Discuss the pros and cons of the NewtonCotes formulae and Ro

Discuss the pros and cons of the Newton-Cotes formulae and Romberg Integration. Provide specific examples and use scholarly sources to support your points.

Solution

The Newton-Cotes formulas are an extremely useful and straightforward family of numerical integration techniques.

To integrate a function f(x) over some interval [a,b] , divide it into ‘n’ equal parts such that f_n = f(x_n)  and h=(b-a)/n . Then find polynomials which approximate the tabulated function, and integrate them to approximate the area under the curve. To find the fitting polynomials, use Lagrange interpolating polynomials. The resulting formulas are called Newton-Cotes formulas, or quadrature formulas.

Newton-Cotes formulas may be \"closed\" if the interval is included in the fit, \"open\" if the points[x₂, x_n-1] are used, or a variation of these two. If the formula uses ‘n’ points (closed or open), then coefficients of terms sums to ‘n-1’.

If the function f(x) is given explicitly instead of simply being tabulated at the values x_i , the best numerical method of integration is called Gaussian quadrature. By picking the intervals at which to sample the function, this procedure produces more accurate approximations (but is significantly more complicated to implement).

Pros : It allows us easily find the values of definite integrations without finding the integral

Cons: it is only used for straight forward numerical integrations for uneven numerical integrations, this method cannot be used

Romberg integration pros and cons

Pros:

1. The successive computation of I(2n) is cheaper and faster when I(n) is already computed.

2. Better accuracy then other methods.

Cons:

1. There is one limitation with this rule. Romberg integration is successful only when integrand satisfies the hypotheses of the Euler-Maclaurin.

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