Numerical Methods207810 marksNumerical: Secant method root findingSimpson's 3/8 rule and composite Simpson's 3/8 rule
Derive the formula for integration using simpsons 3/8 rule. Use Secant Method to estimate the root of equation with initial estimate x₁ = 4 and x₂ = 2, x2 - 4x - 10 = 0.
Derive the formula for integration using simpsons 3/8 rule. Use Secant Method to estimate the root of equation with initial estimate x₁ = 4 and x₂ = 2, $x^2 - 4x - 10 = 0$. [10]
Divide $[a,b]$ into 3 equal sub-intervals with 4 points: $$x0=a,\quad x1=a+h,\quad x2=a+2h,\quad x3=a+3h=b,\qquad h=\frac{b-a}{3}$$ Let $yi=f(xi)$. Use Newton's forward interpolation with $x=x0+th$, $dx=h,dt$: $$f(x)\approx y0+t\Delta y0+\frac{t(t-1)}{2!}\Delta^2 y0+\frac{t(t-1)(t-2)}{3!}\Delta^...