Question

In: Advanced Math

Find the Chebyshev interpolation nodes on the interval [4,12] for an interpolating polynomial of degree 5

Expert Solution

The formula to calculate n Chebyshev nodes on an arbitrary interval [a,b] is given by

Since the interpolating polynomial is of degree 5 we will need 6 data points to find the unique polynomial of degree 5 that interpolates the data. The formula to calculate the six Chebyshev nodes required to find the degree 5 polynomial on the interval [4, 12] is given as

So we start with k = 1 and get

Next we have for k = 2, we get the Chebyshev node

Next for k = 3, the Chebyshev node can be calculated as

For k = 4, we have the fourth Chebyshev node to be calculated as

For k = 5, the fifth Chebyshve node can be calculated as

And lastly for k = 6, we get the final Chebyshev node as

Thus we have the six Chebyshev nodes to find an interpolating polynomial of degree 5 on the interval [4, 12] as

Colby Messinger answered 1 year ago

Consider polynomial interpolation of the function f(x)=1/(1+25x^2) on the interval [-1,1] by (1) an interpolating polynomial...

Consider polynomial interpolation of the function f(x)=1/(1+25x^2) on the interval [-1,1] by (1) an interpolating polynomial determined by m equidistant interpolation points, (2) an interpolating polynomial determined by interpolation at the m zeros of the Chebyshev polynomial T_m(x), and (3) by interpolating by cubic splines instead of by a polynomial. Estimate the approximation error by evaluation max_i |f(z_i)-p(z_i)| for many points z_i on [-1,1]. For instance, you could use 10m points z_i. The cubic spline interpolant can be determined in...

Use Lagrange interpolation to find the polynomial p3(x) of degree 3 or less, that agree with...

Use Lagrange interpolation to find the polynomial p3(x) of degree 3 or less, that agree with the following data: p3(−1) = 3, p3(0) = −4, p3(1) = 5, and p3(2) = −6. Using python to solve

Compute the quartic interpolating polynomial for the Hermite interpolation problem p(0) = 2, p'(0) = -9...

Compute the quartic interpolating polynomial for the Hermite interpolation problem p(0) = 2, p'(0) = -9 p(1) = -4, p'(1) = 4 p(2) = 44 with respect to the Newton basis. Compute the divided differences. Find a quintic interpolating polynomial that additionally satisfies p(3) = 2.

Q11: Use the Lagrange interpolating polynomial of degree three or less and four-digit chopping arithmetic to...

Q11: Use the Lagrange interpolating polynomial of degree three or less and four-digit chopping arithmetic to approximate cos 0.750 using the following values. Find an error bound for the approximation. cos 0.698 = 0.7661 ,cos 0.733 = 0.7432 cos 0.768 = 0.7193 cos 0.803 = 0.6946.

Find the lagrange polynomials that approximate f(x) = x3 a ) Find the linear interpolation polynomial...

Find the lagrange polynomials that approximate f(x) = x3 a ) Find the linear interpolation polynomial P1(x) using the nodes x0= -1 and x1 = 0 b) Find the quadratic interpolation polynomial P2(x) using the nodes x0= -1 and x1 = 0 and x2 = 1 c) Find the cubic interpolation polynomial P3(x) using the nodes x0= -1 and x1 = 0 and x2 = 1 and x3=2 d) Find the linear interpolation polynomial P1(x) using the nodes x0= 1...

How to find the polynomial function with real coefficients, degree 5, zeros 1+i, -3, and 5,...

How to find the polynomial function with real coefficients, degree 5, zeros 1+i, -3, and 5, and P(0)=30 and P(4)= -70?

Prove that the algorithm for computing the coefficients in the Newton form of the interpolating polynomial...

Prove that the algorithm for computing the coefficients in the Newton form of the interpolating polynomial involves n^2 long operations (multiplication and division).

find Lagrange polynomials that approximate f(x)=x^3, a) find the linear interpolation p1(x) using the nodes X0=-1...

find Lagrange polynomials that approximate f(x)=x^3, a) find the linear interpolation p1(x) using the nodes X0=-1 and X1=0 b) find the quadratic interpolation polynomial p2(x) using the nodes x0=-1,x1=0, x2=1 c) find the cubic interpolation polynomials p3(x) using the nodes x0=-1, x1=0 , x2=1 and x3=2. d) find the linear interpolation polynomial p1(x) using the nodes x0=1 and x1=2 e) find the quadratic interpolation polynomial p2(x) using the nodes x0=0 ,x1=1 and x2=2

Find the Taylor polynomial of degree 2 centered at a = 1 for the function f(x)...

Find the Taylor polynomial of degree 2 centered at a = 1 for the function f(x) = e^(2x) . Use Taylor’s Inequality to estimate the accuracy of the approximation e^(2x) ≈ T2(x) when 0.7 ≤ x ≤ 1.3

1. Find the Taylor polynomial of degree ?=3 for ?(?)=?−?22 expanded about ?0=0. 2. Find the...

1. Find the Taylor polynomial of degree ?=3 for ?(?)=?−?22 expanded about ?0=0. 2. Find the error the upper bound of the error term ?5(?) for the polynomial in part (1).