Given the following data table,
x 1.20 1.25 1.3 1.35 1.40 1.45 1.50 f(x) 0.1823 0.2231 0.2624 0.3001 0.3365 0.3716 0.4055
What degree of polynomial required to exactly fit all 7 points to within 3 significant digits?
I want to know the systematic approach to this question, if there is any. I’m also aware of the fact that to fit a polynomial through all seven points require an interpolating polynomial of degree less than or equal to 6, as there are 7 points. But the additional requirement that it has to be within 3 significant digits is something which I’m not so sure about how to deal with. I have initially tried bounding the error formula by (0.5 x 10^-03) and solve for n (the degree of interpolating polynomial), but I’ve reached dead end.
Here is my initial trial of bounding the error formula by (0.5 x 10^-03),
I’ve started out using the error formula E_n (x)=|f(x)-p_n (x)|≤1/(n+1)! |f^((n+1) ) (ζ)| 1/4 n!h^(n+1)<0.5 ×〖10〗^(-3) And since f(x) is a linear function as can be seen from the given data table, I then decided to approximate the derivate using
max┬(x_k≤x≤x_(k+1) )〖|f^" (x)|≈{|(f(x_(k-1) )-2f(x_k )+f(x_(k+1) ))/h^2 |,|(f(x_k )-2f(x_(k+1) )+f(x_(k+2) ))/h^2 |}〗 From which I obtained |D_2 |=6.4 ×〖10〗^(-3) as the approximation to the derivative, then the error formula becomes
E_n (x)=|f(x)-p_n (x)|≤1/(n+1)! |D_2 | 1/4 n!h^(n+1)<0.5 ×〖10〗^(-3) E_n (x)=|f(x)-p_n (x)|≤n!/(n+1)! (6.4 × 〖10〗^(-3))1/4 〖(0.5)〗^(n+1)<0.5 ×〖10〗^(-3)
n!/((n+1).n!) (6.4 × 〖10〗^(-3))1/4 (0.5)^n (0.5)<0.5 ×〖10〗^(-3) 1/((n+1) ) 〖(0.5)〗^n<0.625
ln[〖(0.5)〗^n/((n+1) )]
〖n(ln〗0.5)-ln〖(n+1)
〖n(ln〗0.5)-max┬(0≤n≤6){ln〖(n+1)〗 }〖2.1293.
Is this a valid idea/approach? Or can anyone help me out here by solving this question using a systematic approach?
