Given the data points (-3,5),(-2,5),(-1,3), (0, 1) (a) Find the interpolating polynomial passing through these points....
(a) Find the degree 1 interpolating P(x) through the points (a, f(a)) and (b, f(b)) (b) Develop the following formula by using the interpolating polynomial P1 (x), (c) Find the degree of precision of the approximation, T1 (f) = f(1) + f(-1), for f(r)dr. (a) Find the degree 1 interpolating P(x) through the points (a, f(a)) and (b, f(b)) (b) Develop the following formula by using the interpolating polynomial P1 (x), (c) Find the degree of precision of the approximation,...
12. Given the data set: We want to find the interpolating polynomial of degree 2 through these points. a) Write the interpolating polynomial in Lagrange form b) Write the interpolating polynomial in Newton form.
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
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...
**********************matlab code please******************* 1. Interpolation error of polynomial fit Using 11 equi-distributed points (10 equal segments) in the interval [-1 1], Using Newton's form find and plot the interpolating polynomial p(x) for the function f(x) -1/(125x2). Comment on the large discrepancies between p(x) and the function f(x) that the data came from Write down an expression for the error in the interpolating polynomial above? Which part of the expression is responsible for the large errors observed? 1. Interpolation error of...
Polynomial Interpolation Determine analytically, what is the maximum error in interpolating the function e2x using 5 equispaced points on [-1,11? . Compare this with the upper bound using the 5 roots of T5(x) to interpolate e2* Construct and plot the actual pointwise interpolation error (by sampling at lots of points). Are either of your error bounds close? Polynomial Interpolation Determine analytically, what is the maximum error in interpolating the function e2x using 5 equispaced points on [-1,11? . Compare this...
3. (30 points) Let f(x) = 1/x and data points Zo = 2, x,-3 and x2 = 4. Note that you can use the abscissae to find the corresponding ordinates (a) (8 points) Find by hand the Lagrange form, the standard form, and the Newton form of the interpolating polynomial p2(x) of f(x) at the given points. State which is which! Then, expand out the Newton and Lagrange form to verify that they agree with the standard form of p2...
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,...
Consider the three points (-1,0), (0,1), (2,0) 1. Construct a second degree polynomial P(a) that interpolates the given points. Use Matlab to solve the resulting linear system. 2. Find a piecewise linear function L(x) that interpolates the given points. Consider the three points (-1,0), (0,1), (2,0) 1. Construct a second degree polynomial P(a) that interpolates the given points. Use Matlab to solve the resulting linear system. 2. Find a piecewise linear function L(x) that interpolates the given points.
this is numerical analysis 2. Consider the function f(x) = -21° +1. (a) Calculate the interpolating polynomial pz() for data using the nodes 2o = -1, 11 = 0, 12 = 1. Simplify the polynomial to standard form. Use the error theorem for polynomial interpolation to bound the error f(x) - P2(x) on the interval (-1,2). Is this bound realistic?