This short note provides and proves an easy algorithm to find a basic feasible solution for the Simplex Algorithm. The method uses a rule similar to Bland's rule for the initial phase of the algorithm.

Let E be a compact set in C with connected regular complement and let p(n), n is an element of N, be a sequence of polynomials which converge maximally to a fixed rational function f on E. Then p(n) has n + o(n) interpolation points to f in C and the normalized counting measure nu(n) of these interpolation points (resp. its balayage measure (nu) over cap (n) onto the boundary of E) converges to the equilibrium measure of E as n -> infinity. Furthermore, we prove a complete characterization of maximal convergence by interpolation.

Given a compact set E in \(\mathbb{C}\) and a function f holomorphic on E, we investigate the distribution of zeros of rational uniform approximants \(\{{r}_{n,{m}_{n}}\}\) with numerator degree≤n and denominator degree≤m n , where \({m}_{n} = o(n/\log n)\) as n→∞. We obtain a Jentzsch–Szegő type result, i.e., the zero distribution converges weakly to the equilibrium distribution of the maximal Green domain Eρ(f) of meromorphy of f if f has a singularity of multivalued character on the boundary ∂Eρ(f). Further, we show that any singular point of f on the boundary ∂Eρ(f), that is not a pole, is a limit point of zeros of the sequence \(\{{r}_{n,{m}_{n}}\}\).

Let D be a region, { r n } n ∈ N a sequence of rational functions of degree at most n and let each r n have at most m poles in D , for m ∈ N fixed. We prove that if { r n } n ∈ N converges geometrically to a function f on some continuum S ⊂ D and if the number of zeros of r n in any compact subset of D is of growth o ( n ) as n → ∞ , then the sequence { r n } n ∈ N converges m 1 -almost uniformly to a meromorphic function in D . This result about meromorphic continuation is used to obtain Picard-type theorems for the value distribution of m 1 -maximally convergent rational functions, especially in Padé approximation and Chebyshev rational approximation.

We study the problem of constructing a NURBS (nonuniform rational B-spline curve) with a minimal number of knots forming a given set of continuously resp. smoothly connected circular arcs. In the continuous case, NURBS defined by a minimal number of conditions can always be constructed. In the case when the circular arcs are smoothly connected, we give a characterization of the existence of minimal NURBS.

Let f ∈ L w 1 [−1, 1], let rn,m(f) be the best rational L w 1 -approximation for f with respect to real rational functions of degree at most n in the numerator and of degree at most m in the denominator, let m = m(n), and let limn → ∞ (n-m(n)) = ∞. In this case, we show that the counting measures of certain subsets of sign changes of f-r n,m (f) converge weakly to the equilibrium measure on [−1, 1] as n → ∞. Moreover, we prove estimates for discrepancy between these counting measures and the equilibrium measure.

The distribution of equi-oscillation points (alternation points) for the error in best Chebyshev approximation on [−1,1] by rational functions is investigated. In general, the alternation points need not be dense in [−1,1] when rational functions of degree (n, m) are considered and asymptotically n/m → κ with κ ≥ 1. We show that the asymptotic behavior of the alternation points is closely related to the behavior of the poles of the rational approximants. Hence, poles of the rational approximations are attracting points of alternations such that the well-known equi-distribution for the polynomial case can be heavily disturbed.

Let E be a compact set in C with connected complement and positive logarithmic capacity. For any f continuous on E and analytic in the interior of E , we consider the distribution of extreme points of the error of best uniform polynomial approximation on E . Let Λ =( n j ) be a subsequence of N such that n j +1 / n j →1. If, for n ∈ Λ , A n ( f)⊆ ∂ E denotes the set of extreme points of the error function, we prove that there is a subsequence Λ ′ of Λ such that the distribution of any ( n +2)th Fekete point set F n+2 of A n ( f) tends weakly to the equilibrium distribution on E as n →∞ in Λ ′. Furthermore, we prove a discrepancy result for the distribution of the point sets F n+2 if the boundary of E is smooth enough.

For \(f \in L_{^w }^1 \left[ { - 1{\text{,}}1} \right]\) denote by \(B_{n,1} \left( f \right),n = 0,1,...a\) a polynomial of best weighted \(L^1 \)-approximation to \(f\) on \(\left[ { - 1{\text{,}}1} \right]\) of degree \( \leqq n\). In the present paper, we are dealing with the asymptotic distribution of sign changes of the error functions \(f - B_{n,1} \left( f \right)\).

We investigate a problem posed by Poreda on the behaviour of the strong uniqueness constant with increased polynomial degree. It has been conjectured by Bartelt and Mclaughlin that this constant tends to zero for all non-polynomial functionsIn this paper, we give evidence for this and prove a special result, which we conjecture to be a worst case result. $

We show that interpolation to a function, analytic on a compact setE in the complex plane, can yield maximal convergence only if a subsequence of the interpolation points converges to the equilibrium distribution onE in the weak sense. Furthermore, we will derive a converse theorem for the case when the measure associated with the interpolation points converges to a measure onE, which may be different from the equilibrium measure.

Acknowledgments The author wishes to thank Prof. Dr. H.-P. Blatt for his continuous support of this work. Without his profound knowledge this paper would not have been possible. Also his friendliness, friendship and last, but not least, his patience helped a lot. The author dedicates this paper to his wife and his daughter, who was born during this work, and contributed to it by her smile. Also the University of Eichst att deserves to be mentioned for giving me the possibility to work and teach. The typesetting of this paper and the computations were done on an IBM RS 6000 computer at the Lehrstuhl f ur Angewandte Mathematik. 2 Table of content x1. Introduction x2. Interpolation x3. Proofs of the Results in Section 2 x4. Zeros of Polynomials and Overconvergence x5. Proof of Theorem 4.1 x6. Erd} os-Tur an Theorems x7. Proofs of the Results in Section 6 x8. Estimates for Maximally Convergent Polynomials x9. Random Polynomials x10. Numerical Examples 3 x1. Introduction The relation between zeros of polynomials and approximation theory is the topic of this work. The study of this relation is a very old one. It started with the investigations of Walsh, Fekete and others on interpolation. This makes it clear that this paper should also start with interpolation. The results of Section 2 are extensions of classical results. Their purpose is to study the relation- ship between the location of the interpolation points and the speed of convergence of the interpolating polynomials. In the real case this is expressed by the well known formula

We discuss best segment approximation (with free knots) by polynomials to piecewise analytic functions on a real interval. It is shown that, if the degree of the polynomials tends to infinity and the number of knots is the same as the number of singularities of the function, then the optimal knots converge geometrically fast to the singularities. When the degree is held fixed and the number of knots tends to infinity, we study the asymptotic distribution of the optimal knots.

Erdös and Turán established in [4] a qualitative result on the distribution of the zeros of a monic polynomial, the norm of which is known on [−1, 1]. We extend this result to a polynomial bounded on a systemE of Jordan curves and arcs. If all zeros of the polynomial are real, the estimates are independent of the number of components ofE for any regular compact subsetE ofR. As applications, estimates for the distribution of the zeros of the polynomials of best uniform approximation and for the extremal points of the optimal error curve (generalizations of Kadec's theorem) are given.