nag_complex_qr (f01rcc) finds the  factorization of the complex  by  matrix , where .

The  by  matrix  is factorized as

where  is an  by  unitary matrix and  is an  by  upper triangular matrix with real diagonal elements.

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

  is a scalar for which ,  is a real scalar and  is an  element vector. ,  and  are chosen to annihilate the elements below the triangular part of  and to make the diagonal elements real.

The scalar  and the vector  are returned in the th element of the array theta and in the th column of a, such that , given by

is in  and the elements of  are in . The elements of  are returned in the upper triangular part of .

 is given by

A good background description to the  factorization is given in Dongarra et al. (1979).

"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: .

The computed factors  and  satisfy the relation

where ,  being the machine precision,  is a modest function of  and  and  denotes the spectral (two) norm.

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

Following the use of this function the operations

where  is an  by  matrix, can be performed by calls to f01rdc (nag_complex_apply_q).

The operation  can be obtained by the call:

and  can be obtained by the call:

If  is a one-dimensional array (single column) then the argument  can be replaced by . See f01rdc (nag_complex_apply_q) for further details.

The first  columns of the unitary matrix  can either be obtained by setting  to the first  columns of the unit matrix and using the first of the above two calls, or by calling f01rec (nag_complex_form_q), which overwrites the  columns of  on the first  columns of the array a.  is obtained by the call:

If  is larger than , then  must have been declared to have at least  columns.