Estimates of sensitivities of selected eigenvalues and eigenvectors of real upper quasi-triangular matrix

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NAG[f08qlc] NAG[nag_dtrsna] - Estimates of sensitivities of selected eigenvalues and eigenvectors of real upper quasi-triangular matrix

Calling Sequence

f08qlc(job, how_many, select, t, vl, vr, s, sep, m, 'n'=n, 'mm'=mm, 'fail'=fail)

nag_dtrsna(. . .)

Parameters

job - String;

On entry: indicates whether condition numbers are required for eigenvalues and/or eigenvectors.

Condition numbers for eigenvalues only are computed.

Condition numbers for eigenvectors only are computed.

Condition numbers for both eigenvalues and eigenvectors are computed.

Constraint: "Nag_EigVals", "Nag_EigVecs" or "Nag_DoBoth". .

how_many - String;

On entry: indicates how many condition numbers are to be computed.

Condition numbers for all eigenpairs are computed.

Condition numbers for selected eigenpairs (as specified by select) are computed.

Constraint: "Nag_ComputeAll" or "Nag_ComputeSelected". .

select - Vector(1..dim, datatype=boolean);

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

when ;

otherwise.

On entry: specifies the eigenpairs for which condition numbers are to be computed if . To select condition numbers for the eigenpair corresponding to the real eigenvalue , must be set true. To select condition numbers corresponding to a complex conjugate pair of eigenvalues and , and/or must be set to true.

If , select is not referenced.

t - Matrix(1..dim1, 1..dim2, datatype=float[8], order=order);

Note: this array may be supplied in Fortran_order or C_order , as specified by order. All array parameters must use a consistent order.

On entry: the by upper quasi-triangular matrix in canonical Schur form, as returned by f08pec (nag_dhseqr).

vl - Matrix(1..dim1, 1..dim2, datatype=float[8], order=order);

Note: this array may be supplied in Fortran_order or C_order , as specified by order. All array parameters must use a consistent order.

On entry: if "Nag_EigVals" or "Nag_DoBoth", vl must contain the left eigenvectors of (or of any matrix with orthogonal) corresponding to the eigenpairs specified by how_many and select. The eigenvectors must be stored in consecutive rows or columns of vl, as returned by f08qkc (nag_dtrevc) or f08pkc (nag_dhsein).

If , vl is not referenced.

vr - Matrix(1..dim1, 1..dim2, datatype=float[8], order=order);

Note: this array may be supplied in Fortran_order or C_order , as specified by order. All array parameters must use a consistent order.

On entry: if "Nag_EigVals" or "Nag_DoBoth", vr must contain the right eigenvectors of (or of any matrix with orthogonal) corresponding to the eigenpairs specified by how_many and select. The eigenvectors must be stored in consecutive rows or columns of vr, as returned by f08qkc (nag_dtrevc) or f08pkc (nag_dhsein).

If , vr is not referenced.

s - Vector(1..dim, datatype=float[8]);

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

when "Nag_EigVals" or "Nag_DoBoth";

when .

On exit: the reciprocal condition numbers of the selected eigenvalues if "Nag_EigVals" or "Nag_DoBoth", stored in consecutive elements of the array. Thus , and the th rows or columns of vl and vr all correspond to the same eigenpair (but not in general the th eigenpair unless all eigenpairs have been selected). For a complex conjugate pair of eigenvalues, two consecutive elements of s are set to the same value.

s is not referenced if .

sep - Vector(1..dim, datatype=float[8]);

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

when "Nag_EigVecs" or "Nag_DoBoth";

when .

On exit: the estimated reciprocal condition numbers of the selected right eigenvectors if "Nag_EigVecs" or "Nag_DoBoth", stored in consecutive elements of the array. For a complex eigenvector, two consecutive elements of sep are set to the same value. If the eigenvalues cannot be reordered to compute , then is set to zero; this can only occur when the true value would be very small anyway.

