sgebal.f −

**Functions/Subroutines**

subroutine **SGEBAL** (JOB, N, A, LDA, ILO, IHI, SCALE, INFO)

SGEBAL

**subroutine SGEBAL (characterJOB, integerN, real, dimension( lda, * )A, integerLDA, integerILO, integerIHI, real, dimension( * )SCALE, integerINFO)
SGEBAL**

**Purpose:**

SGEBAL balances a general real 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.

**Parameters:**

*JOB*

JOB is CHARACTER*1

Specifies the operations to be performed on A:

= ’N’: none: simply set ILO = 1, IHI = N, SCALE(I) = 1.0

for i = 1,...,N;

= ’P’: permute only;

= ’S’: scale only;

= ’B’: both permute and scale.

*N*

N is INTEGER

The order of the matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the input matrix A.

On exit, A is overwritten by the balanced matrix.

If JOB = ’N’, A is not referenced.

See Further Details.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,N).

*ILO*

ILO is INTEGER

*IHI*

IHI is INTEGER

ILO and IHI are set to integers such that on exit

A(i,j) = 0 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N.

If JOB = ’N’ or ’S’, ILO = 1 and IHI = N.

*SCALE*

SCALE is REAL array, dimension (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) = P(j) for j = 1,...,ILO-1

= D(j) for j = ILO,...,IHI

= P(j) for j = IHI+1,...,N.

The order in which the interchanges are made is N to IHI+1,

then 1 to ILO-1.

*INFO*

INFO is INTEGER

= 0: successful exit.

< 0: if INFO = -i, the i-th argument had an illegal value.

**Author:**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

November 2011

**Further Details:**

The permutations consist of row and column interchanges which put

the matrix in the form

( T1 X Y )

P A P = ( 0 B Z )

( 0 0 T2 )

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 inv(D) * B * D to make the

1-norms of each row of B and its corresponding column nearly equal.

The output matrix is

( T1 X*D Y )

( 0 inv(D)*B*D inv(D)*Z ).

( 0 0 T2 )

Information about the permutations P and the diagonal matrix D is

returned in the vector SCALE.

This subroutine is based on the EISPACK routine BALANC.

Modified by Tzu-Yi Chen, Computer Science Division, University of

California at Berkeley, USA

Definition at line 161 of file sgebal.f.

Generated automatically by Doxygen for LAPACK from the source code.