Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

  • 0.9
  • 09/09/2020
  • Public Content
Contents

?ptcon

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

Syntax

lapack_int LAPACKE_sptcon
(
lapack_int
n
,
const float*
d
,
const float*
e
,
float
anorm
,
float*
rcond
);
lapack_int LAPACKE_dptcon
(
lapack_int
n
,
const double*
d
,
const double*
e
,
double
anorm
,
double*
rcond
);
lapack_int LAPACKE_cptcon
(
lapack_int
n
,
const float*
d
,
const lapack_complex_float*
e
,
float
anorm
,
float*
rcond
);
lapack_int LAPACKE_zptcon
(
lapack_int
n
,
const double*
d
,
const lapack_complex_double*
e
,
double
anorm
,
double*
rcond
);
Include Files
  • mkl.h
Description
The routine computes the reciprocal of the condition number (in the 1-norm) of a real symmetric or complex Hermitian positive-definite tridiagonal matrix using the factorization
A
 =
L*D*L
T
 for real flavors and
A
 =
L*D*L
H
 for complex flavors or
A
 =
U
T
*D*U
 for real flavors and
A
 =
U
H
*D*U
 for complex flavors computed by
?pttrf
 :
κ
1
(
A
) = ||
A
||
1
 ||
A
-1
||
1
 (since
A
 is symmetric or Hermitian,
κ
∞
(
A
) =
κ
1
(
A
)
).
The norm
||
A
-1
||
 is computed by a direct method, and the reciprocal of the condition number is computed as
rcond
 = 1 / (||
A
|| ||
A
-1
||)
.
Before calling this routine:
  • compute
    anorm
     as
    ||
    A
    ||
    1
     = max
    j
    Σ
    i
     |
    a
    i
    j
    |
  • call
    ?pttrf
     to compute the factorization of
    A
    .
Input Parameters
n
The order of the matrix
A
;
n
≥
 0.
d
Arrays, dimension (
n
).
The array
d
 contains the
n
 diagonal elements of the diagonal matrix
D
 from the factorization of
A
, as computed by
?pttrf
 ;
e
Array, size
(
n
 -1)
.
Contains off-diagonal elements of the unit bidiagonal factor
U
 or
L
 from the factorization computed by
?pttrf
 .
anorm
The 1- 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
4*
n
(
kd
 + 1)
 floating-point operations for real flavors and
16*
n
(
kd
 + 1)
 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