User:Egm6341.s10.team4.anandankala/HW4-3

Question

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 * Discuss the pros and cons of different numerical methods.
 * 1) Taylor's series.
 * 2) Composite Trapezoidal rule.
 * 3) Composite Simpson's rule.
 * 4) Romberg Table (Richardson Extrapolation).
 * 5) Corrected Trapezoidal rule.
 * }.
 * }.

Solution
The pros and cons of the above mentioned methods are discussed below.

Taylors Series:

Advantages:
 * 1) The solution is highly accurate for a small order compared to Trapezoidal and Simpson's rule.
 * 2) It is one step and explicit.

Disadvantages:
 * 1) It needs the explicit form of derivatives of the function.

Composite Trapezoidal rule:

Advantages:
 * 1) The method is quite simple in implementing, compared to other methods.
 * 2) It takes a piecewise linear approximation of the function and hence execution of the solution is faster.
 * 3) The solution is very rapidly convergent.
 * 4) It consists of the fact that weighting coefficients are nearly equal to each other.

Disadvantages:
 * 1) The error is much higher compared to the other methods.
 * 2) This method is less accurate for non linear functions since it has straight line approximation.

Composite Simpson's rule:

Advantages:
 * 1) It assumes piecewise quadratic interpolation and hence the accuracy is much higher compared to trapezoidal rule.
 * 2) It can integrate polynomials up to third order accurately.
 * 3) The error is less compared to that of trapezoidal rule.
 * 4) Weighting coefficients are simple and do not fluctuate in terms of magnitude.

Disadvantages:
 * 1) A large number of ordinates are needed in between the interval.

Romberg Table (Richardson Extrapolation):

Advantages:
 * 1) Error is reduced subsequently as we go from TK(n) to TK+1(n).
 * 2) It is efficient in the sense that TK+1(n) is calculated from the already computed values of TK(n) and TK(2n).
 * 3) The above condition reduces the time and space requirement.

Disadvantages:
 * 1) In Richardson extrapolation, We are neglecting the higher orders of aih2i.

Corrected Trapezoidal rule:

Advantages:
 * 1) When integrals of periodic functions are approximated numerically, Corrected Trapezoidal functions are the best choice.

Disadvantages:
 * 1) This method demands derivatives of the function at the end points of the interval.