Numerical Integration:

Slides:



Advertisements
Similar presentations
Chapter 6 – Polynomial Functions
Advertisements

Mathematics1 Mathematics 1 Applied Informatics Štefan BEREŽNÝ.
Asymptotic error expansion Example 1: Numerical differentiation –Truncation error via Taylor expansion.
NUMERICAL DIFFERENTIATION AND INTEGRATION
Chapter 7 Numerical Differentiation and Integration
Today’s class Romberg integration Gauss quadrature Numerical Methods
Finite element method – basis functions 1 Finite Elements: Basis functions 1-D elements  coordinate transformation  1-D elements  linear basis functions.
Chapter 1 Introduction The solutions of engineering problems can be obtained using analytical methods or numerical methods. Analytical differentiation.
Curve-Fitting Polynomial Interpolation
Interpolation Used to estimate values between data points difference from regression - goes through data points no error in data points.
8-1 Chapter 8 Differential Equations An equation that defines a relationship between an unknown function and one or more of its derivatives is referred.
Chapter 7 Differentiation and Integration
NUMERICAL DIFFERENTIATION The derivative of f (x) at x 0 is: An approximation to this is: for small values of h. Forward Difference Formula.
10.1 – Exponents Notation that represents repeated multiplication of the same factor. where a is the base (or factor) and n is the exponent. Examples:
Quadrature Greg Beckham. Quadrature Numerical Integration Goal is to obtain the integral with as few computations of the integrand as possible.
Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1 Numerical Differentiation and Integration Part 6 Calculus.
Chapter 4, Integration of Functions. Open and Closed Formulas x 1 =a x 2 x 3 x 4 x 5 =b Closed formula uses end points, e.g., Open formulas - use interior.
Numerical Integration In general, a numerical integration is the approximation of a definite integration by a “weighted” sum of function values at discretized.
3. Numerical integration (Numerical quadrature) .
Numerical Computation
1 Numerical Analysis Lecture 12 Numerical Integration Dr. Nader Okasha.
1 Numerical Integration Section Why Numerical Integration? Let’s say we want to evaluate the following definite integral:
Taylor Series & Error. Series and Iterative methods Any series ∑ x n can be turned into an iterative method by considering the sequence of partial sums.
1.2 – Evaluate and Simplify Algebraic Expressions A numerical expression consists of numbers, operations, and grouping symbols. An expression formed by.
Meeting 11 Integral - 3.
Taylor’s Polynomials & LaGrange Error Review
Consider the following: Now, use the reciprocal function and tangent line to get an approximation. Lecture 31 – Approximating Functions
MA/CS 375 Fall MA/CS 375 Fall 2002 Lecture 31.
Algebra 2.  Warm Up  A monomial is an expression that is either a real number, a variable or a product of real numbers and variables.  A polynomial.
Sequences Math 4 MM4A9: Students will use sequences and series.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1 Chapter 22.
5. Integration 2.Quadrature as Box Counting 3.Algorithm: Trapezoid Rule 4.Algorithm: Simpson’s Rule 5.Integration Error 6.Algorithm: Gaussian Quadrature.
Chap. 11 Numerical Differentiation and Integration
Integrating Rational Functions by Partial Fractions Objective: To make a difficult/impossible integration problem easier.
The purpose of Chapter 5 is to develop the basic principles of numerical integration Usefule Words integrate, integral 积分(的), integration 积分(法), quadrature.
Do Now: Evaluate each expression for x = -2. Aim: How do we work with polynomials? 1) -x + 12) x ) -(x – 6) Simplify each expression. 4) (x + 5)
Partial Fractions. Idea behind partial fraction decomposition Suppose we have the following expression Each of the two fractions on the right is called.
Notes Over 4.3 Evaluate Determinants of 2 x 2 Matrices
“In the fall of 1972 President Nixon announced that the rate of increase of inflation was decreasing. This was the first time a sitting president used.
Quadratic Functions 2A Polynomials. A polynomial in x is an expression that contains only non-negative, whole number powers of x. The degree of a polynomial.
Quadrature – Concepts (numerical integration) Don Allen.
CHAPTER 3 NUMERICAL METHODS
ASV Chapters 1 - Sample Spaces and Probabilities
Chapter 4, Integration of Functions
NUMERICAL DIFFERENTIATION Forward Difference Formula
TOPIC : 7 NUMERICAL METHOD.
INTEGRATION & TECHNIQUES OF INTEGRATION
Chapter 7 Numerical Differentiation and Integration
Lecture 25 – Power Series Def: The power series centered at x = a:
DIFFERENTIATION & INTEGRATION
Chapter 22.
Numerical Analysis Lecture 27.
Class Notes 18: Numerical Methods (1/2)
Chapter 23.
Start with a square one unit by one unit:
Taylor Polynomials & Approximation (9.7)
Chapter 7 Numerical Differentiation and Integration
Today’s class Multiple Variable Linear Regression
Basic Calculus Review: Infinite Series
ASV Chapters 1 - Sample Spaces and Probabilities
Numerical differentiation
Section 11.6 – Taylor’s Formula with Remainder
Summation Formulas Constant Series.
Objectives Classify polynomials and write polynomials in standard form. Evaluate polynomial expressions.
SKTN 2393 Numerical Methods for Nuclear Engineers
Polynomial Functions 1 Definitions 2 Degrees 3 Graphing.
equivalent expression
CLASSIFYING POLYNOMIAL
Numerical Integration
Unit 13 Pretest.
Presentation transcript:

Numerical Integration: Approximating an integral by a sum

Integrals as areas

Integrals as areas Approximate the integral by a finite sum of areas rectangles

Integrals as areas Approximate the integral by a finite sum of areas trapeziums

Integrals as areas Trapezium rule: Associated error:

Integrals as areas Composite trapezium rule:

Integrals as areas Simpson’s rule:

Integrals as areas the rectangle approximation takes the function to be constant in the interval the trapezium rule uses linear interpolation between points Simpson’s rule uses polynomial (quadratic) interpolation between the points and has an associated error these should be compared to a Taylor expansion, the first term is a constant, the second term is linear, the third is quadratic…

Simpson’s rule: derivation by method of undetermined coefficients express the integral as this must hold for polynomials of degree two or less. In particular, it must hold for but we already know the left hand side for these

Simpson’s rule: derivation by method of undetermined coefficients Now calculate the right-hand-side: Solving these gives

Romberg integration: integration by iteration make repeated use of the trapezium rule etc. this is equivalent to etc. this gives a series of approximations which can be used to extrapolate to give the answer

Romberg integration: doing the extrapolation the algorithm takes the form of a triangle, Rjk, where we start with R00 and work down R00 R10 R11 R20 R21 R22 R30 R31 R32 R33 … … are the approximations, repeat until

Romberg integration: doing the extrapolation The left hand column are given by the trapezium rule starting with To work from the left to the right for a given column use

so Example,

We end up with the following triangle Example, We end up with the following triangle First approximation Second approximation Third approximation

Infinite ranged integrals: evaluate an integral over an infinite range the previous methods would lead to an infinite sum so cannot be used transform the integral into a finite ranged integral, e.g. change the variable in the second integral (prove this)