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Chapter5 - University of Houston Department of Mathematics...

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University of Houston, Department of Mathematics Numerical Analysis, Fall 2005 Chapter 5 Numerical integration Problem: Let f : [ a , b ] lR lR be a piecewise continuous function. Compute the integral I ( f ) = b integraldisplay a f ( x ) dx . If I ( f ) can not be determined in closed form, we have to use a numerical method which is is known as numerical quadrature resp. numerical integration . 5.1 Newton-Cotes formulas Example 5.1 Trapezoidal sum T 3 T 4 a=x 0 2 T 0 T 1 T x 5 = b x y f(x) x 1 x 2 x 3 x 4 Idea: Partition of the domain of integration [ a , b ] into n subintervals a = x 0 < x 1 < ... < x n = b of length h i := x i + 1 x i and approximation of I ( f ) by the sum of the trapezoids T ( n ) = n 1 i = 0 T i , T i = h i 2 [ f ( x i ) + f ( x i + 1 )] = ˆ I ( f ) = n i = 0 λ i f ( x i ) , λ 0 = h 0 2 , λ i = ( h i 1 + h i ) 2 , λ n = h n 1 2 .
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University of Houston, Department of Mathematics Numerical Analysis, Fall 2005 Definition 5.2 Quadrature formula A finite sum of weighted function values of the form ˆ I ( f ) := n summationdisplay i = 0 λ i f ( x i ) for the approximation of I ( f ) = b integraltext a f ( x ) dx is called a quadrature formula . The points x i , 0 i n , are referred to as the nodes and the numbers λ i , 0 i n , are called the weights of the qua- drature formula. Remark 6.3 Quadrature formulas based on interpolation In the special case n = 1 , the trapezoidal sum reduces to the trapezoidal rule ˆ I ( f ) = b a 2 [ f ( a ) + f ( b )] , which can be obtained formally by replacing the integrand by its linear interpolant ˆ f := p 1 ( f ) = x a b a [ f ( b ) f ( a )] + f ( a ) = ˆ I ( f ) = I ( ˆ f ) = b integraldisplay a ˆ f ( x ) dx .
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University of Houston, Department of Mathematics Numerical Analysis, Fall 2005 Idea: Replacement of the integrand by its polynomial interpolant with respect to ( x i , f ( x i ) , 0 i n : ˆ f ( x ) = p n ( f ) = n summationdisplay i = 0 f ( x i ) L i , n ( x ) , where L i , n ( · ) , 0 i n , denote the Lagrangian fundamental polynomials. Definition 5.4 Newton-Cotes formulas The quadrature formulas given by ˆ I ( f ) = ( b a ) n summationdisplay i = 0 α in f ( x i ) , α in := 1 b a b integraldisplay a L i , n ( x ) dx , 0 i n are called Newton-Cotes formulas . The weights α in , 0 i n , are dubbed Newton-Cotes weights . The error I ( f ) I ( ˆ f ) is referred to as quadrature error .
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University of Houston, Department of Mathematics Numerical Analysis, Fall 2005 Remark 5.5 Computation of the Newton-Cotes weights In case of equidistant partitions of mesh width h := ( b a ) / n , the substitution t := ( x a ) / h = n ( x a ) / ( b a ) yields: α in = 1 b a b integraldisplay a productdisplay j negationslash = i x x j x i x j dx = 1 n n integraldisplay 0 productdisplay j negationslash = i t j i j dt , and hence: n summationdisplay i = 0 α in = 1 , α n i , n = α i , n , 0 i n . The following table contains the Newton-Cotes formulas for n = 1 , 2 , 3 , 4 : n α in error name 1 1 2 1 2 h 3 / 12 f ( 2 ) ( ξ ) trapezoidal rule 2 1 6 2 3 1 6 h 5 / 90 f ( 4 ) ( ξ ) Simpson’s rule 3 1 8 3 8 3 8 1 8 3h 5 / 80 f ( 4 ) ( ξ ) Kepler’s rule 4 7 90 16 45 2 15 16 45 7 90 8h 7 / 945 f ( 6 ) ( ξ ) Milne’s rule Observe that for n > 6 negative weights occur ( cancellation! ).
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University of Houston, Department of Mathematics
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