Trapezoidal And Simpson’s Rules, Power Tools In The World Of Numerical Calculations Report Samples
Introduction & Background
Receiving what you need in the fastest possible way is an essential part of the world we are living today. However, we might give up some insignificant aspects in the way of reaching to the solution for our problem. Sometimes, the common methods and tools of solving typical problems might not work for some special cases and then we need to find an alternative to give the most of what we are in demand. In the world of mathematics, the numerical integration is one of those powerful tools that come to rescue in such situations.
Numerical integration, also known as numerical quadrature for one-dimensional integrals, includes algorithms and methods which compute approximately the numerical value of an integral. Term “Quadrature” originates in classic action of finding a square whose area equals to the area under the curve of a function. The main reasons why we utilize numerical integrations is the impossibility or infeasibility of evaluating definite integral of many functions since they have no explicit antiderivative or whose antiderivative is not easy to obtain. Another reason is the occasional need for integration of a function which is only known at some tabulated data points. Also, there are some situations where the analytically evaluated integral is much more difficult to use than the numerically calculated one; the case where the antiderivative is calculated as an infinite series or product might be good example for such situations. Almost in every rule or procedure used for numerical integrations, algebraic polynomials are used to approximate an arbitrary set of data. There are two main reasons for that; first, there’s always a polynomial for a defined function on a closed interval which is arbitrarily close to the function at every point in the interval; second, the polynomials are among the easiest functions to evaluate derivatives and integrals. Many methods are available for approximating the integral of a function over a given domain to a desired precision including Trapezoidal rule and Simpson’s rule which are commonly used owing to their simplicity. The formulae for these two rules are generated based on first and second Lagrange polynomials with equally-spaced nodes. Later on, we will discuss these two classic methods and find the error associated with them.
The Trapezoidal rule is derived by using the first Lagrange (linear) polynomial. The geometric expression for this method is approximation of the area under the curve y=f(x) over a given interval by the area of a trapezoid (figure 1).
Figure 1: Geometric derivation of the Trapezoidal rule
The mathematical expression for this rule is
and the associated error with this method for a given ξ value ranging over the interval is
where h=a-b is the interval length. For more accurate approximation, the interval is divided into some subintervals to give the composite trapezoidal rule whose express is as follows.
The Simpson’s rule is derived by using the second Lagrange (quadratic) polynomial at points a, a+b2, b (figure 2).
Figure 2: Geometric expression of the Simpson’s rule
The Simpson’s rule is expressed by following formula.
The associated error with this method for a given ξ value ranging over the interval is
where h=a-b is the interval length. Similar to Trapezoidal rule, composite Simpson’s rule can be also derived by applying the Simpson’s over short subintervals.
2.1. The exact solution of the definte integral
2.2. Approximation by Trapezoidal Rule
The associated error with this rule is calculated as follows.
The associated error with this rule is calculated as follows.
The Absolute ERR= |Exact solution-Approximate solution|=|0.3059537409-0.3041421422|=1.8115987×10-3
The Relative ERR%=The Absolute ERRExact solution×100=1.8115987×10-30.3059537409=0.592115231%
2.3. Approximation by Simpson’s Rule
The Absolute ERR= |Exact solution-Approximate solution|=|0.3059537409-0.3059278283|=2.59126×10-5
The Relative ERR%=The Absolute ERRExact solution×100=2.59126×10-50.3059537409=8.46945029×10-3%
In this Project, we used the Trapezoidal and Simpson’s rule as methods of numerical integration to approximate the definite integral of a given function. We first calculated the integral analytically and then compared with the results obtained from these two rules by dividing the interval into 6 subintervals. The relative errors for the Simpson’s and Trapezoidal rules are 0.846945029×10-2% and 0.592115231% ,respectively, which are very good and within the acceptable range for many applications including engineering fields. The results prove the validity of this statement that the Simpson’s rule is two orders more accurate since its error is an order of ∆x5 while the error associated with Trapezoidal rule is an order of ∆x3 (2.59126×10-5 <1.8115987×10-3), where ∆x is the length of subintervals.
As we have now completed the task, we realize how close an approximate method, or sometimes a simplified one, might be to the exact solution we seek, which may save plenty of time and energy and provide us with almost what we need.
Burden, R., & Faires, J. (2010). Numerical analysis (9th ed.). Boston: Thomson/Brooks/Cole.
Levy, D. (2010). Introduction to Numerical Analysis. Department of Mathematics and Center for Scientific Computation and Mathematical Modeling (CSCAMM), University of Maryland.
Davis, P., & Rabinowitz, P. (1984). Methods of numerical integration (2nd ed.). New York: Academic Press.
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