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4. The Runge Example. Let f(x) = 1 / (1 + 25x^2) x ? [-1, 1]. (3) Using your Barycentric Formula code (Prob. 3) and (4) and (5) below, evaluate and plot the interpolating polynomial p_n of f corresponding to (a) The equidistributed nodes x_j = -1 + j(2/n), j = 0, …, n for n = 4, 8, and 12. (b) the Chebyshev nodes x_j = cos(j??/n), j = 0, …, n for n = 4, 8, 12, and 100. As seen in class, for equidistributed nodes one can use the barycentric weights ?_j = (-1)^j (n over j) j = 0, …, n, (4) and for the Chebyshev nodes we can use ?_j = { 1/2(-1)^j for j = 0 or j = n, (-1)^j j = 1, …, n - 1. (5) Make sure to employ (4) and (5) in your Barycentric Formula code for this prob- lem. To plot the corresponding p_n evaluate this at sufficiently large number of points n_e as in Prob. 2. Note that your Barycentric Formula cannot be used to evaluate p_n when x coincides with an interpolating node! Plot also f for compar- ison. (c) Plot the error e_n = f - p_n for (a) and (b) and comment on the results. (d) Repeat (a) for f(x) = e^-x^2 for x ? [-1, 1] and comment on the result.

          4. The Runge Example. Let
f(x) = 1 / (1 + 25x^2) x ? [-1, 1]. (3)
Using your Barycentric Formula code (Prob. 3) and (4) and (5) below, evaluate and
plot the interpolating polynomial p_n of f corresponding to
(a) The equidistributed nodes x_j = -1 + j(2/n), j = 0, …, n for n = 4, 8, and 12.
(b) the Chebyshev nodes x_j = cos(j??/n), j = 0, …, n for n = 4, 8, 12, and 100.
As seen in class, for equidistributed nodes one can use the barycentric weights
?_j = (-1)^j (n over j) j = 0, …, n, (4)
and for the Chebyshev nodes we can use
?_j = { 1/2(-1)^j for j = 0 or j = n,
(-1)^j j = 1, …, n - 1. (5)
Make sure to employ (4) and (5) in your Barycentric Formula code for this prob-
lem. To plot the corresponding p_n evaluate this at sufficiently large number of
points n_e as in Prob. 2. Note that your Barycentric Formula cannot be used to
evaluate p_n when x coincides with an interpolating node! Plot also f for compar-
ison.
(c) Plot the error e_n = f - p_n for (a) and (b) and comment on the results.
(d) Repeat (a) for f(x) = e^-x^2 for x ? [-1, 1] and comment on the result.

4. The Runge Example. Let
f(x) = 1 / (1 + 25x^2) x ? [-1, 1]. (3)
Using your Barycentric Formula code (Prob. 3) and (4) and (5) below, evaluate and
plot the interpolating polynomial pn of f corresponding to
(a) The equidistributed nodes xj = -1 + j(2/n), j = 0, …, n for n = 4, 8, and 12.
(b) the Chebyshev nodes xj = cos(j??/n), j = 0, …, n for n = 4, 8, 12, and 100.
As seen in class, for equidistributed nodes one can use the barycentric weights
?j = (-1)^j (n over j) j = 0, …, n, (4)
and for the Chebyshev nodes we can use
?j = 1/2(-1)^j for j = 0 or j = n,
(-1)^j j = 1, …, n - 1. (5)
Make sure to employ (4) and (5) in your Barycentric Formula code for this prob-
lem. To plot the corresponding pn evaluate this at sufficiently large number of
points ne as in Prob. 2. Note that your Barycentric Formula cannot be used to
evaluate pn when x coincides with an interpolating node! Plot also f for compar-
ison.
(c) Plot the error en = f - pn for (a) and (b) and comment on the results.
(d) Repeat (a) for f(x) = e^-x^2 for x ? [-1, 1] and comment on the result.

Added by Christopher P.

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