Chapter5 - Universityof Houston, Department of Mathematics...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 Newton-Cotes 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 Newton-Cotes 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 Newton-Cotes formulas . The weights α in , ≤ i ≤ n , are dubbed Newton-Cotes 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 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 productdisplay j negationslash = i t − j i − j dt , and hence: n summationdisplay i = α in = 1 , α n − i , n = α i ,...
View Full Document

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.

Page1 / 27

Chapter5 - Universityof Houston, Department of Mathematics...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online