3. For any function ƒ and n + 1 distinct nodes x0, …, xn, Lagrange interpolation factors f = P +

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3. For any function ƒ and n + 1 distinct nodes $x_0, \dots, x_n$, Lagrange interpolation factors $f = P + R$ where $P: x \mapsto \sum_{i=0}^n f(x_i) \prod_{\substack{j=0 \ j \neq i}}^n \frac{x - x_j}{x_i - x_j}$ and R is the approximation error. The following Matlab function, function a = linterp_poly(X, Y) a = zeros(1, numel(X)); for i = 1:numel(X) aa = poly(X([1:i-1 i+1:end])); a = a + Y(i) * aa / polyval(aa, X(i)); end end represents $P(x) = \sum_{k=0}^n a_k x^k$ as its coefficients a = [$a_n, a_{n-1}, \dots, a_1, a_0$] (this solves Worksheet 7 Q2.) (a) Write a Matlab function that uses P to approximate $f^{(d)}(x)$ for any positive integer d and real number x. The function signature should be function y = num_diff(X, Y, x, d) and should invoke both linterp_poly (see above) and Matlab's polyder. Test your code with by con- structing a problem where you expect the answer to be exact (up to rounding errors). What happens when d equals or exceeds the number of given interpolation nodes (n + 1)?

          3. For any function ƒ and n + 1 distinct nodes $x_0, \dots, x_n$, Lagrange interpolation factors $f = P + R$ where

$P: x \mapsto \sum_{i=0}^n f(x_i) \prod_{\substack{j=0 \ j \neq i}}^n \frac{x - x_j}{x_i - x_j}$

and R is the approximation error. The following Matlab function,
function a = linterp_poly(X, Y)
a = zeros(1, numel(X));
for i = 1:numel(X)
    aa = poly(X([1:i-1 i+1:end]));
    a = a + Y(i) * aa / polyval(aa, X(i));
end
end
represents $P(x) = \sum_{k=0}^n a_k x^k$ as its coefficients a = [$a_n, a_{n-1}, \dots, a_1, a_0$] (this solves Worksheet 7 Q2.)
(a) Write a Matlab function that uses P to approximate $f^{(d)}(x)$ for any positive integer d and real number x.
The function signature should be
function y = num_diff(X, Y, x, d)
and should invoke both linterp_poly (see above) and Matlab's polyder. Test your code with by con-
structing a problem where you expect the answer to be exact (up to rounding errors). What happens when
d equals or exceeds the number of given interpolation nodes (n + 1)?

3. For any function ƒ and n + 1 distinct nodes x0, …, xn, Lagrange interpolation factors f = P + R where

P: x ↦∑i=0^n f(xi) ∏j=0 j ≠ i^n (x - xj)/(xi - xj)

and R is the approximation error. The following Matlab function,
function a = linterppoly(X, Y)
a = zeros(1, numel(X));
for i = 1:numel(X)
aa = poly(X([1:i-1 i+1:end]));
a = a + Y(i) * aa / polyval(aa, X(i));
end
end
represents P(x) = ∑k=0^n ak x^k as its coefficients a = [an, an-1, …, a1, a0] (this solves Worksheet 7 Q2.)
(a) Write a Matlab function that uses P to approximate f^(d)(x) for any positive integer d and real number x.
The function signature should be
function y = numdiff(X, Y, x, d)
and should invoke both linterppoly (see above) and Matlab's polyder. Test your code with by con-
structing a problem where you expect the answer to be exact (up to rounding errors). What happens when
d equals or exceeds the number of given interpolation nodes (n + 1)?

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