Simpson's Rule

Simpson's rule method is accredited to Thomas Simpson, a mathematician of Leicestershire, England. It is a method for numerical integration, i e. numerical approximation of definite integrals. It can be derived by 3 methods, namely,

• Quadratic interpolation: This is derived by replacing the integrand function with the quadratic polynomial that assumes the same values as the integrand function at the end points and the midpoint. Lagrange Polynomial Interpolation to derive an expression for the above polynomial. An easy calculation can be carried out easily if one primarily finds by scaling that there is no loss of generality in assuming the lower endpoint as -1 and the upper end point as +1.
• Averaging the midpoint and the trapezoidal rules: Next derivation builds Simpson's rule from the mid point rule and the trapezoidal rule. The leading error term vanishes by taking the weighted average and the result is exactly Simpson's rule. By using alternative approximation, ie by trapezoidal rule with two times as many points, it is easy to eliminate another error term by taking a suitable weighted average. This method is Romberg's method.
• Undetermined co-efficient: The last derivation begins from ansatz, meaning onsets. Formally it is a prepared guess that could be affirmed later by its results. The coefficients could be fixed by necessity that this approximation be exact for all quadratic polynomials. This derives Simpson's rule.

We have to note that Simpson's rule gives exact results for all three or less polynomial because the error term takes the fourth derivatives of the function.

Newton's Method:

This is also known as Newton-Raphson method. This yielded the name after Issac Newton and Joseph Raphson. This may be the most well known method for calculating better approximations successively to the roots of a real-valued function. If the iteration starts sufficiently close to the required root, then Newton's method could often converge very quickly. When iteration starts away from the required root, then unfortunately the method could lead easily to an unwary user astray with minimum warning. Thus, good usage of the method leads it in a formal way that also finds and might overcome possible convergence failures. This method also can be used to find the reciprocal of a number by using only subtraction and multiplication. This method also can be used to find a minimum/maximum value of a function.

Practical Usage:

Newton's method is a highly powerful method-in common the convergence leads to quadratic. The error evolved is necessarily squared at each stage. The difficulties involved with this method are as follows: Newton's method

• needs that the derivative be estimated directly.
• May fail to converge if the initial value is very far from the true zero.
• May fail to converge if the derivative is not continuous.
• Will fail in cases where the derivative is zero.
• May overshoot the desired root if the derivative is close to zero and the tangent line is almost horizontal.
• May have convergence rate as merely linear if the root being expected has multiplicity greater than one.
• Works good for functions with low curvature.
• Would find the root after a single iteration, if the linear function is with zero curvature.

Large scale developers prefer secant method over Newton's method. In use, the vantage of maintaining a smaller code base normally outweighs the top convergence characteristics of Newton's method.