Eigenvalues and Schur factorization of real upper Hessenberg matrix reduced from real general matrix

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NAG[f08pec] NAG[nag_dhseqr] - Eigenvalues and Schur factorization of real upper Hessenberg matrix reduced from real general matrix

Calling Sequence

f08pec(job, compz, ilo, ihi, h, wr, wi, z, 'n'=n, 'fail'=fail)

nag_dhseqr(. . .)

Parameters

job - String;

On entry: indicates whether eigenvalues only or the Schur form is required.

Eigenvalues only are required.

The Schur form is required.

Constraint: "Nag_EigVals" or "Nag_Schur". .

compz - String;

On entry: indicates whether the Schur vectors are to be computed.

No Schur vectors are computed (and the array z is not referenced).

The Schur vectors of are computed (and the array z is initialized by the function).

The Schur vectors of are computed (and the array z must contain the matrix on entry).

Constraint: "Nag_NotZ", "Nag_UpdateZ" or "Nag_InitZ". .

ilo - integer;
ihi - integer;

On entry: if the matrix has been balanced by f08nhc (nag_dgebal), then ilo and ihi must contain the values returned by that function. Otherwise, ilo must be set to 1 and ihi to n.

Constraint: and . .

h - 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 Hessenberg matrix , as returned by f08nec (nag_dgehrd).

On exit: if , the array contains no useful information.

If , h is overwritten by the upper quasi-triangular matrix from the Schur decomposition (the Schur form) unless .

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

Note: the dimension, dim, of the array wr and wi must be at least .

On exit: the real and imaginary parts, respectively, of the computed eigenvalues, unless (in which case see Section [Error Indicators and Warnings]). Complex conjugate pairs of eigenvalues appear consecutively with the eigenvalue having positive imaginary part first. The eigenvalues are stored in the same order as on the diagonal of the Schur form (if computed); see Section [Further Comments] for details.

z - 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 , z must contain the orthogonal matrix from the reduction to Hessenberg form.

If , z need not be set.

On exit: if "Nag_UpdateZ" or "Nag_InitZ", z contains the orthogonal matrix of the required Schur vectors, unless .

If , z is not referenced.

'n'=n - integer; (optional)

Default value: the dimension of the array h.

On entry: , the order of the matrix .

Constraint: . .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_dhseqr (f08pec) computes all the eigenvalues and, optionally, the Schur factorization of a real Hessenberg matrix or a real general matrix which has been reduced to Hessenberg form.

Description

nag_dhseqr (f08pec) computes all the eigenvalues and, optionally, the Schur factorization of a real upper Hessenberg matrix :

where is an upper quasi-triangular matrix (the Schur form of ), and is the orthogonal matrix whose columns are the Schur vectors . See Section [Further Comments] for details of the structure of .

The function may also be used to compute the Schur factorization of a real general matrix which has been reduced to upper Hessenberg form :

[[A=QHQ^T , Q ],[=(Q,Z)T(Q,Z)^T ,]]

In this case, after f08nec (nag_dgehrd) has been called to reduce to Hessenberg form, f08nfc (nag_dorghr) must be called to form explicitly; is then passed to nag_dhseqr (f08pec), which must be called with .

The function can also take advantage of a previous call to f08nhc (nag_dgebal) which may have balanced the original matrix before reducing it to Hessenberg form, so that the Hessenberg matrix has the structure:

([[H[11],H[12],H[13]],[,H[22],H[23]],[,,H[33]]])

where and are upper triangular. If so, only the central diagonal block (in rows and columns to ) needs to be further reduced to Schur form (the blocks and are also affected). Therefore the values of and can be supplied to nag_dhseqr (f08pec) directly. Also, f08njc (nag_dgebak) must be called after this function to permute the Schur vectors of the balanced matrix to those of the original matrix. If f08nhc (nag_dgebal) has not been called however, then must be set to 1 and to . Note that if the Schur factorization of is required, f08nhc (nag_dgebal) must not be called with "Nag_Schur" or "Nag_DoBoth", because the balancing transformation is not orthogonal.

nag_dhseqr (f08pec) uses a multishift form of the upper Hessenberg algorithm, due to Bai and Demmel (1989). The Schur vectors are normalized so that , but are determined only to within a factor .

Error Indicators and Warnings

"NE_ALLOC_FAIL"

Dynamic memory allocation failed.

"NE_BAD_PARAM"

On entry, argument had an illegal value.

"NE_CONVERGENCE"

The algorithm has failed to find all the eigenvalues after a total of iterations.

"NE_INT"

On entry, . Constraint: .

"NE_INT_3"

On entry, , , . Constraint: and .

"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 Schur factorization is the exact factorization of a nearby matrix , where and is the machine precision. If is an exact eigenvalue, and is the corresponding computed value, then where is a modestly increasing function of , and is the reciprocal condition number of . The condition numbers may be computed by calling f08qlc (nag_dtrsna).

Further Comments

The total number of floating-point operations depends on how rapidly the algorithm converges, but is typically about:

if only eigenvalues are computed;

if the Schur form is computed;

if the full Schur factorization is computed.

The Schur form has the following structure (referred to as canonical Schur form).

If all the computed eigenvalues are real, is upper triangular, and the diagonal elements of are the eigenvalues; , for and .

If some of the computed eigenvalues form complex conjugate pairs, then has 2 by 2 diagonal blocks. Each diagonal block has the form

$(Matrix(2, 2, {(1, 1) = t[i^2], (1, 2) = t[i, i+1], (2, 1) = t[i+1, i], (2, 2) = t[i+1, i+1]})) = (Matrix(2, 2, {(1, 1) = alpha, (1, 2) = beta, (2, 1) = gamma, (2, 2) = alpha}))$

where . The corresponding eigenvalues are ; ; ; .

The complex analogue of this function is f08psc (nag_zhseqr).

Examples

job := "Nag_Schur":
compz := "Nag_InitZ":
n := 4:
ilo := 1:
ihi := 4:
h := Matrix([[0.35, -0.116, -0.3886, -0.2942], [-0.514, 0.1225, 0.1004, 0.1126], [0, 0.6443, -0.1357, -0.0977], [0, 0, 0.4262, 0.1632]], datatype=float[8]):
wr := Vector(4, datatype=float[8]):
wi := Vector(4, datatype=float[8]):
z := Matrix([[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]], datatype=float[8]):
NAG:-f08pec(job, compz, ilo, ihi, h, wr, wi, z, 'n' = n):

wr;

wi;

	See Also
	Bai Z and Demmel J W (1989) On a block implementation of Hessenberg multishift iteration Internat. J. High Speed Comput. 1 97–112 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.