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
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

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 reserved 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