Matrix(1..n, 1..tda, datatype=float[8], order=C_order);

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NAG[f02bjc] NAG[nag_real_general_eigensystem] - All eigenvalues and optionally eigenvectors of real generalized eigenproblem, by algorithm

Calling Sequence

f02bjc(a, b, tol, alfa, beta, wantv, iter, 'n'=n, 'tda'=tda, 'tdb'=tdb, 'v'=v, 'tdv'=tdv, 'fail'=fail)

nag_real_general_eigensystem(. . .)

Parameters

a - Matrix(1..n, 1..tda, datatype=float[8], order=C_order);

On entry: the by matrix .

On exit: the array is overwritten.

b - Matrix(1..n, 1..tdb, datatype=float[8], order=C_order);

On entry: the by matrix .

On exit: the array is overwritten.

tol - float;

On entry: the tolerance used to determine negligible elements.

An element will be considered negligible if it is less than tol times the norm of its matrix.

machine precision is used in place of tol.

A value of tol greater than machine precision may result in faster execution but less accurate results.

alfa - Vector(1..n, datatype=complex[8]);

On exit: , for .

beta - Vector(1..n, datatype=float[8]);

On exit: , for .

wantv - boolean;

On entry: wantv must be set to true if the eigenvectors are required. If wantv is set to false then the array v is not referenced.

iter - Vector(1..n, datatype=integer[kernelopts('wordsize')/8]);

On exit: contains the number of iterations needed to obtain the th eigenvalue. Note that the eigenvalues are obtained in reverse order, starting with the th.

'n'=n - integer; (optional)

Default value: the first dimension of the arrays a, b, alfa, beta, v, iter and the second dimension of the arrays a, b, alfa, beta, v, iterthe arrays a, b.

On entry: , the order of the matrices and .

Constraint: . .

'tda'=tda - integer; (optional)

On entry: the second dimension of the array a as declared in the function from which nag_real_general_eigensystem (f02bjc) is called.

Constraint: . .

'tdb'=tdb - integer; (optional)

On entry: the second dimension of the array b as declared in the function from which nag_real_general_eigensystem (f02bjc) is called.

Constraint: . .

'v'=v - Matrix(1..n, 1..tdv, datatype=float[8], order=C_order); (optional)

On exit: if , then

if the th eigenvalue is real, the th column of v contains its eigenvector;

if the th and th eigenvalues form a complex pair, the th and th columns of v contain the real and imaginary parts of the eigenvector associated with the first eigenvalue of the pair. The conjugate of this vector is the eigenvector for the conjugate eigenvalue.

Each eigenvector is normalized so that the component of largest modulus is real and the sum of squares of the moduli equal one.

If , v is not referenced and may be set to the null pointer, i.e., (double *)0.

'tdv'=tdv - integer; (optional)

On entry: the second dimension of the array v as declared in the function from which nag_real_general_eigensystem (f02bjc) is called.

Constraint: if , . .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_real_general_eigensystem (f02bjc) calculates all the eigenvalues and, if required, all the eigenvectors of the generalized eigenproblem where and are real, square matrices, using the algorithm.

Description

All the eigenvalues and, if required, all the eigenvectors of the generalized eigenproblem where and are real, square matrices, are determined using the algorithm. The algorithm consists of four stages:

a. is reduced to upper Hessenberg form and at the same time is reduced to upper triangular form.

b. is further reduced to quasi-triangular form while the triangular form of is maintained.

c. The quasi-triangular form of is reduced to triangular form and the eigenvalues extracted.

d. This function does not actually produce the eigenvalues , but instead returns and such that

The division by becomes the responsibility of the user's program, since may be zero indicating an infinite eigenvalue. Pairs of complex eigenvalues occur with and complex conjugates, even though and are not conjugate.

e. If the eigenvectors are required (), they are obtained from the triangular matrices and then transformed back into the original co-ordinate system.

Error Indicators and Warnings

"NE_INT_ARG_LT"

On entry, n must not be less than 1: .

"NE_ITERATIONS_QZ"

More than iterations are required to determine all the diagonal 1 by 1 or 2 by 2 blocks of the quasi-triangular form in the second step of the algorithm. This failure occurs at the th eigenvalue, . and are correct for but v does not contain any correct eigenvectors.

The value of will be returned in member fail.errnum of the NAG error structure provided NAGERR_DEFAULT is not used as the error argument.

Accuracy

The computed eigenvalues are always exact for a problem where and are both of the order of , tol being defined as in Section [Parameters] and being the machine precision.

Note: interpretation of results obtained with the algorithm often requires a clear understanding of the effects of small changes in the original data. These effects are reviewed in Wilkinson (1979), in relation to the significance of small values of and . It should be noted that if and are both small for any , it may be that no reliance can be placed on any of the computed eigenvalues . The user is recommended to study Wilkinson (1979) and, if in difficulty, to seek expert advice on determining the sensitivity of the eigenvalues to perturbations in the data. .

	Further Comments
	The time taken by nag_real_general_eigensystem (f02bjc) is approximately proportional to and also depends on the value chosen for argument tol.

Examples

n := 4:
tda := 4:
tdb := 4:
tol := 1.111307226797642e-16:
wantv := true:
tdv := 4:
a := Matrix([[3.9, 12.5, -34.5, -0.5], [4.3, 21.5, -47.5, 7.5], [4.3, 21.5, -43.5, 3.5], [4.4, 26, -46, 6]], datatype=float[8], order='C_order'):
b := Matrix([[1, 2, -3, 1], [1, 3, -5, 4], [1, 3, -4, 3], [1, 3, -4, 4]], datatype=float[8], order='C_order'):
alfa := Vector(4, datatype=complex[8]):
beta := Vector(4, datatype=float[8]):
v := Matrix(4, 4, datatype=float[8], order='C_order'):
iter := Vector(4, datatype=integer[kernelopts('wordsize')/8]):
NAG:-f02bjc(a, b, tol, alfa, beta, wantv, iter, 'n' = n, 'tda' = tda, 'tdb' = tdb, 'v' = v, 'tdv' = tdv):

alfa;

beta;

iter;

	See Also
	Moler C B and Stewart G W (1973) An algorithm for generalized matrix eigenproblems SIAM J. Numer. Anal. 10 241–256 Ward R C (1975) The combination shift algorithm SIAM J. Numer. Anal. 12 835–853 Wilkinson J H (1979) Kronecker's canonical form and the algorithm Linear Algebra Appl. 28 285–303 f02 Chapter Introduction. NAG Toolbox Overview. NAG Web Site.