## Developer Reference

• 0.10
• 10/21/2020
• Public Content
Contents

# ?gbcon

Estimates the reciprocal of the condition number of a band matrix in the 1-norm or the infinity-norm.

## Syntax

Include Files
• mkl.h
Description
The routine estimates the reciprocal of the condition number of a general band matrix
A
in the 1-norm or infinity-norm:
κ
1
(
A
) = ||
A
||
1
||
A
-1
||
1
=
κ
(
A
T
) =
κ
(
A
H
)
κ
(
A
) = ||
A
||
||
A
-1
||
=
κ
1
(
A
T
) =
κ
1
(
A
H
).
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
?gbtrf
to compute the
LU
factorization of
A
.
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
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.
kl
The number of subdiagonals within the band of
A
;
kl
0.
ku
The number of superdiagonals within the band of
A
;
ku
0.
ldab
The leading dimension of the array
ab
. (
ldab
2*
kl
+
ku
+1).
ipiv
Array, size at least
max(1,
n
)
. The
ipiv
array, as returned by
?gbtrf
.
ab
The array
ab
of size max(1,
ldab
*
n
)
contains the factored band matrix
A
, as returned by
?gbtrf
.
anorm
The norm of the
original
matrix
A
.
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
or
A
H
*
x
=
b
; the number is usually 4 or 5 and never more than 11. Each solution requires approximately
2
n
(
ku
+ 2
kl
)
floating-point operations for real flavors and
8
n
(
ku
+ 2
kl
)
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 reserverd 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