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Unformatted text preview: Universityof 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 NewtonCotes formulas Example 5.1 Trapezoidal sum T 3 T 4 a=x 2 T 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 < x 1 <... < x n = b of length h i := x i + 1 − x i and approximation of I ( f ) by the sumof the trapezoids T ( n ) = n − 1 ∑ i = T i , T i = h i 2 [ f ( x i ) + f ( x i + 1 )] = ⇒ ˆ I ( f ) = n ∑ i = λ i f ( x i ) , λ = h 2 , λ i = ( h i − 1 + h i ) 2 , λ n = h n − 1 2 . Universityof Houston, Department of Mathematics Numerical Analysis, Fall 2005 Definition 5.2 Quadrature formula Afinite sumof weighted function values of the form ˆ I ( f ) := n summationdisplay i = 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 , ≤ i ≤ n , are referred to as the nodes and the numbers λ i , ≤ 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 sumreduces 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 . Universityof 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 ) , ≤ i ≤ n : ˆ f ( x ) = p n ( f ) = n summationdisplay i = f ( x i ) L i , n ( x ) , where L i , n ( · ) , ≤ i ≤ n , denote the Lagrangian fundamental polynomials. Definition 5.4 NewtonCotes formulas The quadrature formulas given by ˆ I ( f ) = ( b − a ) n summationdisplay i = α in f ( x i ) , α in := 1 b − a b integraldisplay a L i , n ( x ) dx , ≤ i ≤ n are called NewtonCotes formulas . The weights α in , ≤ i ≤ n , are dubbed NewtonCotes weights . The error I ( f ) − I ( ˆ f ) is referred to as quadrature error . Universityof Houston, Department of Mathematics Numerical Analysis, Fall 2005 Remark 5.5 Computation of the NewtonCotes 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 productdisplay j negationslash = i t − j i − j dt , and hence: n summationdisplay i = α in = 1 , α n − i , n = α i ,...
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This note was uploaded on 02/21/2012 for the course ENG 3000 taught by Professor Staff during the Spring '12 term at University of Houston.
 Spring '12
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