Contents

# ?pbcon

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

## Syntax

Include Files
• mkl.h
Description
The routine estimates the reciprocal of the condition number of a symmetric (Hermitian) positive-definite band 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
?pbtrf
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.
kd
The number of superdiagonals or subdiagonals in the matrix
A
;
kd
0.
ldab
The leading dimension of the array
ab
. (
ldab
kd
+1).
ab
The array
ab
of size max(1,
ldab
*
n
)
contains the factored matrix
A
in band form, as returned by
?pbtrf
.
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
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.