Contents

# ?pbequ

Computes row and column scaling factors intended to equilibrate a symmetric (Hermitian) positive-definite band matrix and reduce its condition number.

## Syntax

Include Files
• mkl.h
Description
The routine computes row and column scalings intended to equilibrate a symmetric (Hermitian) positive definite band matrix
A
and reduce its condition number (with respect to the two-norm).
The output array
s
returns scale factors such that contains These factors are chosen so that the scaled matrix
B
with elements
b
i
j
=
s
[i-1]*
a
i
j
*
s
[j-1]
has diagonal elements equal to 1. This choice of
s
puts the condition number of
B
within a factor
n
of the smallest possible condition number over all possible diagonal scalings.
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 whether the upper or lower triangular part of
A
is stored in the array
ab
:
If
uplo
=
'U'
, the array
ab
stores the upper triangular part of the matrix
A
.
If
uplo
=
'L'
, the array
ab
stores the lower triangular part of the matrix
A
.
n
The order of matrix
A
;
n
0
.
kd
The number of superdiagonals or subdiagonals in the matrix
A
;
kd
0.
ab
Array, size
max(1,
ldab
*
n
)
.
The array
ap
contains either the upper or the lower triangular part of the matrix
A
(as specified by
uplo
) in band storage (see Matrix Storage Schemes).
ldab
The leading dimension of the array
ab
;
ldab
kd
+1.
Output Parameters
s
Array, size (
n
).
If
info
= 0
, the array
s
contains the scale factors for
A
.
scond
If
info
= 0
,
scond
contains the ratio of the smallest
s
[i]
to the largest
s
[i]
.
amax
Absolute value of the largest element of the matrix
A
.
Return Values
This function returns a value
info
.
If
info
= 0
, the execution is successful.
If
info
=
-i
, parameter
i
had an illegal value.
If
info
=
i
, the
i
-th diagonal element of
A
is nonpositive.
Application Notes
If
scond
0.1
and
amax
is neither too large nor too small, it is not worth scaling by
s
.
If
amax
is very close to
SMLNUM
or very close to
BIGNUM
, the matrix
A
should be scaled.

#### Product and Performance Information

1

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