Contents

# p?gebal

Balances a general real/complex matrix.

## Syntax

Include Files
• mkl_scalapack.h
Description
p?gebal
balances a general real/complex matrix
A
. This involves, first, permuting
A
by a similarity transformation to isolate eigenvalues in the first 1 to
ilo
-1 and last
ihi
+1 to
n
elements on the diagonal; and second, applying a diagonal similarity transformation to rows and columns
ilo
to
ihi
to make the rows and columns as close in norm as possible. Both steps are optional.
Balancing may reduce the 1-norm of the matrix, and improve the accuracy of the computed eigenvalues and/or eigenvectors.
Input Parameters
job
(global )
Specifies the operations to be performed on
a
:
= 'N': none: simply set
ilo
= 1,
ihi
=
n
,
scale
[
i
] = 1.0 for
i
= 0,...,
n
-1
;
= 'P': permute only;
= 'S': scale only;
= 'B': both permute and scale.
n
(global )
The order of the matrix
A
(
n
0).
a
(local ) Pointer into the local memory to an array of size
lld_a
* LOC
c
(
n
)
This array contains the local pieces of global input matrix
A
.
desca
(global and local) array of size
dlen_
.
The array descriptor for the distributed matrix
A
.
OUTPUT Parameters
a
On exit,
a
is overwritten by the balanced matrix
A
.
If
job
= 'N',
a
is not referenced.
See Notes for further details.
ilo
,
ihi
(global )
ilo
and
ihi
are set to integers such that on exit matrix elements
A
(
i
,
j
) are zero if
i
>
j
and
j
= 1,...,
ilo
-1 or
i
=
ihi
+1,...,
n
.
If
job
= 'N' or 'S',
ilo
= 1 and
ihi
=
n
.
scale
(global ) array of size
n
.
Details of the permutations and scaling factors applied to
a
. If
p
j
is the index of the row and column interchanged with row and column
j
and
d
j
is the scaling factor applied to row and column
j
, then
scale
[
j
-1] =
p
j
for
j
= 1,...,
ilo
-1,
ihi
+1,..., n
scale
[
j
-1] =
d
j
for
j
=
ilo
,...,
ihi
The order in which the interchanges are made is
n
to
ihi
+1, then 1 to
ilo
-1.
info
(global )
= 0: successful exit.
< 0: if
info
= -
i
, the
i
-th argument had an illegal value.
Application Notes
The permutations consist of row and column interchanges which put the matrix in the form
where
T1
and
T2
are upper triangular matrices whose eigenvalues lie along the diagonal. The column indices
ilo
and
ihi
mark the starting and ending columns of the submatrix B. Balancing consists of applying a diagonal similarity transformation
D
-1
B
D
to make the 1-norms of each row of
B
and its corresponding column nearly equal. The output matrix is