Numerical Methods/Numerical Integration

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Often, we need to find the integral of a function that may be difficult to integrate analytically (ie, as a definite integral) or impossible (the function only existing as a table of values).

Some methods of approximating said integral are listed below.

Trapezoidal Rule[edit]

Consider some function, possibly unknown, , with known values over the interval [a,b] at n+1 evenly spaced points xi of spacing , and .

Further, denote the function value at the ith mesh point as .

Using the notion of integration as "finding the area under the function curve", we can denote the integral over the ith segment of the interval, from to as:

= (1)

Since we may not know the antiderivative of , we must approximate it. Such approximation in the Trapezoidal Rule, unsurprisingly, involves approximating (1) with a trapezoid of width h, left height , right height . Thus,

(1) = (2)

(2) gives us an approximation to the area under one interval of the curve, and must be repeated to cover the entire interval.

For the case where n = 2,

= (3)


Collecting like terms on the right hand side of (3) gives us:

or

Now, substituting in for h and cleaning up,


To motivate the general version of the trapezoidal rule, now consider n = 4,

Following a similar process as for the case when n=2, we obtain

Proceeding to the general case where n = N,

Trapizoidruleexample.jpg This is an example of what the trapezoidal rule would represent graphicly, here .


Example[edit]

Approximate to within 5%.

First, since the function can be exactly integrated, let us do so, to provide a check on our answer.

= (4)

We will start with an interval size of 1, only considering the end points.

(4)

Relative error =

Hmm, a little high for our purposes. So, we halve the interval size to 0.5 and add to the list

(4)

Relative error =

Still above 0.01, but vastly improved from the initial step. We continue in the same fashion, calculating and , rounding off to four decimal places.

(4)

Relative error =

We are well on our way. Continuing, with interval size 0.125 and rounding as before,


(4)

Relative error =

Since our relative error is less than 5%, we stop.

Error Analysis[edit]

Let y=f(x) be continuous,well-behaved and have continuous derivatives in [x0,xn]. We expand y in a Taylor series about x=x0,thus-

Simpson's Rule[edit]

Consider some function possibily unknown with known values over the interval [a,b] at n+1 evently spaced points then it defined as

where and and .

Example[edit]

Evaluate by taking ( must be even)

Solution: Here

Since & so

Now when then

And since , therefore for , , , , , the corresponding values are , , , , ,

Incomplete ... Completed soon

Error Analysis[edit]

Simpson's 3/8[edit]

The numerical integration technique known as "Simpson's 3/8 rule" is credited to the mathematician Thomas Simpson (1710-1761) of Leicestershire, England. His also worked in the areas of numerical interpolation and probability theory.


Theorem (Simpson's 3/8 Rule) Consider over , where , , and . Simpson's 3/8 rule is

   .   

This is an numerical approximation to the integral of over and we have the expression

   .  

The remainder term for Simpson's 3/8 rule is , where lies somewhere between , and have the equality

   .


Proof Simpson's 3/8 Rule Simpson's 3/8 Rule


Composite Simpson's 3/8 Rule

   Our next method of finding the area under a curve  is by approximating that curve with a series of cubic segments that lie above the intervals  .  When several cubics are used, we call it the composite Simpson's 3/8 rule.  


Theorem (Composite Simpson's 3/8 Rule) Consider over . Suppose that the interval is subdivided into subintervals of equal width by using the equally spaced sample points for . The composite Simpson's 3/8 rule for subintervals is

   .  

This is an numerical approximation to the integral of over and we write

   .  


Proof Simpson's 3/8 Rule Simpson's 3/8 Rule


Remainder term for the Composite Simpson's 3/8 Rule

Corollary (Simpson's 3/8 Rule: Remainder term) Suppose that is subdivided into subintervals of width . The composite Simpson's 3/8 rule

   .  

is an numerical approximation to the integral, and

   .  

Furthermore, if , then there exists a value with so that the error term has the form

   .  

This is expressed using the "big " notation .


Remark. When the step size is reduced by a factor of the remainder term should be reduced by approximately .


Algorithm Composite Simpson's 3/8 Rule. To approximate the integral

   ,  


by sampling at the equally spaced sample points for , where . Notice that and .


Animations (Simpson's 3/8 Rule Simpson's 3/8 Rule). Internet hyperlinks to animations.


Computer Programs Simpson's 3/8 Rule Simpson's 3/8 Rule


Mathematica Subroutine (Simpson's 3/8 Rule). Object oriented programming.


Example 1. Numerically approximate the integral by using Simpson's 3/8 rule with m = 1, 2, 4. Solution 1.


Example 2. Numerically approximate the integral by using Simpson's 3/8 rule with m = 10, 20, 40, 80, and 160. Solution 2.


Example 3. Find the analytic value of the integral (i.e. find the "true value"). Solution 3.


Example 4. Use the "true value" in example 3 and find the error for the Simpson' 3/8 rule approximations in example 2. Solution 4.


Example 5. When the step size is reduced by a factor of the error term should be reduced by approximately . Explore this phenomenon. Solution 5.


Example 6. Numerically approximate the integral by using Simpson's 3/8 rule with m = 1, 2, 4. Solution 6.


Example 7. Numerically approximate the integral by using Simpson's 3/8 rule with m = 10, 20, 40, 80, and 160. Solution 7.


Example 8. Find the analytic value of the integral (i.e. find the "true value"). Solution 8.


Example 9. Use the "true value" in example 8 and find the error for the Simpson's 3/8 rule approximations in example 7. Solution 9.


Example 10. When the step size is reduced by a factor of the error term should be reduced by approximately . Explore this phenomenon. Solution 10.


Various Scenarios and Animations for Simpson's 3/8 Rule.

Example 11. Let over . Use Simpson's 3/8 rule to approximate the value of the integral. Solution 11.


Animations (Simpson's 3/8 Rule Simpson's 3/8 Rule). Internet hyperlinks to animations.


Research Experience for Undergraduates

Simpson's Rule for Numerical Integration Simpson's Rule for Numerical Integration Internet hyperlinks to web sites and a bibliography of articles.

Headline text[edit]

Example[edit]

Error Analysis[edit]

References[edit]

Eric W. Weisstein. "Trapezoidal Rule." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/TrapezoidalRule.html



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