Computes selected eigenvalues and eigenvectors of a real general matrix

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NAG[f02ecc] NAG[nag_real_eigensystem_sel] - Computes selected eigenvalues and eigenvectors of a real general matrix

Calling Sequence

f02ecc(crit, a, wl, wu, m, w, v, 'n'=n, 'tda'=tda, 'mest'=mest, 'tdv'=tdv, 'fail'=fail)

nag_real_eigensystem_sel(. . .)

Parameters

crit - String;

On entry: indicates the criterion for selecting eigenvalues:

if , then eigenvalues are selected according to their moduli: .

if , then eigenvalues are selected according to their real parts: .

Constraint: "Nag_Select_Modulus" or "Nag_Select_RealPart". .

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

On entry: the by general matrix .

On exit: a contains the Hessenberg form of the balanced input matrix (see Section [Further Comments]).

wl - float;
wu - float;

On entry: and , the lower and upper bounds on the criterion for the selected eigenvalues.

Constraint: . .

m - assignable;

Note: On exit the variable m will have a value of type integer.

On exit: , the number of eigenvalues actually selected.

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

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

On exit: the first m elements of w hold the values of the selected eigenvalues; elements from the index m to contain the other eigenvalues. Complex conjugate pairs of eigenvalues are stored in consecutive elements of the array, with the eigenvalue having the positive imaginary part first.

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

On exit: v contains the selected eigenvectors, with the th column holding the eigenvector associated with the eigenvalue (stored in ).

'n'=n - integer; (optional)

Default value: the first dimension of the arrays a, w, v and the second dimension of the arrays a, w, vthe array a.

On entry: , the order of the matrix .

Constraint: . .

'tda'=tda - integer; (optional)

On entry: the second dimension of the array a as declared in the function from which nag_real_eigensystem_sel (f02ecc) is called.

Constraint: . .

'mest'=mest - integer; (optional)

Default value: the second dimension of the array v.

On entry: mest must be an upper bound on , the number of eigenvalues and eigenvectors selected. No eigenvectors are computed if .

Constraint: . .

'tdv'=tdv - integer; (optional)

On entry: the second dimension of the array v as declared in the function from which nag_real_eigensystem_sel (f02ecc) is called.

Constraint: . .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_real_eigensystem_sel (f02ecc) computes selected eigenvalues and eigenvectors of a real general matrix.

Description

nag_real_eigensystem_sel (f02ecc) computes selected eigenvalues and the corresponding right eigenvectors of a real general matrix :

Eigenvalues may be selected either by modulus, satisfying:

or by real part, satisfying:

Note that even though is real, and may be complex. If is an eigenvector corresponding to a complex eigenvalue , then the complex conjugate vector is the eigenvector corresponding to the complex conjugate eigenvalue . The eigenvalues in a complex conjugate pair and are either both selected or both not selected.

Error Indicators and Warnings

"NE_2_REAL_ARG_LE"

On entry, while . These arguments must satisfy .

"NE_ALLOC_FAIL"

Dynamic memory allocation failed.

"NE_BAD_PARAM"

On entry, argument crit had an illegal value.

"NE_EIGVEC"

Inverse iteration failed to compute all the specified eigenvectors. If an eigenvector failed to converge, the corresponding column of v is set to zero.

"NE_INT_ARG_LT"

On entry, n must not be less than 0: .

"NE_QR_FAIL"

The QR algorithm failed to compute all the eigenvalues. No eigenvectors have been computed.

"NE_REQD_EIGVAL"

There are more than mest eigenvalues in the specified range. The actual number of eigenvalues in the range is returned in m. No eigenvectors have been computed.

Rerun with the second dimension of .

Accuracy

If is an exact eigenvalue, and is the corresponding computed value, then

|(lambda˜)[i],-,lambda[i]|<=(c(n)epsilon||A'||[2])/(s[i])

where is a modestly increasing function of , is the machine precision, and is the reciprocal condition number of ; is the balanced form of the original matrix , and .

If is the corresponding exact eigenvector, and is the corresponding computed eigenvector, then the angle between them is bounded as follows:

theta((x˜)[i],,,x[i])<=(c(n)epsilon||A'||[2])/((sep)[i])

where is the reciprocal condition number of .

Further Comments

nag_real_eigensystem_sel (f02ecc) first balances the matrix, using a diagonal similarity transformation to reduce its norm; and then reduces the balanced matrix to upper Hessenberg form , using an orthogonal similarity transformation: . The function uses the Hessenberg algorithm to compute all the eigenvalues of , which are the same as the eigenvalues of . It computes the eigenvectors of which correspond to the selected eigenvalues, using inverse iteration. It premultiplies the eigenvectors by to form the eigenvectors of ; and finally transforms the eigenvectors to those of the original matrix .

Each eigenvector (real or complex) is normalized so that , and the element of largest absolute value is real and positive.

The inverse iteration function may make a small perturbation to the real parts of close eigenvalues, and this may shift their moduli just outside the specified bounds. If you are relying on eigenvalues being within the bounds, you should test them on return from nag_real_eigensystem_sel (f02ecc).

The time taken by the function is approximately proportional to .

The function can be used to compute all eigenvalues and eigenvectors, by setting wl large and negative, and wu large and positive.

Examples

crit := "Nag_Select_Modulus":
n := 4:
tda := 4:
wl := 0.2:
wu := 0.5:
mest := 3:
tdv := 3:
a := Matrix([[0.35, 0.45, -0.14, -0.17], [0.09, 0.07000000000000001, -0.54, 0.35], [-0.44, -0.33, -0.03, 0.17], [0.25, -0.32, -0.13, 0.11]], datatype=float[8], order='C_order'):
w := Vector(4, datatype=complex[8]):
v := Matrix(4, 3, datatype=complex[8], order='C_order'):
NAG:-f02ecc(crit, a, wl, wu, m, w, v, 'n' = n, 'tda' = tda, 'mest' = mest, 'tdv' = tdv):

	See Also
	Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore f02 Chapter Introduction. NAG Toolbox Overview. NAG Web Site.