By Samuel Daniel Conte

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4: Interpolation using an increasing number of interpolation points Given distinct points x 0 , x 1 , x 2 , . . and the value! f(x0), f(x1), f(x2), . . of a function f(x) at these points. Also, given a point For k = 0, 1, 2, . . , until satisfied, do: This algorithm generates the entries of the divided-difference table for f(x) at x0 , x1 , x2 , . . a diagonal at a time. During the calculation of the upward diagonal emanating from f[xk+1] is calculated up to and including the number f[x0, .

R. [Hint: Under these conditions, p(x) can be written with q(x) some polynomial. 1-3 Find the first derivative of at x = 2. 1-3 at x = 2. 1-3. [Hint: The Taylor expansion for a polynomial around a point c is just the Newton form for this polynomial with centers c, c, c, c, . . 2. 1, p(x) = (x - z1)q(x), Now, to finish the proof by induction on the number k of zeros in the given sequence, prove that z2, . . , zk is necessarily a sequence of zeros (counting multiplicity) of q(x). For this, assume that the number z occurs exactly j times in the sequence z2, .

Specifically, we would like to measure how fast sequences converge. 6 SOME COMMENTS ON CONVERGENCE OF SEQUENCES 21 for some constant K and all sufficiently large n. 26) Thus while sin The order notation appears customarily only on the right-hand side of an equation and serves the purpose of describing the essential feature of an error term without bothering about multiplying constants or other detail. , the series converges to as fast as 1/n (goes to zero) but no faster. A convergence order or rate of l/n is much too slow to be useful in calculations.