Solving Interpolation Error w/ ≤ 5*10-8 Precision

Click For Summary
SUMMARY

The discussion focuses on solving interpolation errors for the function f(x) = cos(x) over the interval [0, π]. The goal is to determine the step size h such that the interpolation error remains within the threshold of ≤ 5*10-8. The user has calculated the third derivative f'''(x) for x in [0, π] to be ≤ 1, which is a crucial step in estimating the error. The challenge lies in evaluating the expression |(x - xi+1)(x - xi)(x - xi-1)| to find the appropriate value of h.

PREREQUISITES
  • Understanding of numerical interpolation methods
  • Knowledge of Taylor series and derivatives
  • Familiarity with error analysis in numerical methods
  • Basic calculus, specifically dealing with trigonometric functions
NEXT STEPS
  • Research the error bounds for polynomial interpolation
  • Learn about the Lagrange interpolation formula
  • Study the implications of the Mean Value Theorem in interpolation
  • Explore numerical methods for estimating derivatives
USEFUL FOR

Students and professionals in mathematics, particularly those focused on numerical analysis and interpolation techniques, will benefit from this discussion.

peripatein
Messages
868
Reaction score
0
Hi,

Homework Statement


Interpolation for the function f(x)=cos(x) for evenly distributed values of x in [0,π] (h=xi+1-xi, xi=ih, i=0,1,...,πh) is carried out. Also known are f(xi).
I am asked to determine the value of h so that the interpolation's error is ≤ 5*10-8.

Homework Equations




The Attempt at a Solution


I have found f'''(x) for x in [0,π] to be ≤ 1.
Now I am stuck with evaluating |(x-xi+1)(x-xi)(x-xi-1)|.
I'd truly appreciate some assistance. Thanks in advance!
 
Physics news on Phys.org
I have managed on my own with this one. Thank you in any case :).
 

Similar threads

Matlab Function for Composite Simpson
  • · Replies 3 ·
Replies
3
Views
2K
Matlab Plotting Points and a Cubic Polynomial that passes through them
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Determination of error in interpolating polynomial
  • · Replies 5 ·
Replies
5
Views
3K
PLL - How to find all the gains of a PI corrector and fix Ki ? MATLAB
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad 2nd order ODE's, variation of parameters and the notorious constraint
  • · Replies 5 ·
Replies
5
Views
2K
MHB Calculating Integral with Simpson's Rule for Error < $0.5\cdot 10^{-3}$
  • · Replies 6 ·
Replies
6
Views
2K
Filtering problem. Help with the output
  • · Replies 6 ·
Replies
6
Views
2K
Bifurcation values for logistic map
  • · Replies 5 ·
Replies
5
Views
2K
MHB Interpolating Points with Continuous Modular Functions?
  • · Replies 5 ·
Replies
5
Views
2K
MATLAB: Finding the 5th Root using Newton's Method
  • · Replies 2 ·
Replies
2
Views
8K