Consider a table values of f(x) = sinx for x on [0,1.58] with a spacing h=0.01. Bound the error of quadratic interpolation in this table
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Consider a table values of f(x) = sinx for x on [0,1.58] with a spacing h=0.01. Bound the error o...
<4> Consider a table values of f(x) = in(x) for x on (8,16) with a spacing of h. Bound the error of linear interpolation in this table. How small h should be chosen if this interpolation error is to be less than 104 ?
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...
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...
3. Determine (analytically) the spacing h in a table of evenly spaced values of the function f(x)vx between 1 and 2, so that interpolation with a) degree polynomial b) 2"d degree polynomial in this table will yield a desired accuracy of at least N places after zero. Choose your own N.
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. Consider the function f(x) = -0.1.24 – 0.15x3 – 0.522 – 0.25x + 1.2. (a) Obtain the analytical expression (i.e. True or Exact Solution) for the first derivative, Eval- uate its value at 1 =0.5. Box your answer and label it as fexact- (b) Now assume the function is discretized on a grid with uniform spacing of h. Evaluate your finite difference approximation at x = 0.5 using central differencing with step sizes starting at 1 and re- duced...
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...
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?
Section A Q1 0 Using the following Taylor series expansion: f(x+h) = f(x)+hf'(x)+22 h 3! f"(x)+ (+0) (1.1) 4! show that the central finite difference formula for the first derivative can be written as: f'(x)= f(x+h)-f(x-1) + ch" +0(hº) (1.2) 2h Determine cp and of the derived equation. [4 marks] Consider the function: f(x) = sin +COS (1.3) 2 2 Let x =ih with n=0.25, give your answer in 3 decimals for (ii) to (vi): (ii) Evaluate f(x) for i...
Problem 2 (hand-calculation): Consider the function f(x) tabulated in table 1. Apply improved trapezoid rule to estimate the integral, If) J ) dz, by using the following number of subintervals, n (a) n-3. Use grid points at i0, 4, 8 and 12 (b) n- 6. Use grid points at i0, 2,4, 6, 8, 10 and 12 (c) n = 12, Use all grid points For each part, compute the integral, T(f) and the corresponding absolute error Er(f), and the error...