The linear inequality method for rational approximation with interpolation

Journal of Computational and Applied Mathematics 10 (1984) 383 North-Holland

383

Letter Section

The linear inequality method for rational approximation with interpolation Charles B. D U N H A M Computer Science Department, University of Western Ontario, London, Ontario N6A 5B9, Canada

Writing all the inequalities as linear inequalities yields

f(z)q(z) Received 15 April 1983

f ( x ) q ( x ) - p ( x ) <~Eq(x),

Abstract: The linear inequality method for rational minimax approximation is extended to cover approximation under Lagrange-type interpolatory constraints. Let X be a finite set and for h a function on X define Ilhll= m a x ( I h ( x ) l : x ~ g ) . Let R be a set of generalized rationals ( p / q : q > 0) on X, as treated in [2, Chapter 5]. Let Z be a subset of X. The problem of approximation with (Lagrange-type) interpolation is, given a function f on X, to choose p * / q * to minimize I l f - p / q l l , subject to the constraints q > 0 and

p(z)/q(z)=f(z),

z~Z.

(1)

Such a ratio p * / q * is called best to f with interpolation on Z [1, 50fq. A n algorithm for ordinary Chebyshev approximation that can be adapted is the linear inequality method, as described in [2, p. 170]. It might be helpful for the reader to check Cheney's a c c o u n t at this time. Given ~ > 0, we seek p / q satisfying the above constraints such that

[[f - p/ql[ <~~.

(2)

As in [2], the constraint q > 0 can be replaced by the constraint q >~ 1. Further, as in [2], (2) can be replaced by

- E q <~f q - p <~cq, while the constraint (1) becomes f(z)q(z)-p(z)=O,

- p ( z ) = 0,

z~Z.

-f(x)q(x)

+ p ( x ) <~cq(x),

-q(x)

<~1,

z~Z, x~X-Z, xEX-Z,

(3)

xEX.

The system of inequalities (3) is either consistent or inconsistent. If it is consistent, we reduce c to try to get a m i n i m u m e for which (3) is consistent. If (3) is inconsistent, we increase ~. A modification of the approximation problem is to choose functions u and v such that 0 < u ~< v and then constrain q to satisfy u ~< q ~< v. This p r o b l e m (without interpolation) was studied by the author in [3]. This problem with interpolation has not hitherto been considered. The algorithm is the same except that the b o t t o m inequality of (3) is replaced by q ( x ) < v(x),

-q(x)<

-u(x),

x • X,

x~X.

References

[1] B.L. Chalmers and G.D. Taylor, Uniform approximation with constraints, Jber. Deutsch. Math.-Verein. 81 (1979) 49-86.

[2] E.W. Cheney, Introduction to Approximation Theory (McGraw-Hill, New York, 1966). [3] C.B. Dunham, Chebyshev approximation by rationals with

constrained denominators, J. Approx. Theory, to appear. [4] C.M. Lee and F.D.K. Roberts, A comparison of algorithms for rational loo approximation, Math. Comput. 27 (1973) 111-121.

0377-0427/84/$3.00 © 1984, Elsevier Science Publishers B.V. (North-Holland)