10992
Extended Formulas (Closed)
Лекция
Математика и математический анализ
Extended Formulas Closed If we use equation 2.5 N 1 times to do the integration in the intervals x1; x2; x2; x3; xN 1; xN and then add the results we obtain an extendedr or compositer formula for the integral from x1 to xN. Extended trapezoidal rule: In this method the area under the curve is approximated by sums of trapezoids areas under the curve see Fig. 2.3.. Figure 2.3. Extended trapezoidal rule. Trapezoid formul...
Английский
20130403
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If we use equation (2.5) N – 1 times to do the integration in the intervals (x1; x2); (x2; x3); … (xN  1; xN), and then add the results, we obtain an “extended” or “composite” formula for the integral from x1 to xN.
Extended trapezoidal rule:
In this method the area under the curve is approximated by sums of trapezoids areas under the curve (see Fig. 2.3.).
Figure 2.3. Extended trapezoidal rule.
Trapezoid formula is
(2.9)
Here we have written the error estimate in terms of the interval b – a and the number of points N instead of in terms of h. This is clearer, since one is usually holding a and b fixed and wanting to know (e.g.) how much the error will be decreased by taking twice as many steps (in this case, it is by a factor of 4). In subsequent equations we will show only the scaling of the error term with the number of steps.
If we apply equation (2.5) to successive, nonoverlapping pairs of intervals, we get the extended Simpson’s rule:
(2.10)
Notice that the 2/3, 4/3 alternations continues throughout the interior of the evaluation. Many people believe that the wobbling alternation somehow contains deep information about the integral of their function that is not apparent to mortal eyes. In fact, the alternation is an artifact of using the building block (2.6).
The trapezoidal rule is two orders lower than Simpson’s rule, however, its contribution to the integral goes down as an additional power of N (since it is used only twice, not N times). This makes the resulting formula of degree one less than Simpson.
We can construct open and semiopen extended formulas by adding the closed formulas, evaluated for the second and subsequent steps, to the extrapolative open formulas for the first step. The resulting formula for an interval open at both ends is as follows. This one is known as the extended midpoint rule
(2.11)
This rule is shown in Figure 2.4.
Figure 2.4. Midpoint extended formula with equally spaced abscissas compute the integral of a function y = x3.
Example
Suppose we want to evaluate the integral The definite integral is the area under the curve y = ex, above the xaxis and between the lines x = 1 and
x = 2.
It is obvious the above approximation is an underestimate for the actual value. The actual value can be easily obtained as follows
We can find an approximation by Left Sum Approximation formula:
(2.12)
Figure 2.5. Left Sum Approximation.
In this rule we divide the interval [a, b] into subintervals with same length (h). Construct rectangle over each subinterval, the height of the rectangle being the function value at the left end point of the subintervals. The sum of the areas of these rectangles will provide an approximation to the area under the curve on the interval (see Fig 2.5).
Right Sum Approximation
We can find another approximation to the area under the graph. Divide the interval [a, b] into subintervals and construct rectangle over each subinterval, the height of the rectangle being the function value at the right end point of the subinterval.
The sum of the areas of these rectangles will provide an approximation to the area under the curve and is shown in Fig. 2.6.
Figure 2.6. Right Sum Approximation.
Right Sum Approximation formula is:
(2.13)
The following table gives the number of subintervals and the corresponding approximations.
Number of subintervals 
Approximate value of integral 

Left Sum Approximation 
Midpoint Approximation 
Right Sum Approximation 
Trapezoidal Approximation 
Simpson’s Approximation 

10 
4.44112722 
4.66882868 
4.90820465 
4.25683746 
4.67077443 
100 
4.64745932 
4.67075481 
4.69416706 
4.62903035 
4.67077427 
In this table we can compare accuracy of different approximation formulas and see how the increase in the number of subintervals affects the approximation.
As the number of subintervals increases, the approximations gets better.
Suppose that we pick N random points, uniformly distributed in an interval [a, b]. Call them x1, x2, ... xN. Then the basic theorem of Monte Carlo integration estimates the integral of a function f over the interval
[a, b],
. (2.14)
In formula (2.14) we note that each random point xi is actually a random variable which is uniformly distributed on [a, b]
. (2.15)
Uniform deviates are just random numbers that lie within a specified range [0, 1], with any one number in the range just as likely as any other. Then f(xi) is also a random variable as a function of random point xi.
There is an another advanced technique of Monte Carlo integration (see Fig. 2.7).
Figure 2.7. Monte Carlo integration.
Random points are chosen within the area A. The integral of the function f is estimated as the area of A multiplied by the fraction of random points that fall below the curve f:
. (2.16)
Where S(A) is the area of A; n is the number of random points that lies below the curve f; N is the total number of random points.
The Monte Carlo method thus becomes a deterministic quadrature scheme — albeit a simple one — whose fractional error O(.) decreases at least as fast as .
1. What is a problem of a numerical integration?
2. What is possible to say about error of left sum, right sum and midpoint approximation formulas?
3. How to increase an accuracy of calculations in the problem of numerical integration?
4. Compare both advantages and defects of trapezoidal and Monte Carlo integration formulas.
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