Six Myths of Polynomial Interpolation and Quadrature

I t is a pleasure to o er this essay for Mathematics Today as a record of my Summer Lecture on 29 June 2011 at the Royal Society. Polynomials are as basic a topic as any in mathematics, and for numerical mathematicians like me, they are the starting point of numerical methods that in some cases go back centuries, like quadrature formulae for numerical integration and Newton iterations for finding roots. You would think that by now, the basic facts about computing with polynomials would be widely understood. In fact, the situation is almost the reverse. There are indeed widespread views about polynomials, but some of the important ones are wrong, founded on misconceptions entrenched by generations of textbooks. Since 2006, my colleagues and I have been solving mathematical problems with polynomials using the Chebfun software system (www.maths.ox.ac.uk/chebfun). We have learned from daily experience how fast and reliable polynomials are. This entirely positive record has made me curious to try to pin down where these misconceptions come from, and this essay is an attempt to summarise some of my findings. Full details, including precise definitions and theorems, can be found in my draft book Approximation Theory and Approximation Practice, available at www.maths.ox.ac.uk/~trefethen. The essay is organised around ‘six myths’. Each myth has some truth in it – mathematicians rarely say things that are simply false! Yet each one misses something important. Throughout the discussion, f is a continuous function defined on the interval [ 1, 1], n + 1 distinct points x0, . . . , xn in [ 1, 1] are given, and p