Matrix(1..m, 1..tda, datatype=float[8], order=C_order);

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NAG[f01qcc] NAG[nag_real_qr] - factorization of real by matrix

Calling Sequence

f01qcc(a, zeta, 'm'=m, 'n'=n, 'tda'=tda, 'fail'=fail)

nag_real_qr(. . .)

Parameters

a - Matrix(1..m, 1..tda, datatype=float[8], order=C_order);

On entry: the leading by part of the array a must contain the matrix to be factorized.

On exit: the by upper triangular part of a will contain the upper triangular matrix and the by strictly lower triangular part of a will contain details of the factorization as described in Section [Description]

zeta - Vector(1..n, datatype=float[8]);

On exit: contains the scalar for the th transformation. If then zeta, otherwise contains as described in Section [Description] and is always in the range .

'm'=m - integer; (optional)

Default value: the first dimension of the array a.

On entry: , the number of rows of .

Constraint: . .

'n'=n - integer; (optional)

Default value: the first dimension of the array zeta and the second dimension of the array zetathe array a.

On entry: , the number of columns of .

When then an immediate return is effected.

Constraint: . .

'tda'=tda - integer; (optional)

On entry: the second dimension of the array a as declared in the function from which nag_real_qr (f01qcc) is called.

Constraint: . .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_real_qr (f01qcc) finds the factorization of the real by matrix , where .

Description

The by matrix is factorized as

[[A=Q([[R],[0]]), m>n ],[A=QR, m=n ]]

where is an by orthogonal matrix and is an by upper triangular matrix. The factorization is obtained by Householder's method. The th transformation matrix, , which is used to introduce zeros into the th column of is given in the form

Q[k] = (Matrix(2, 2, {(1, 1) = I, (1, 2) = 0, (2, 1) = 0, (2, 2) = T[k]}))

where

u[k] = binomial(Zeta[k], z[k])

is a scalar and is an element vector. and are chosen to annihilate the elements below the triangular part of .

The vector is returned in the th element of the array zeta and in the th column of a, such that is in and the elements of are in . The elements of are returned in the upper triangular part of a. is given by

Good background descriptions to the factorization are given in Dongarra et al. (1979) and Golub and Van Loan (1996).

	Error Indicators and Warnings
	"NE_2_INT_ARG_LT" On entry, while . These arguments must satisfy . "NE_INT_ARG_LT" On entry, n must not be less than 0: .

	Accuracy
	The computed factors and satisfy the relation where , and is the machine precision, is a modest function of and and denotes the spectral (two) norm.

	Further Comments
	The approximate number of floating-point operations is given by .

Examples

>	m := 5: n := 3: tda := 3: a := Matrix([[2, 2.5, 2.5], [2, 2.5, 2.5], [1.6, -0.4, 2.8], [2, -0.5, 0.5], [1.2, -0.3, -2.9]], datatype=float[8], order='C_order'): zeta := Vector(3, datatype=float[8]): NAG:-f01qcc(a, zeta, 'm' = m, 'n' = n, 'tda' = tda):

zeta;

	See Also
	Dongarra J J, Moler C B, Bunch J R and Stewart G W (1979) LINPACK Users' Guide SIAM, Philadelphia Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore Wilkinson J H (1965) The Algebraic Eigenvalue Problem Oxford University Press, Oxford f01 Chapter Introduction. NAG Toolbox Overview. NAG Web Site.