Q2. Let f(x) = √
x. Using equally spaced nodes on the interval [0.25, 1], what is
the upper bound for
the error of the cubic Newton interpolation.
Q2. Let f(x) = √ x. Using equally spaced nodes on the interval [0.25, 1], what...
6. Consider f(x)-sinx and evenly spaced nodes 0-0 < xīく… < Zn-2T. Let P(z) be the piecewise cubic interpolant given values and first derivatives of f at the nodes. (a) In the case n = 100, use calculus and the error formula 4! where 1 E [xi,Ti+1], to bound the absolute error lf(1)-P(1) (b) For arbitrary x E [0.2 , use error bounds to determine n ensuring that If(x)- P(x) s 10-10
6. Consider f(x)-sinx and evenly spaced nodes 0-0
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...
We want to produce an evenly spaced table of values for the function f(x) sin(x) for x E [0,Tt/2] such that, with cubic interpolation, we can give the values of the function at any point in the interval with an error less than 5 10-12. That means finding a number n such that with h = π/2n and Xk-kh, k-0, , n the cubic interpolation polynomial with the interpolation points XK-1,XK, X+1 XK+2 for x has an error less than...
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...
Problem 4 Let f(x) be a cubic polynomial defined on interval
[−1, 1]. Determine a Gaussian intergration formula with minimal
number of nodes such that the integral formula Xn i=0 f(xi)wi is
exact for cubic polynomials.
Problem 4 Let f(z) be a cubic polynomial defined on interval [-1,. Determine a Gaussian intergration formula with minimal number of nodes such that the integral formula is exact for cubic polynomials.
Problem 4 Let f(z) be a cubic polynomial defined on interval [-1,....
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?
2. Graph the functions f(x)x(x 1)(x-2) ..(x- k) for k- 1,2,..,10. (These are examples of the polynomials occurring in the error formula for polynomial interpolation.) We want to produce an evenly spaced table of values for the function f(x) sin(x) for x E [O,T/2] such that, with cubic interpolation, we can give the values of the function at any point in the interval with an error less than 5 10-12. That means finding a number n such that with h-/2n...
1. Consider the polynonial Pl (z) of degree 4 interpolating the function f(x) sin(x) on the interval n/4,4 at the equidistant points r--r/4, xi =-r/8, x2 = 0, 3 π/8, and x4 = π/4. Estimate the maximum of the interpolation absolute error for x E [-r/4, π/4 , ie, give an upper bound for this absolute error maxsin(x) P(x) s? Remark: you are not asked to give the interpolation polynomial P(r).
1. Consider the polynonial Pl (z) of degree 4...
(35) Problem 5. 2) e Consider using the composite trapezoidal rule T7, with n equally spaced u rule Tn with n equally spaced subintervals to estimate I-In zdz Give a rigorous error bound for 1-Tr. Using the rigorous error n should be in order that li-Tal 3x 10-0
(35) Problem 5. 2) e Consider using the composite trapezoidal rule T7, with n equally spaced u rule Tn with n equally spaced subintervals to estimate I-In zdz Give a rigorous error...
Determine the step size h that can be used in the tabulation of f(x)= (1 + x) in the interval [0,1] at equally spaced nodal points so that the truncation error of linear interpolation is bounded by than 5x10-5 (6 marks)