If , sep is not referenced.

m - assignable;

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

On exit: , the number of elements of s and/or sep actually used to store the estimated condition numbers. If , m is set to .

'n'=n - integer; (optional)

Default value: the dimension of the array t.

On entry: , the order of the matrix .

Constraint: . .

'mm'=mm - integer; (optional)

Default value: the second dimension of the arrays vl, vr.

On entry: the number of elements in the arrays s and sep, and the number of rows or columns (depending on the value of storage order) in the arrays vl and vr (if used). The precise number required, , is if ; if , is obtained by counting 1 for each selected real eigenvalue, and 2 for each selected complex conjugate pair of eigenvalues (see select), in which case .

Constraint: . .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_dtrsna (f08qlc) estimates condition numbers for specified eigenvalues and/or right eigenvectors of a real upper quasi-triangular matrix.

Description

nag_dtrsna (f08qlc) estimates condition numbers for specified eigenvalues and/or right eigenvectors of a real upper quasi-triangular matrix in canonical Schur form. These are the same as the condition numbers of the eigenvalues and right eigenvectors of an original matrix (with orthogonal ), from which may have been derived.

nag_dtrsna (f08qlc) computes the reciprocal of the condition number of an eigenvalue as

s[i]=(|v^H,u|)/(||u||[E]||v||[E])

where and are the right and left eigenvectors of , respectively, corresponding to . This reciprocal condition number always lies between zero (i.e., ill-conditioned) and one (i.e., well-conditioned).

An approximate error estimate for a computed eigenvalue is then given by

epsilon*LinearAlgebra[Norm](T)/s[i]

where is the machine precision.

To estimate the reciprocal of the condition number of the right eigenvector corresponding to , the function first calls f08qfc (nag_dtrexc) to reorder the eigenvalues so that is in the leading position:

The reciprocal condition number of the eigenvector is then estimated as , the smallest singular value of the matrix . This number ranges from zero (i.e., ill-conditioned) to very large (i.e., well-conditioned).

An approximate error estimate for a computed right eigenvector corresponding to is then given by

epsilon*LinearAlgebra[Norm](T)/sep[i]

	Error Indicators and Warnings
	"NE_ALLOC_FAIL" Dynamic memory allocation failed. "NE_BAD_PARAM" On entry, argument had an illegal value. "NE_INT" On entry, . Constraint: . "NE_INT_2" On entry, , . Constraint: . "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 consult NAG for assistance.

	Accuracy
	The computed values may over estimate the true value, but seldom by a factor of more than 3.

	Further Comments
	For a description of canonical Schur form, see the document for f08pec (nag_dhseqr). The complex analogue of this function is f08qyc (nag_ztrsna).

Examples

job := "Nag_DoBoth":
how_many := "Nag_ComputeAll":
n := 4:
mm := 4:
selector := Vector([false], datatype=boolean):
t := Matrix([[0.7995, -0.1144, 0.006, 0.0336], [0, -0.0994, 0.2478, 0.3474], [0, -0.6483, -0.0994, 0.2026], [0, 0, 0, -0.1007]], datatype=float[8]):
vl := Matrix([[1, 0, 0, 0], [-0.1101757724599591, 0.8492971136777001, 0, 0], [-0.02369735945664465, 0, 0.5250761453388552, 0], [-0.01052671448401023, -0.2630232022148211, 0.7369767977851788, 1]], datatype=float[8]):
vr := Matrix([[1, 0.06811593844054897, 0.02369735945664464, 0.008112867113990377], [0, 0.6182478862610575, 0, 0.2206494686069181], [0, 0, 1, -1], [0, 0, 0, 0.7124731021612292]], datatype=float[8]):
s := Vector(4, datatype=float[8]):
sep := Vector(4, datatype=float[8]):
NAG:-f08qlc(job, how_many, selector, t, vl, vr, s, sep, m, 'n' = n, 'mm' = mm):

sep;

	See Also
	Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore f08 Chapter Introduction. NAG Toolbox Overview. NAG Web Site.