lapack_int
lda
,
float
anorm
,
float*
rcond
);
lapack_int LAPACKE_dpocon
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
const double*
a
,
lapack_int
lda
,
double
anorm
,
double*
rcond
);
lapack_int LAPACKE_cpocon
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
const lapack_complex_float*
a
,
lapack_int
lda
,
float
anorm
,
float*
rcond
);
lapack_int LAPACKE_zpocon
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
const lapack_complex_double*
a
,
lapack_int
lda
,
double
anorm
,
double*
rcond
);
Include Files
  • mkl.h
Description
The routine estimates the reciprocal of the condition number of a symmetric (Hermitian) positive-definite matrix
A
:
κ
1
(
A
) = ||
A
||
1
 ||
A
-1
||
1
 (since
A
 is symmetric or Hermitian,
κ
∞
(
A
) =
κ
1
(
A
)
).
An estimate is obtained for
||
A
-1
||
, and the reciprocal of the condition number is computed as
rcond
 = 1 / (||
A
|| ||
A
-1
||)
.
Before calling this routine:
  • compute
    anorm
     (either
    ||
    A
    ||
    1
     = max
    j
    Σ
    i
     |
    a
    i
    j
    |
     or
    ||
    A
    ||
    ∞
     = max
    i
    Σ
    j
     |
    a
    i
    j
    |)
  • call
    ?potrf
     to compute the Cholesky factorization of
    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'
,
A
 is factored as
A
 =
U
T
*
U
 for real flavors or
A
 =
U
H
*
U
 for complex flavors, and
U
 is stored.
If
uplo
 =
'L'
,
A
 is factored as
A
 =
L
*
L
T
 for real flavors or
A
 =
L
*
L
H
 for complex flavors, and
L
 is stored.
n
The order of the matrix
A
;
n
≥
 0.
a
The array
a
of size max(1,
lda
*
n
)
 contains the factored matrix
A
, as returned by
?potrf
.
lda
The leading dimension of
a
;
lda
≥
 max(1,
n
)
.
anorm
The norm of the
original
 matrix
A
(see
Description
)
.
Output Parameters
rcond
An estimate of the reciprocal of the condition number. The routine sets
rcond
 =0 if the estimate underflows; in this case the matrix is singular (to working precision). However, anytime
rcond
 is small compared to 1.0, for the working precision, the matrix may be poorly conditioned or even singular.
Return Values
This function returns a value
info
.
If
info
 = 0
, the execution is successful.
If
info
 =
-i
, parameter
i
 had an illegal value.
Application Notes
The computed
rcond
 is never less than
r
 (the reciprocal of the true condition number) and in practice is nearly always less than 10
r
. A call to this routine involves solving a number of systems of linear equations
A
*
x
 =
b
; the number is usually 4 or 5 and never more than 11. Each solution requires approximately
2
n
2
 floating-point operations for real flavors and
8
n
2
 for complex flavors.

Product and Performance Information

1

Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex.