Problem 3. Let f(r) be the piecewise linear function on [-1, 1] that consists of four line segments connecting the points (-1,-1) and (-0.5, 1); (-0.5, 1) and (0, -1); (0, -1) and (0.5, 1); (0.5,1) and (1,-1). Write a code that computes the barycentric Lagrange form of the interpolating polynomial pn (x) of degree n for n+ 1 Chebyshev nodes. The code should produce pictures similar to Fig. 5.1 in Berrut et al for your piecewise function f(x). Submit your pictures for p20 (x), and p100(x). Exercise 6.9 (Lyche and Merrien) and the paper by Berrut et al are useful. Problem 4. Add a few lines to your code in Problem 3 to compute the numer- ical error of approximation: en := _max | f(x) - pn(x)|. One way to do it is to introduce a very fine uniform grid {t}, where m is very large. Then max f(x)-Pn(x) max f(k) - Pn(k). Submit a table that has two columns: n, en, for n = 2¹, 1 = 1,...,7. The table will have seven rows. Problem 5. Based on the table in Problem 4, decide on numerical convergence of pn to f. Prove or disprove that f(x) is Lipschitz continuous. Based on your answer, discuss theoretical convergence of pn to f.
Problem 3. Let f(r) be the piecewise linear function on [-1, 1] that consists of four line segments connecting the points (-1,-1) and (-0.5, 1); (-0.5, 1) and (0, -1); (0, -1) and (0.5, 1); (0.5,1) and (1,-1). Write a code that computes the barycentric Lagrange form of the interpolating polynomial pn (x) of degree n for n+ 1 Chebyshev nodes. The code should produce pictures similar to Fig. 5.1 in Berrut et al for your piecewise function f(x). Submit your pictures for p20 (x), and p100(x). Exercise 6.9 (Lyche and Merrien) and the paper by Berrut et al are useful. Problem 4. Add a few lines to your code in Problem 3 to compute the numer- ical error of approximation: en := _max | f(x) - pn(x)|. One way to do it is to introduce a very fine uniform grid {t}, where m is very large. Then max f(x)-Pn(x) max f(k) - Pn(k). Submit a table that has two columns: n, en, for n = 2¹, 1 = 1,...,7. The table will have seven rows. Problem 5. Based on the table in Problem 4, decide on numerical convergence of pn to f. Prove or disprove that f(x) is Lipschitz continuous. Based on your answer, discuss theoretical convergence of pn to f.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Problem 3. Let f(r) be the piecewise linear function on [1,1] that consists
of four line segments connecting the points
(-1,-1) and (-0.5, 1); (-0.5, 1) and (0, -1);
(0, -1) and (0.5, 1); (0.5,1) and (1,-1).
Write a code that computes the barycentric Lagrange form of the interpolating
polynomial pn (x) of degree n for n+ 1 Chebyshev nodes. The code should
produce pictures similar to Fig. 5.1 in Berrut et al for your piecewise function
f(x). Submit your pictures for p20 (x), and p100 (x). Exercise 6.9 (Lyche and
Merrien) and the paper by Berrut et al are useful.
Problem 4. Add a few lines to your code in Problem 3 to compute the numer-
ical error of approximation:
en = max f(x) - pn (x).
One way to do it is to introduce a very fine uniform grid {t}, where m is
very large. Then
max f(x) pn(x) max f(k) - Pn (Tk)|.
-1551
1≤k≤m
....
Submit a table that has two columns: n, en, for n = 2', l = 1,.
will have seven rows.
7. The table
Problem 5. Based on the table in Problem 4, decide on numerical convergence
of pn to f. Prove or disprove that f(x) is Lipschitz continuous. Based on your
answer, discuss theoretical convergence of pn to f.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2dde791b-4e50-43ae-95b2-a5c8fabc78e8%2Fb5d4201e-77e9-486d-a1f2-d4d40f459886%2Fuh4mgjk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 3. Let f(r) be the piecewise linear function on [1,1] that consists
of four line segments connecting the points
(-1,-1) and (-0.5, 1); (-0.5, 1) and (0, -1);
(0, -1) and (0.5, 1); (0.5,1) and (1,-1).
Write a code that computes the barycentric Lagrange form of the interpolating
polynomial pn (x) of degree n for n+ 1 Chebyshev nodes. The code should
produce pictures similar to Fig. 5.1 in Berrut et al for your piecewise function
f(x). Submit your pictures for p20 (x), and p100 (x). Exercise 6.9 (Lyche and
Merrien) and the paper by Berrut et al are useful.
Problem 4. Add a few lines to your code in Problem 3 to compute the numer-
ical error of approximation:
en = max f(x) - pn (x).
One way to do it is to introduce a very fine uniform grid {t}, where m is
very large. Then
max f(x) pn(x) max f(k) - Pn (Tk)|.
-1551
1≤k≤m
....
Submit a table that has two columns: n, en, for n = 2', l = 1,.
will have seven rows.
7. The table
Problem 5. Based on the table in Problem 4, decide on numerical convergence
of pn to f. Prove or disprove that f(x) is Lipschitz continuous. Based on your
answer, discuss theoretical convergence of pn to f.
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