Numerical methods. Note 2

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Integration I

Numerical Methods. Note~ « 2 »

Integration. Classical formulas for equally spaced abscissas.

The classical n-point formula gives an estimate of an integral I_n ≈ ∫₀^h f(x)dx using the values of the integrand at n-points typically equally spaced within the interval (0,h). Due to round-off errors n cannot be too large, say n<10 for single and n<20 for double precision.
Closed (Newton-Cotes) formulae use the end-points of the interval

1-point (rectangular rule): I₁ = h f(^h/₂), err = ¹/₁₂ h³ [^{f ''(0)}/_2!]
2-point (trapezoidal rule): I₂ = ¹/₂ h f(0) + ¹/₂ h f(h), err = ^-1/₆ h³ [^{f ''{0}}/_2!]
3-point (Simpson's rule): I₃ = ²/₃ I₁ + ¹/₃ I₂, err = ¹/₁₅h⁵ [^{f ''''(0)}/_3!]

Open formulae use only the inner points of the interval:

2-point: J₂ = ¹/₂ h f(¹/₃ h) + ¹/₂ h f(²/₃ h)
4-point: J₄ = ¹/₃ h f(¹/₆ h) + ¹/₆ h f(¹/₃ h) + ¹/₆ h f(²/₃ h) + ¹/₃ h f(⁵/₆ h)

Problems

Make a function OPEN24 that estimates the integral J=∫_a^b f(x)dx and the the integration error ERR using the 2-4 point open formulae:
```
function [J,ERR] = OPEN24 ("f",a,b)
   calculate J₂ and J₄ 
   J=J₄ ; ERR=abs(J₄-J₂)
```

Make a function DUMBINT that estimates the integral J=∫_a^b f(x)dx and the integration error by subdividing the interval (a,b) into N sub-intervals and using OPEN24 for each sub-interval:

function [J,err] = DUMBINT ("f",a,b,N)
   subdivide (a,b) into N sub-intervals (a_i,b_i)
   for each sub-interval calculate the sub-integral and sub-error [S_i,e_i] = OPEN24("f",a_i,b_i)
   calculate the global-integral J=∑_iS_i and global-error err=√(∑_ie_i²)

Using your program calculate the integral
∫_-20²⁰ [¹/_1+x²] dx
and estimate the number of points (integrand evaluations) necessary to achieve the accuracy of err = 0.0001.