nag_dggbal (f08whc) : NAG Library, Mark 24

3 Description

Balancing may reduce the

1

-norms of the matrices and improve the accuracy of the computed eigenvalues and eigenvectors in the real generalized eigenvalue problem

A x = λ B x .

nag_dggbal (f08whc) is usually the first step in the solution of the above generalized eigenvalue problem. Balancing is optional but it is highly recommended.

The term ‘balancing’ covers two steps, each of which involves similarity transformations on

A

and

B

. The function can perform either or both of these steps. Both steps are optional.

The function first attempts to permute

A

and

B

to block upper triangular form by a similarity transformation:

P A P^{T} = F = (\begin{matrix} F_{11} & F_{12} & F_{13} \\ F_{22} & F_{23} \\ F_{33} \end{matrix})

P B P^{T} = G = (\begin{matrix} G_{11} & G_{12} & G_{13} \\ G_{22} & G_{23} \\ G_{33} \end{matrix})

where

P

is a permutation matrix,

F_{11}

F_{33}

G_{11}

and

G_{33}

are upper triangular. Then the diagonal elements of the matrix pairs

(F_{11}, G_{11})

and

(F_{33}, G_{33})

are generalized eigenvalues of

(A, B)

. The rest of the generalized eigenvalues are given by the matrix pair

(F_{22}, G_{22})

which are in rows and columns

i_{lo}

i_{hi}

. Subsequent operations to compute the generalized eigenvalues of

(A, B)

need only be applied to the matrix pair

(F_{22}, G_{22})

; this can save a significant amount of work if

i_{lo} > 1

and

i_{hi} < n

. If no suitable permutation exists (as is often the case), the function sets

i_{lo} = 1

and

i_{hi} = n

The function applies a diagonal similarity transformation to

(F, G)

, to make the rows and columns of

(F_{22}, G_{22})

as close in norm as possible:

D F \hat{D} = (\begin{matrix} I & 0 & 0 \\ 0 & D_{22} & 0 \\ 0 & 0 & I \end{matrix}) (\begin{matrix} F_{11} & F_{12} & F_{13} \\ F_{22} & F_{23} \\ F_{33} \end{matrix}) (\begin{matrix} I & 0 & 0 \\ 0 & {\hat{D}}_{22} & 0 \\ 0 & 0 & I \end{matrix})

D G \hat{D} = (\begin{matrix} I & 0 & 0 \\ 0 & D_{22} & 0 \\ 0 & 0 & I \end{matrix}) (\begin{matrix} G_{11} & G_{12} & G_{13} \\ G_{22} & G_{23} \\ G_{33} \end{matrix}) (\begin{matrix} I & 0 & 0 \\ 0 & {\hat{D}}_{22} & 0 \\ 0 & 0 & I \end{matrix})

This transformation usually improves the accuracy of computed generalized eigenvalues and eigenvectors.

5 Arguments

1: order – Nag_OrderTypeInput

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: job – Nag_JobTypeInput

On entry: specifies the operations to be performed on matrices

A

and

B

$job = Nag_DoNothing$: No balancing is done. Initialize $ilo = 1$ , $ihi = n$ , $lscale [i - 1] = 1.0$ and $rscale [i - 1] = 1.0$ , for $i = 1, 2, \dots, n$ .
$job = Nag_Permute$: Only permutations are used in balancing.
$job = Nag_Scale$: Only scalings are are used in balancing.
$job = Nag_DoBoth$: Both permutations and scalings are used in balancing.

Constraint:

job = Nag_DoNothing

Nag_Permute

Nag_Scale

Nag_DoBoth

3: n – IntegerInput

On entry:

n

, the order of the matrices

A

and

B

Constraint:

n \geq 0

4: a[

\dim

] – doubleInput/Output

Note: the dimension, dim, of the array a must be at least

\max (1, pda \times n)

Where

A (i, j)

appears in this document, it refers to the array element

$a [(j - 1) \times pda + i - 1]$ when $order = Nag_ColMajor$ ;
$a [(i - 1) \times pda + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

n

n

matrix

A

On exit: a is overwritten by the balanced matrix. If

job = Nag_DoNothing

, a is not referenced.

