T47:02 731 VPN 97% 5 TOⓇ + : 4. Find the Hermite interpolating polynomial which interpolates...
T47:02 731 VPN 97% 5 TOⓇ + : 6. Find the natural cubic spline to approximate f(0) = 32 – 1 at 3o = 0,01 = 0.25, 12 = 0.5, 23 = 0.75 and 24 = 1. 13
Given the data points (-3,5),(-2,5),(-1,3), (0, 1) (a) Find the interpolating polynomial passing through these points. (b) Using your polynomial from (a), evaluate P(1). (c) This polynomial interpolates the function f(x) = 24. Find an upper bound for the approximation in part (b).
4) (a) Find the function P(x) = a + bcos(m) + c sin(TX), which interpolates the data: 0.5 Ly Hint find a, b, and c so that P(O) 2, P(0.5)-5 and P(1)-4] (b) Find a Lagrange's quadratic polynomial interpolating the above data. 1 4
Please answer problem 4, thank you.
2. The polynomial p of degree n that interpolates a given function f at n+1 prescribed nodes is uniquely defined. Hence, there is a mapping f -> p. Denote this mapping by L and show that rl Show that L is linear; that is, 3. Prove that the algorithm for computing the coefficients ci in the Newton form of the interpolating polynomial involves n long operations (multiplications and divisions 4. Refer to Problem 2,...
4. For the following table, answer the questions.
(1) Find the cubic Newton’s interpolating polynomial using the
first four data points and estimate the function value at x=2.5
with the interpolating polynomial.
(2) Find the quartic Newton’s interpolating polynomial using the
five data points and estimate the function value at x=2.5 with the
interpolating polynomial.
(3) Find the bases functions of Lagrange interpolation, Li(x)
(i=1,2,…,5), and estimate the function value at x=2.5 with the
Lagrange interpolating polynomial.
3 5 1...
6. (25 pts) Find the osculating polynomial, P, interpolating the following table of data, and evaluate P(1): -1 2 f(x) f'(x -4 2 1 5 -4 f"(x) -12
6. (25 pts) Find the osculating polynomial, P, interpolating the following table of data, and evaluate P(1): -1 2 f(x) f'(x -4 2 1 5 -4 f"(x) -12
Step by step working is highly necessary and required, please Obtain the interpolating polynomial for the following points (3, 5), (5, 9), (7, 13), (9, 20), (11, 30) using Newton's forward difference formula and hence, evaluate f(5.9)
Problem 2. Given the data points (xi. yi), with xi 2 02 4 yil 5 1 1.25 find the following interpolating polynomials, and use MATLAB to graph both the interpolating polynomials and the data points: a) The piecewise linear Lagrange interpolating polynomialx) b) The piecewise quadratic Lagrange interpolating polynomial q(x) c) Newton's divided difference interpolation pa(x) of degree s 4
Problem 2. Given the data points (xi. yi), with xi 2 02 4 yil 5 1 1.25 find the following...
Using Lagrange interpolation, find degree two interpolating polynomial if following points are known (0, 1, 5), (2, 0, −3), (1, 2, 8), (−2, −1, 10), (−1, 0, 5
Using Lagrange interpolation, find degree two interpolating polynomial if following points are known (0, 1, 5), (2, 0, −3), (1, 2, 8), (−2, −1, 10), (−1, 0, 5) (2,3,1)