Orthogonal reduction of a pair of real general matrices to generalized upper Hessenberg form

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NAG[f08wec] NAG[nag_dgghrd] - Orthogonal reduction of a pair of real general matrices to generalized upper Hessenberg form

Calling Sequence

f08wec(compq, compz, ilo, ihi, a, b, q, z, 'n'=n, 'fail'=fail)

nag_dgghrd(. . .)

Parameters

compq - String;

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

Do not compute .

The orthogonal matrix is returned.

q must contain an orthogonal matrix , and the product is returned.

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

compz - String;

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

Do not compute .

The orthogonal matrix is returned.

z must contain an orthogonal matrix , and the product is returned.

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

ilo - integer;
ihi - integer;

On entry: and as determined by a previous call to f08whc (nag_dggbal). Otherwise, they should be set to and , respectively.

Constraints:

if , ;

if , and .

a - 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 matrix of the matrix pair . Usually, this is the matrix returned by f08agc (nag_dormqr).

On exit: is overwritten by the upper Hessenberg matrix .

b - 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 upper triangular matrix of the matrix pair . Usually, this is the matrix returned by the factorization function f08aec (nag_dgeqrf).

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

q - 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 , q must contain an orthogonal matrix .

If , q is not referenced.

On exit: if , q contains the orthogonal matrix .

Iif , q is overwritten by .

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 an orthogonal matrix .

If , z is not referenced.

On exit: if , z contains the orthogonal matrix .

If , z is overwritten by .

'n'=n - integer; (optional)

Default value: the first dimension of the array z.

On entry: , the order of the matrices and .

Constraint: . .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_dgghrd (f08wec) reduces a pair of real matrices , where is upper triangular, to the generalized upper Hessenberg form using orthogonal transformations.

Description

nag_dgghrd (f08wec) is the third step in the solution of the real generalized eigenvalue problem

The (optional) first step balances the two matrices using f08whc (nag_dggbal). In the second step, matrix is reduced to upper triangular form using the factorization function f08aec (nag_dgeqrf) and this orthogonal transformation is applied to matrix by calling f08agc (nag_dormqr).

nag_dgghrd (f08wec) reduces a pair of real matrices , where is upper triangular, to the generalized upper Hessenberg form using orthogonal transformations. This two-sided transformation is of the form

[[Q^TAZ=H],[Q^TBZ=T]]

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

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

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 orthogonal transformations which are backward stable.

	Further Comments
	This function is usually followed by f08xec (nag_dhgeqz) which implements the algorithm for computing generalized eigenvalues of a reduced pair of matrices. The complex analogue of this function is f08wsc (nag_zgghrd).

Examples

compq := "Nag_NotQ":
compz := "Nag_NotZ":
n := 5:
ilo := 1:
ihi := 5:
a := Matrix([[-2.189810707792021, -3.860243234889781, -0.7832784454794537, -1.868357725686654, -5.398725629594945], [-0.6852853888947079, -3.288784152686995, -1.438693822365931, -6.499961199366684, -30.43126686913852], [-0.4675345497569942, -0.9460296237838147, 0.05890247520307763, 1.68772079635865, 11.83388168132396], [0.1275112017793676, 0.08562118570231017, -0.08916578356814751, -0.5024032653213368, -1.524287626980205], [-0.01628919921740115, 0.0108594661449348, 0.01466027929566115, 0.03366434504929487, -0.05049651757395134]], datatype=float[8]):
b := Matrix([[-1.424780684877501, -4.604217385613993, -1.088588592238918, -2.347027886812987, -4.825304043612226], [0.4124084319199036, -3.498740096950594, -2.422558190366834, -9.651981131278539, -28.51976667786654], [0.04124084319199037, 0.1104973584069892, 0.6928828841397811, 3.972051174738771, 13.55775258164351], [0.04124084319199037, 0.2780329357492619, 0.015085691276236, 0.3071315763993982, 1.792521439103921], [0.04124084319199037, 0.6131040904338072, -0.8759673660952852, 0.1236241913362016, 0.03257839843480646]], datatype=float[8]):
q := Matrix([[0]], datatype=float[8]):
z := Matrix([[0]], datatype=float[8]):
NAG:-f08wec(compq, compz, ilo, ihi, a, b, q, z, 'n' = n):

	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.