Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.
ζk is a scalar, wk is a (k-1) element vector and zk is an (n-m) element vector. wk must be supplied in the kth row of A in elements Ak1,…,Akk-1. zk must be supplied in the kth row of A in elements Akm+1,…,Akn and ζk must be supplied either in Akk or in ZETAk, depending upon the parameter WHERET.
4 References
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
5 Parameters
1: WHERET – CHARACTER*1Input
On entry: indicates where the elements of ζ are to be found.
Note: the second dimension of the array A
must be at least
max1,N.
On entry: the leading m by m strictly lower triangular part of the array A, and the m by (n-m) rectangular part of A with top left-hand corner at element A1M+1 must contain details of the matrix P. In addition, when WHERET='I', then the diagonal elements of A must contain the elements of ζ.
On exit: the first NROWP rows of the array A are overwritten by the first NROWP rows of the n by n orthogonal matrix PT.
6: LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F01QKF is called.
Constraint:
LDA≥max1,M,NROWP.
7: ZETA(*) – double precision arrayInput
Note: the dimension of the array ZETA
must be at least
max1,M if WHERET='S', and at least 1 otherwise.
On entry: with WHERET='S', the array ZETA must contain the elements of ζ. If ZETAk=0.0 then Pk is assumed to be I, otherwise ZETAk is assumed to contain ζk.
When WHERET='I', the array ZETA is not referenced.
Note: the dimension of the array WORK
must be at least
maxM-1,NROWP-M,1.
9: IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to 0, -1 or 1. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
On exit: IFAIL=0 unless the routine detects an error (see Section 6).
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1 or 1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is 0. When the value -1 or 1 is used it is essential to test the value of IFAIL on exit.
6 Error Indicators and Warnings
If on entry IFAIL=0 or -1, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
IFAIL=-1
On entry,
WHERET≠'I' or 'S',
or
M<0,
or
N<M,
or
NROWP<0 or NROWP>N,
or
LDA<maxM,NROWP.
7 Accuracy
The computed matrix P satisfies the relation
P=Q+E,
where Q is an exactly orthogonal matrix and
E≤cε,
ε is the machine precision (see X02AJF), c is a modest function of n, and . denotes the spectral (two) norm. See also Section 7 in F01QJF.
8 Further Comments
The approximate number of floating-point operations is given by
23m3n-m2ℓ-m-mℓ-m,if ℓ≥m, and 23ℓ23n-ℓ,if ℓ<m.
9 Example
This example obtains the 5 by 5 orthogonal matrix P following the RQ factorization of the 3 by 5 matrix A given by