Lagrange interpolation is well-known to be susceptible to Runge's phenomenon. This is the occurrence of large oscillations in the interpolating polynomial near the endpoints of a set of data points, when the data points origin has no oscillatory behaviour there. Runge studied the function f(x) 1 1+25x2 (3.3) on the interval [1, 1], using data points interpolated at n + 1 equidistant points, xk, going from x01 to xn = 1. 1. Write a function xeq(n) which, for a given n, returns an array of n + 1 equidistant points in [−1,1]. 2. Produce a plot illustrating the Runge phenomenon for a given set of equidistant points (i.e., for n fixed). The figure should show the functions f(x) and P(x) for x = [-1, 1], as well as the set of discrete points (xk, f(x)), k = 0,,n, used to construct P. Comment on your results. 3. Produce a second figure showing the polynomials interpolating f at n + 1 equidistant points, for a range of values of n. How does P(x) change as the number of interpolation points increases?
Lagrange interpolation is well-known to be susceptible to Runge's phenomenon. This is the occurrence of large oscillations in the interpolating polynomial near the endpoints of a set of data points, when the data points origin has no oscillatory behaviour there. Runge studied the function f(x) 1 1+25x2 (3.3) on the interval [1, 1], using data points interpolated at n + 1 equidistant points, xk, going from x01 to xn = 1. 1. Write a function xeq(n) which, for a given n, returns an array of n + 1 equidistant points in [−1,1]. 2. Produce a plot illustrating the Runge phenomenon for a given set of equidistant points (i.e., for n fixed). The figure should show the functions f(x) and P(x) for x = [-1, 1], as well as the set of discrete points (xk, f(x)), k = 0,,n, used to construct P. Comment on your results. 3. Produce a second figure showing the polynomials interpolating f at n + 1 equidistant points, for a range of values of n. How does P(x) change as the number of interpolation points increases?
Algebra for College Students
10th Edition
ISBN:9781285195780
Author:Jerome E. Kaufmann, Karen L. Schwitters
Publisher:Jerome E. Kaufmann, Karen L. Schwitters
Chapter9: Polynomial And Rational Functions
Section9.4: Graphing Polynomial Functions
Problem 47PS
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![Lagrange interpolation is well-known to be susceptible to Runge's phenomenon. This is the
occurrence of large oscillations in the interpolating polynomial near the endpoints of a set of data
points, when the data points origin has no oscillatory behaviour there. Runge studied the function
f(x)
1
1+25x2
(3.3)
on the interval [−1, 1], using data points interpolated at n + 1 equidistant points, xk, going from
xo 1 to xn = 1.
1. Write a function xeq(n) which, for a given n, returns an array of n + 1 equidistant points in
[−1,1].
2. Produce a plot illustrating the Runge phenomenon for a given set of equidistant points (i.e.,
for n fixed). The figure should show the functions f(x) and P(x) for x = [−1,1], as well as
the set of discrete points (xk, f(x)), k = 0,...,n, used to construct P. Comment on your
results.
3. Produce a second figure showing the polynomials interpolating f at n + 1 equidistant points,
for a range of values of n. How does P(x) change as the number of interpolation points
increases?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3011c556-643e-4a01-b0e0-55d8cf24eddf%2F9d9c482a-c084-48ed-b17d-b73821066cf7%2Fccbp7uh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Lagrange interpolation is well-known to be susceptible to Runge's phenomenon. This is the
occurrence of large oscillations in the interpolating polynomial near the endpoints of a set of data
points, when the data points origin has no oscillatory behaviour there. Runge studied the function
f(x)
1
1+25x2
(3.3)
on the interval [−1, 1], using data points interpolated at n + 1 equidistant points, xk, going from
xo 1 to xn = 1.
1. Write a function xeq(n) which, for a given n, returns an array of n + 1 equidistant points in
[−1,1].
2. Produce a plot illustrating the Runge phenomenon for a given set of equidistant points (i.e.,
for n fixed). The figure should show the functions f(x) and P(x) for x = [−1,1], as well as
the set of discrete points (xk, f(x)), k = 0,...,n, used to construct P. Comment on your
results.
3. Produce a second figure showing the polynomials interpolating f at n + 1 equidistant points,
for a range of values of n. How does P(x) change as the number of interpolation points
increases?
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