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Evaluating integrals are common in engineering problems. Sometimes, it is difficult to evaluate complex integrals. The need for Gauss Quadrature Therefore,

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Presentation on theme: "Evaluating integrals are common in engineering problems. Sometimes, it is difficult to evaluate complex integrals. The need for Gauss Quadrature Therefore,"— Presentation transcript:

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2 Evaluating integrals are common in engineering problems. Sometimes, it is difficult to evaluate complex integrals. The need for Gauss Quadrature Therefore, techniques are devised in order to simplify these.

3 From Trapezoidal rule, Trapezoidal rule evaluates an integral by approximating the function of the curve to a straight line. And then finding the area under the straight line.

4 Larger error Reduced error (Using Trapezoidal rule) (Taking an improved estimate)

5 Method of Undetermined Coefficients Constant function Linear function Quadratic function Cubic function

6 For x­ d =-1, For x­ d =1, Then, Therefore, When differentiated, Then, the above two equations can be substituted in to the equation to be integrated. Assuming that a new variable x d is related to the original x in a linear fashion

7 Table for finding Weighting factors and Function Arguments

8 Example 1: Use three-point formula to evaluate it. Solution: Given: a=-1 and b=1; f(x) = x+3x 2 +x 5 From table for three points, C 0 = C 1 = C 2 = X 0 = X 1 =0 X 2 = I=C 0 f(X 0 ) +C 1 f(X 1 ) +C 2 f(X 2 ) f (X 0 )=.7465 f(X 1 )=0 f(X 2 )= I=[ * * * ]= 2.001

9 Example 2: Given: a=-1 and b=1; f(x) = x+3x 2 +x 5 From table for three points, C 0 = C 1 = C 2 = X 0 = X 1 =0 X 2 = I=C 0 f(X 0 ) +C 1 f(X 1 ) +C 2 f(X 2 ) X 0 `= ((b-a)/2+ X 0 (b+a)/2 ) = X 1 `= ((b-a)/2+ X 1 (b+a)/2) =7 X 2 `= ((b-a)/2+ X 2 (b+a)/2)= f (X 0 `)= f (X 1 `)=16961 f (X 2 `)= I=3[ * * * ] I= 166,

10 Error Analysis for Gauss Quadrature

11 See this


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