1.The function first attempts to permute $A$ to block upper triangular form by a similarity transformation:
$P A P^{T} = A^{'} = (\begin{matrix} A_{11}^{'} & A_{12}^{'} & A_{13}^{'} \\ 0 & A_{22}^{'} & A_{23}^{'} \\ 0 & 0 & A_{33}^{'} \end{matrix})$
where $P$ is a permutation matrix, and $A_{11}^{'}$ and $A_{33}^{'}$ are upper triangular. Then the diagonal elements of $A_{11}^{'}$ and $A_{33}^{'}$ are eigenvalues of $A$ . The rest of the eigenvalues of $A$ are the eigenvalues of the central diagonal block $A_{22}^{'}$ , in rows and columns $i_{lo}$ to $i_{hi}$ . Subsequent operations to compute the eigenvalues of $A$ (or its Schur factorization) need only be applied to these rows and columns; 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 $i_{hi} = n$ , and $A_{22}^{'}$ is the whole of $A$ .

2.The function applies a diagonal similarity transformation to

A^{'}

, to make the rows and columns of

A_{22}^{'}

as close in norm as possible:

A^{''} = D A^{'} D^{- 1} = (\begin{matrix} I & 0 & 0 \\ 0 & D_{22} & 0 \\ 0 & 0 & I \end{matrix}) (\begin{matrix} A_{11}^{'} & A_{12}^{'} & A_{13}^{'} \\ 0 & A_{22}^{'} & A_{23}^{'} \\ 0 & 0 & A_{33}^{'} \end{matrix}) (\begin{matrix} I & 0 & 0 \\ 0 & D_{22}^{- 1} & 0 \\ 0 & 0 & I \end{matrix}) .

This scaling can reduce the norm of the matrix (i.e.,

‖ A_{22}^{''} ‖ < ‖ A_{22}^{'} ‖

) and hence reduce the effect of rounding errors on the accuracy of computed eigenvalues and eigenvectors.

4 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Arguments

1: $order$ – Nag_OrderType Input

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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $job$ – Nag_JobType Input

On entry: indicates whether

A

is to be permuted and/or scaled (or neither).

$job = Nag_DoNothing$: $A$ is neither permuted nor scaled (but values are assigned to ilo, ihi and scale).
$job = Nag_Permute$: $A$ is permuted but not scaled.
$job = Nag_Scale$: $A$ is scaled but not permuted.
$job = Nag_DoBoth$: $A$ is both permuted and scaled.

Constraint:

job = Nag_DoNothing

Nag_Permute

Nag_Scale

Nag_DoBoth

3: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

4: $a [\dim]$ – double Input/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 \times n

matrix

A

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

job = Nag_DoNothing

, a is not referenced.

5: $pda$ – Integer Input

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: $ilo$ – Integer * Output

7: $ihi$ – Integer * Output

On exit: the values

i_{lo}

and

i_{hi}

such that on exit

A (i, j)

is zero if

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

8: $scale [n]$ – double Output

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

A

. More precisely, if

p_{j}

is the index of the row and column interchanged with row and column

j

and

d_{j}

is the scaling factor used to balance row and column

j

then

scale [j - 1] = {\begin{cases} p_{j}, & j = 1, 2, \dots, i_{lo} - 1 \\ d_{j}, & j = i_{lo}, i_{lo} + 1, \dots, i_{hi} and \\ p_{j}, & j = i_{hi} + 1, i_{hi} + 2, \dots, n . \end{cases}

The order in which the interchanges are made is

n

i_{hi} + 1

then

1

i_{lo} - 1

9: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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: $pda > 0$ .
NE_INT_2: On entry, $pda = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pda \geq \max (1, n)$ .
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

7 Accuracy

The errors are negligible.

8 Parallelism and Performance

f08nhc 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 X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

If the matrix

A

is balanced by f08nhc, then any eigenvectors computed subsequently are eigenvectors of the matrix

A^{''}

(see Section 3) and hence f08njc must then be called to transform them back to eigenvectors of

A

If the Schur vectors of

A

are required, then this function must not be called with

job = Nag_Scale

Nag_DoBoth

, because then the balancing transformation is not orthogonal. If this function is called with

job = Nag_Permute

, then any Schur vectors computed subsequently are Schur vectors of the matrix

A^{''}

, and f08njc must be called (with

side = Nag_RightSide

) to transform them back to Schur vectors of

A

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

n^{2}

The complex analogue of this function is f08nvc.

10 Example

This example computes all the eigenvalues and right eigenvectors of the matrix

A

, where

A = (\begin{array}{r} 5.14 & 0.91 & 0.00 & - 32.80 \\ 0.91 & 0.20 & 0.00 & 34.50 \\ 1.90 & 0.80 & - 0.40 & - 3.00 \\ - 0.33 & 0.35 & 0.00 & 0.66 \end{array}) .

The program first calls f08nhc to balance the matrix; it then computes the Schur factorization of the balanced matrix, by reduction to Hessenberg form and the

Q R

algorithm. Then it calls f08qkc to compute the right eigenvectors of the balanced matrix, and finally calls f08njc to transform the eigenvectors back to eigenvectors of the original matrix

A

f08nh: FL CL CPP AD

NAG CL Interfacef08nhc (dgebal)

▸▿ 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

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
f08nhc (dgebal)