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11) The first divided difference, at any rate, is a ratio of differences. 2-1 Prove that (x - xn). 2-l as 22-2 Calculate the limit of the formula for while all other points remain fixed. 2-3 Prove that the polynomial of degree < n which interpolates f(x) at n + 1 distinct points is f(x) itself in case f(x) is a polynomial of degree < n. 2-4 Prove that the kth divided difference p[x0, . . , xk] of a polynomial p(x) of degree < k is independent of the interpolation points x0, xl, . . , xk. 2-5 Prove that the kth divided difference of a polynomial of degree < k is 0.

1, for n = 4. The entries in the table are calculated, for example, column by column, according to the following algorithm. 2: Divided-difference table Given the first two columns of the table, containing x 0 , x 1 , . . , x n and, correspondingly, If this algorithm is carried out by hand, the following directions might be helpful. Draw the two diagonals from the entry to be calculated through its two neighboring entries to the left. If these lines terminate at f[xi] and f[xj], respectively, divide the difference of the two neighboring entries by the corresponding difference x j - x i to get the desired entry.

You find only additions/subtractions and multiplications involving X and numerical constants in that subprogram, with X appearing as a factor less than r times). How many function values would you have to check before you could be sure that the routine does indeed do what it is supposed to do (assuming no rounding errors in the calculation)? 1-9 For each of the following power series, exploit the idea of nested multiplication to find an efficient way for their evaluation. ) . 2 EXISTENCE AND UNIQUENESS OF THE INTERPOLATING POLYNOMIAL Let x0, x1, .

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