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Chapter 7 Numerical Differentiation and Integration
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INTRODUCTION DIFFERENTIATION USING DIFFERENCE OPREATORS DIFFERENTIATION USING INTERPOLATION RICHARDSON’S EXTRAPOLATION METHOD NUMERICAL INTEGRATION
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NEWTON-COTES INTEGRATION FORMULAE
THE TRAPEZOIDAL RULE ( COMPOSITE FORM ) SIMPSON’S RULES ROMBERG’S INTEGRATION DOUBLE INTEGRATION
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Basic Issues in Integration
What does an integral represent? = AREA = VOLUME
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Basic definition of an integral::
= = sum of Height x Width
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Objective: Evaluate I = without doing calculation analytically. When would we want to do this?
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1. Integrand is too complicated to integrate analytically.
2. Integrand is not precisely defined by an equation,i.e., we are given a set of data (xi,ƒ(xi)), i=1,...,n. All methods are applicable to integrands that are functions. Some are applicable to tabulated values.
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Key concepts: Integration is a summing process. Thus virtually all numerical approximations can be represented by I = =
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where: x = weights xi = sampling points Et = truncation error 2. Closed & Open forms: Closed forms include the end points a & b in xi. Open forms do not.
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NUMERICAL INTEGRATION
Consider the definite integral
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where f (x) is known either explicitly or is given as a table of values corresponding to some values of x, whether equispaced or not. Integration of such functions can be carried out using numerical techniques.
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Of course, we assume that the function to be integrated is smooth and Riemann integrable in the interval of integration. In the following section, we shall develop Newton-Cotes
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formulae based on interpolation which form the basis for trapezoidal rule and Simpson’s rule of numerical integration.
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NEWTON-COTES INTERGRATION FORMULAE
In this method, as in the case of numerical differentiation, we shall approximate the given tabulated function, by a polynomial Pn(x) and then integrate this polynomial.
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Suppose, we are given the data (xi, yi), i = 0(1)n, at equispaced points with spacing h = xi+1 – xi, we can represent the polynomial by any standard interpolation polynomial. Suppose, we use Lagrangian approximation, then we have
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with associated error given by
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where and
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Then, we obtain an equivalent integration formula to the definite integral in the form
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where ck are the weighting coefficients given by
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which are also called Cotes numbers
which are also called Cotes numbers. Let the equispaced nodes are defined by
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so that xk – x1 = ( k – 1)h etc. Now, we shall change the variable x to p such that, x = x0 + ph, then we can rewrite equations.
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as
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and
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or
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Also, noting that dx = h dp. The limits of the integral in Equation
change from 0 to n and equation reduces to
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The error in approximating the integral can be obtained from
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Where x0 < ξ < xn. For illustration, consider the cases for n = 1, 2; For which we get
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and
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Thus, the integration formula is found to be
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This equation represents the Trapezoidal rule in the interval [x0, x1] with error term. Geometrically, it represents an area between the curve y = f (x), the x-axis and the ordinates erected at x = x0 ( = a) and x = x1 as shown in the figure.
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yn-1 y3 y2 y1 y0 yn xn = b xn-1 x3 x2 x1 x0 = a X O Y (x2, y2) (x1, y1) (x0, y0) y = f(x)
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This area is approximated by the trapezium formed by replacing the curve with its secant line drawn between the end points (x0, y0) and (x1, y1).
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For n =2, We have
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and the error term is given by
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Thus, for n = 2, the integration takes the form
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This is known as Simpson’s 1/3 rule
This is known as Simpson’s 1/3 rule. Geometrically, this equation represents the area between the curve y = f (x), the x-axis and the ordinates at x = x0 and x2 after replacing the arc of the curve between (x0, y0) and (x2, y2) by an arc of a quadratic polynomial as in the figure
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xn = b xn-1 x3 x2 x1 x0 = a X O Y (x2, y2) (x0, y0) y2 y1 y0 y = f(x)
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Thus Simpson’s 1/3 rule is based on fitting three points with a quadratic.
Similarly, for n = 3, the integration is found to be
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This is known as Simpson’s 3/8 rule, which is based on fitting four points by a cubic. Still higher order Newton-Cotes integration formulae can be derived for large values of n.
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But for all practical purposes,
Simpson’s 1/3 rule is found to be sufficiently accurate.
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