Contents

# ?gtcon

Estimates the reciprocal of the condition number of a tridiagonal matrix.

## Syntax

Include Files
• mkl.h
Description
The routine estimates the reciprocal of the condition number of a real or complex tridiagonal matrix
A
in the 1-norm or infinity-norm:
κ
1
(
A
) = ||
A
||
1
||
A
-1
||
1
κ
(
A
) = ||
A
||
||
A
-1
||
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
?gttrf
to compute the
LU
factorization of
A
.
Input Parameters
norm
Must be
'1'
or
'O'
or
'I'
.
If
norm
=
'1'
or
'O'
, then the routine estimates the condition number of matrix
A
in 1-norm.
If
norm
=
'I'
, then the routine estimates the condition number of matrix
A
in infinity-norm.
n
The order of the matrix
A
;
n
0.
dl
,
d
,
du
,
du2
Arrays:
dl
(
n
-1)
,
d
(
n
)
,
du
(
n
-1)
,
du2
(
n
-2)
.
The array
dl
contains the
(
n
- 1)
multipliers that define the matrix
L
from the
LU
factorization of
A
as computed by
?gttrf
.
The array
d
contains the
n
diagonal elements of the upper triangular matrix
U
from the
LU
factorization of
A
.
The array
du
contains the (
n
- 1) elements of the first superdiagonal of
U
.
The array
du2
contains the (
n
- 2) elements of the second superdiagonal of
U
.
ipiv
Array, size (
n
). The array of pivot indices, as returned by
?gttrf
.
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
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

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Notice revision #20110804