| SUBROUTINE F01BLF ( | M, N, T, A, LDA, AIJMAX, IRANK, INC, D, U, LDU, DU, IFAIL) |
| INTEGER | M, N, LDA, IRANK, INC(N), LDU, IFAIL |
| double precision | T, A(LDA,N), AIJMAX(N), D(M), U(LDU,N), DU(N) |
Householder's factorization with column interchanges is used in the decomposition
F=QU, where
F is
A with its columns permuted,
Q is the first
r columns of an
m by
m orthogonal matrix and
U is an
r by
n upper-trapezoidal matrix of rank
r. The pseudo-inverse of
F is given by
X where
If the matrix is found to be of maximum rank,
r=n,
U is a nonsingular
n by
n upper-triangular matrix and the pseudo-inverse of
F simplifies to
X=U-1QT. The transpose of the pseudo-inverse of
A is overwritten on
A.
Peters G and Wilkinson J H (1970) The least-squares problem and pseudo-inverses
Comput. J. 13 309–316
Wilkinson J H and Reinsch C (1971)
Handbook for Automatic Computation II, Linear Algebra Springer–Verlag
If on entry
IFAIL=0 or
-1, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
For most matrices the pseudo-inverse is the best possible having regard to the condition of
A and the choice of
T. Note that only the singular value decomposition method can be relied upon to give maximum accuracy for the precision of computation used and correct determination of the condition of a matrix (see
Wilkinson and Reinsch (1971)).
The most difficult practical problem is the determination of the rank of the matrix (see pages 314–315 of
Peters and Wilkinson (1970)); only the singular value decomposition method gives a reliable indication of rank deficiency (see pages 134–151 of
Wilkinson and Reinsch (1971) and
F08KBF (DGESVD)). In F01BLF a tolerance,
T, is used to recognize ‘zero’ elements in the remaining matrix at each step in the factorization. The value of
T should be set at
n times the bound on possible errors in individual elements of the original matrix. If the elements of
A vary widely in their orders of magnitude, of course this presents severe difficulties. Sound decisions can only be made by somebody who appreciates the underlying physical problem.
A complete program follows which outputs the maximum of the moduli of the ‘remaining’ elements at each step in the factorization, the rank, as determined by the given value of
T, and the transposed pseudo-inverse. Data and results are given for an example which is a
6 by
5 matrix of deficient rank in which the last column is a linear combination of the other four. Using
T=119ε (119 is the norm of the matrix) the rank is correctly determined as
4 and the pseudo-inverse is computed to full implementation accuracy.