Contents

# ?ptcon

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

## Syntax

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
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

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