5: pda – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array a.

Constraint:

pda \geq \max (1, n)

6: b[

\dim

] – doubleInput/Output

Note: the dimension, dim, of the array b must be at least

\max (1, pdb \times n)

Where

B (i, j)

appears in this document, it refers to the array element

$b [(j - 1) \times pdb + i - 1]$ when $order = Nag_ColMajor$ ;
$b [(i - 1) \times pdb + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

n

n

matrix

B

On exit: b is overwritten by the balanced matrix. If

job = Nag_DoNothing

, b is not referenced.

7: pdb – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array b.

Constraint:

pdb \geq \max (1, n)

8: ilo – Integer *Output

9: ihi – Integer *Output

On exit:

i_{lo}

and

i_{hi}

are set such that

A (i, j) = 0

and

B (i, j) = 0

i > j

and

1 \leq j < i_{lo}

i_{hi} < i \leq n

job = Nag_DoNothing

Nag_Scale

i_{lo} = 1

and

i_{hi} = n

10: lscale[n] – doubleOutput

On exit: details of the permutations and scaling factors applied to the left side of the matrices

A

and

B

. If

P_{i}

is the index of the row interchanged with row

i

and

d_{i}

is the scaling factor applied to row

i

, then

$lscale [i - 1] = P_{i}$ , for $i = 1, 2, \dots, i_{lo} - 1$ ;
$lscale [i - 1] = d_{i}$ , for $i = i_{lo}, \dots, i_{hi}$ ;
$lscale [i - 1] = P_{i}$ , for $i = i_{hi} + 1, \dots, n$ .

The order in which the interchanges are made is

n

i_{hi} + 1

, then

1

i_{lo} - 1

11: rscale[n] – doubleOutput

On exit: details of the permutations and scaling factors applied to the right side of the matrices

A

and

B

P_{j}

is the index of the column interchanged with column

j

and

{\hat{d}}_{j}

is the scaling factor applied to column

j

, then

$rscale [j - 1] = P_{j}$ , for $j = 1, 2, \dots, i_{lo} - 1$ ;
$rscale [j - 1] = {\hat{d}}_{j}$ , for $j = i_{lo}, \dots, i_{hi}$ ;
$rscale [j - 1] = P_{j}$ , for $j = i_{hi} + 1, \dots, n$ .

The order in which the interchanges are made is

n

i_{hi} + 1

, then

1

i_{lo} - 1

12: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.

NE_BAD_PARAM

On entry, argument

⟨value⟩

had an illegal value.

NE_INT

On entry,

n = ⟨value⟩

.
Constraint:

n \geq 0

On entry,

pda = ⟨value⟩

.
Constraint:

pda > 0

On entry,

pdb = ⟨value⟩

.
Constraint:

pdb > 0

NE_INT_2

On entry,

pda = ⟨value⟩

and

n = ⟨value⟩

.
Constraint:

pda \geq \max (1, n)

On entry,

pdb = ⟨value⟩

and

n = ⟨value⟩

.
Constraint:

pdb \geq \max (1, n)

NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

8 Parallelism and Performance

nag_dggbal (f08whc) is not threaded by NAG in any implementation.

nag_dggbal (f08whc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

nag_dggbal (f08whc) is usually the first step in computing the real generalized eigenvalue problem but it is an optional step. The matrix

B

is reduced to the upper triangular form using the

Q R

factorization function nag_dgeqrf (f08aec) and this orthogonal transformation

Q

is applied to the matrix

A

by calling nag_dormqr (f08agc). This is followed by nag_dgghrd (f08wec) which reduces the matrix pair into the generalized Hessenberg form.

If the matrix pair

(A, B)

is balanced by this function, then any generalized eigenvectors computed subsequently are eigenvectors of the balanced matrix pair. In that case, to compute the generalized eigenvectors of the original matrix, nag_dggbak (f08wjc) must be called.

The total number of floating-point operations is approximately proportional to

n^{2}

The complex analogue of this function is nag_zggbal (f08wvc).

NAG Library Function Document

nag_dggbal (f08whc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

NAG Library Function Documentnag_dggbal (f08whc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

NAG Library Function Document

nag_dggbal (f08whc)