Unitary reduction of a pair of complex general matrices to generalized upper Hessenberg form

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NAG[f08wsc] NAG[nag_zgghrd] - Unitary reduction of a pair of complex general matrices to generalized upper Hessenberg form

Calling Sequence

f08wsc(compq, compz, n, ilo, ihi, a, b, q, z, 'fail'=fail)

nag_zgghrd(. . .)

Parameters

compq - String;

On entry: specifies the form of the computed unitary matrix .

Do not compute .

The unitary matrix is returned.

q must contain a unitary matrix , and the product is returned.

Constraint: "Nag_NotQ", "Nag_InitQ" or "Nag_UpdateSchur". .

compz - String;

On entry: specifies the form of the computed unitary matrix .

Do not compute .

The unitary matrix is returned.

z must contain a unitary matrix , and the product is returned.

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

n - integer;

On entry: , the order of the matrices and .

Constraint: . .

ilo - integer;
ihi - integer;

On entry: and as determined by a previous call to f08wvc (nag_zggbal). Otherwise, they should be set to and , respectively.

Constraints:

if , ;

if , and .

a - Matrix(1..dim1, 1..dim2, datatype=complex[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 matrix of the matrix pair . Usually, this is the matrix returned by f08auc (nag_zunmqr).

On exit: is overwritten by the upper Hessenberg matrix .

b - Matrix(1..dim1, 1..dim2, datatype=complex[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 upper triangular matrix of the matrix pair . Usually, this is the matrix returned by the factorization function f08asc (nag_zgeqrf).

On exit: the array b is overwritten by the upper triangular matrix .

q - Matrix(1..dim1, 1..dim2, datatype=complex[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 , q must contain a unitary matrix .

If , q is not referenced.

On exit: if , q contains the unitary matrix .

Iif , q is overwritten by .

z - Matrix(1..dim1, 1..dim2, datatype=complex[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 a unitary matrix .

If , z is not referenced.

On exit: if , z contains the unitary matrix .

If , z is overwritten by .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_zgghrd (f08wsc) reduces a pair of complex matrices , where is upper triangular, to the generalized upper Hessenberg form using unitary transformations.

Description

nag_zgghrd (f08wsc) is usually the third step in the solution of the complex generalized eigenvalue problem

The (optional) first step balances the two matrices using f08wvc (nag_zggbal). In the second step, matrix is reduced to upper triangular form using the factorization function f08asc (nag_zgeqrf) and this unitary transformation is applied to matrix by calling f08auc (nag_zunmqr).

nag_zgghrd (f08wsc) reduces a pair of complex matrices , where is triangular, to the generalized upper Hessenberg form using unitary transformations. This two-sided transformation is of the form

[[Q^HAZ=H],[Q^HBZ=T]]

where is an upper Hessenberg matrix, is an upper triangular matrix and and are unitary matrices determined as products of Givens rotations. They may either be formed explicitly, or they may be post multiplied into input matrices and , so that

[[Q[1]AZ[1]^H=(Q[1],Q)H(Z[1],Z)^H ],[Q[1]BZ[1]^H=(Q[1],Q)T(Z[1],Z)^H ]]

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_3"

On entry, , , . Constraint: if , and .

On entry, , , . Constraint: if , .

"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 reduction to the generalized Hessenberg form is implemented using unitary transformations which are backward stable.

	Further Comments
	This function is usually followed by f08xsc (nag_zhgeqz) which implements the algorithm for computing generalized eigenvalues of a reduced pair of matrices. The real analogue of this function is f08wec (nag_dgghrd).

Examples

compq := "Nag_NotQ":
compz := "Nag_NotZ":
n := 4:
ilo := 1:
ihi := 4:
a := Matrix([[-2.867890104378766 -1.594524360587801*I , -4.687788932894172 -1.335343722471056*I , -0.7905009462560725 +0.07944449994448058*I , -1.439015977717754 +0.8017696696524528*I ], [-2.515655332627996 +0.5597332416374018*I , -4.543587844404469 +0.03162421545477236*I , -0.9404703494183806 -0.1305718159625312*I , -2.391088344496183 -0.5474958021881112*I ], [0.9137894874444604 -0.09397719654058648*I , 1.001627625946434 +0.1053096496371167*I , -0.09620027967881147 +0.03545150990580874*I , -1.284793228244642 +0.04644135395523489*I ], [-0.08371480021894684 +0.004924400012879077*I , -0.0426781334449532 -0.01149026669671836*I , 0.0233909000611764 -0.002215980005795651*I , 0.09323530691051354 -0.0001641466670958877*I ]], datatype=complex[8]):
b := Matrix([[-1.774823934929885 +0*I , -4.744132549876073 +0.8845947866158473*I , -0.918964392974807 +0.3876440848354796*I , -1.548322594662642 +1.005733563126935*I ], [0.3603832255487836 +0.3603832255487836*I , -2.271717062641916 +0*I , -0.8105776924797566 -0.1026200900444423*I , -2.074151659012492 -0.4057262470104193*I ], [0.03603832255487836 +0.07207664510975673*I , 0.1493318102558879 -0.03084332363005017*I , -0.3174420096804637 +0*I , -1.435800103589889 -0.02051468561236519*I ], [0.03603832255487836 +0.1081149676646351*I , 0.3578350975272399 -0.04233510073426493*I , 0.9002179361271463 -0.07846747522359941*I , -0.0428422801120493 +0*I ]], datatype=complex[8]):
q := Matrix([[0 +0*I ]], datatype=complex[8]):
z := Matrix([[0 +0*I ]], datatype=complex[8]):
NAG:-f08wsc(compq, compz, n, ilo, ihi, a, b, q, z):

	See Also
	Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore Moler C B and Stewart G W (1973) An algorithm for generalized matrix eigenproblems SIAM J. Numer. Anal. 10 241–256 f08 Chapter Introduction. NAG Toolbox Overview. NAG Web Site.