User:Egm6341.s11.team2.cleveland/HW5HW5

=Problem 5.7: =

Problem Statement:
Discuss the pros and cons of the following quadrature methods:


 * 1) Taylor's Series
 * 2) Composite Trapezoidal Rule
 * 3) Composite Simpson's Rule
 * 4) Romberg Table (Including Richardson's Extrapolation)
 * 5) Corrected Trapezoidal Rule

Pros:

 * 1) Local behavior about the point of expansion is very accurate, fast and computationally tractable.
 * 2) The number of operations are typically small for higher order approximation of simple functions, such as polynomials.
 * 3) With the aid of symbolic packages complex function behavior is easily found and accurate.

Cons:

 * 1) One needs to know the "smoothness" and global behavior of the function since computing derivative of an unbounded solution would be pointless.
 * 2) The number of operations may be large for a complex functions. For example, functions containing non-distributable products, quotients or composite trigonometric and exponential, to name a few.
 * 3) In addition, oscillatory functions, especially those having a small period(high frequency) are more cumbersome to approximate using a Taylor's Series.

Pros:

 * 1) Code implementation is very easy.
 * 2) Convergence for periodic functions is very quick when choosing to integrate over integer multiple of the period.
 * 3) The function (in the integrand) need only be twice continuously differentiable on the domain.

Cons:

 * 1) Steep concavity (or convexity) of the integrand can result in under or overestimates due to the second order continuously differentiable criteria. This means for a concave up integrand, and thus a positive second derivative, one will observe negative error and therefore an overestimate. Similarly, for a concave down (convex) integrand, and thus a negative second derivative, one will observe positive error and therefor an underestimate.
 * 2) Add more here!!!!!!!!!!!!!!!!!!!!