Orthogonal reduction of real general matrix to upper Hessenberg form

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      - f08aec (nag_dgeqrf)
      - f08afc (nag_dorgqr)
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      - f08mec (nag_dbdsqr)
      - f08msc (nag_zbdsqr)
      - f08nec (nag_dgehrd)
      - f08nfc (nag_dorghr)
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      - f08nhc (nag_dgebal)
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      - f08nsc (nag_zgehrd)
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NAG[f08nec] NAG[nag_dgehrd] - Orthogonal reduction of real general matrix to upper Hessenberg form

Calling Sequence

f08nec(ilo, ihi, a, tau, 'n'=n, 'fail'=fail)

nag_dgehrd(. . .)

Parameters

ilo - integer;
ihi - integer;

On entry: if has been output 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.

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 by general matrix .

On exit: is overwritten by the upper Hessenberg matrix and details of the orthogonal matrix .

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

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

On exit: further details of the orthogonal matrix .

'n'=n - integer; (optional)

Default value: the dimension of the array a.

On entry: , the order of the matrix .

Constraint: . .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_dgehrd (f08nec) reduces a real general matrix to Hessenberg form.

Description

nag_dgehrd (f08nec) reduces a real general matrix to upper Hessenberg form by an orthogonal similarity transformation: .

The matrix is not formed explicitly, but is represented as a product of elementary reflectors (see the f08 Chapter Introduction for details). Functions are provided to work with in this representation (see Section [Further Comments]).

The function can take advantage of a previous call to f08nhc (nag_dgebal), which may produce a matrix with the structure:

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

where and are upper triangular. If so, only the central diagonal block , in rows and columns to , needs to be reduced to Hessenberg form (the blocks and will also be affected by the reduction). Therefore the values of and determined by f08nhc (nag_dgebal) can be supplied to the function directly. If f08nhc (nag_dgebal) has not previously been called however, then must be set to 1 and to .

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 computed Hessenberg matrix is exactly similar to a nearby matrix , where is a modestly increasing function of , and is the machine precision. The elements of themselves may be sensitive to small perturbations in or to rounding errors in the computation, but this does not affect the stability of the eigenvalues, eigenvectors or Schur factorization.

Further Comments

The total number of floating-point operations is approximately , where ; if and , the number is approximately .

To form the orthogonal matrix nag_dgehrd (f08nec) may be followed by a call to f08nfc (nag_dorghr):

nag_dorghr (ilo,ihi,a,tau,'n'=n)

To apply to an by real matrix nag_dgehrd (f08nec) may be followed by a call to f08ngc (nag_dormhr). For example,

nag_dormhr ("Nag_LeftSide","Nag_NoTrans",ilo,ihi,a,tau,c,'m'=m,'n'=n)

forms the matrix product .

The complex analogue of this function is f08nsc (nag_zgehrd).

Examples

>	n := 4: ilo := 1: ihi := 4: 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]): tau := Vector(3, datatype=float[8]): NAG:-f08nec(ilo, ihi, a, tau, 'n' = n):

tau;

	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.