Integration. Topic: Trapezoidal Rule. Major: General Engineering. Author: Autar Kaw, Charlie Barker.
|
|
- Simon Simon
- 5 years ago
- Views:
Transcription
1 Integration Topic: Trapezoidal Rule Major: General Engineering Author: Autar Kaw, Charlie Barker 1
2 What is Integration Integration: The process of measuring the area under a function plotted on a graph. Where: f(x) is the integrand a= lower limit of integration b= upper limit of integration 2
3 Basis of Trapezoidal Rule Trapezoidal Rule is based on the Newton-Cotes Formula that states if one can approximate the integrand as an n th order polynomial where and 3
4 Basis of Trapezoidal Rule Then the integral of that function is approximated by the integral of that n th order polynomial. Trapezoidal Rule assumes n=1, that is, the area under the linear polynomial, 4
5 Derivation of the Trapezoidal Rule 5
6 Method Derived From Geometry The area under the curve is a trapezoid. The integral 6
7 Example 1 The vertical distance covered by a rocket from t=8 to t=30 seconds is given by: a) Use single segment Trapezoidal rule to find the distance covered. b) Find the true error, for part (a). c) Find the absolute relative true error, for part (a). 7
8 Solution a) 8
9 Solution (cont) a) b) The exact value of the above integral is 9
10 Solution (cont) b) c) The absolute relative true error,, would be 10
11 Multiple Segment Trapezoidal Rule In Example 1, the true error using single segment trapezoidal rule was large. We can divide the interval [8,30] into [8,19] and [19,30] intervals and apply Trapezoidal rule over each segment. 11
12 With Multiple Segment Trapezoidal Rule Hence: 12
13 Multiple Segment Trapezoidal Rule The true error is: The true error now is reduced from -807 m to -205 m. Extending this procedure to divide the interval into equal segments to apply the Trapezoidal rule; the sum of the results obtained for each segment is the approximate value of the integral. 13
14 Multiple Segment Trapezoidal Rule Divide into equal segments as shown in Figure 4. Then the width of each segment is: The integral I is: Figure 4: Multiple (n=4) Segment Trapezoidal Rule 14
15 Multiple Segment Trapezoidal Rule The integral I can be broken into h integrals as: Applying Trapezoidal rule on each segment gives: 15
16 Example 2 The vertical distance covered by a rocket from to seconds is given by: a) Use two-segment Trapezoidal rule to find the distance covered. b) Find the true error, for part (a). c) Find the absolute relative true error, for part (a). 16
17 Solution a) The solution using 2-segment Trapezoidal rule is 17
18 Solution (cont) Then: 18
19 Solution (cont) b) The exact value of the above integral is so the true error is 19
20 Solution (cont) c) The absolute relative true error,, would be 20
21 Solution (cont) Table 1 gives the values obtained using multiple segment Trapezoidal rule for: n Value E t Table 1: Multiple Segment Trapezoidal Rule Values 21
22 Example 3 Use Multiple Segment Trapezoidal Rule to find the area under the curve. from to Using two segments, we get and 22
23 Solution Then: 23
24 Solution (cont) So what is the true value of this integral? Making the absolute relative true error: 24
25 Solution (cont) Table 2: Values obtained using Multiple Segment Trapezoidal Rule for: 25 n Approximate Value % % % % % % %
26 Error in Multiple Segment Trapezoidal Rule The true error for a single segment Trapezoidal rule is given by: where is some point in What is the error, then in the multiple segment Trapezoidal rule? It will be simply the sum of the errors from each segment, where the error in each segment is that of the single segment Trapezoidal rule. The error in each segment is 26
27 Error in Multiple Segment Trapezoidal Rule Similarly: It then follows that: 27
28 Error in Multiple Segment Trapezoidal Rule Hence the total error in multiple segment Trapezoidal rule is. The term is an approximate average value of the Hence: 28
29 Error in Multiple Segment Trapezoidal Rule Below is the table for the integral as a function of the number of segments. You can visualize that as the number of segments are doubled, the true error gets approximately quartered. n Value
30 Integration Topic: Simpson s 1/3 rd Rule Major: General Engineering 30
31 Basis of Simpson s 1/3 rd Rule Trapezoidal rule was based on approximating the integrand by a first order polynomial, and then integrating the polynomial in the interval of integration. Simpson s 1/3rd rule is an extension of Trapezoidal rule where the integrand is approximated by a second order polynomial. Hence Where is a second order polynomial. 31
32 Basis of Simpson s 1/3 rd Rule Choose and as the three points of the function to evaluate a 0, a 1 and a 2. 32
33 Basis of Simpson s 1/3 rd Rule Solving the previous equations for a 0, a 1 and a 2 give 33
34 Basis of Simpson s 1/3 rd Rule Then 34
35 Basis of Simpson s 1/3 rd Rule Substituting values of a 0, a 1, a 2 give Since for Simpson s 1/3rd Rule, the interval [a, b] is broken into 2 segments, the segment width 35
36 Basis of Simpson s 1/3 rd Rule Hence Because the above form has 1/3 in its formula, it is called Simpson s 1/3rd Rule. 36
37 Example 1 The distance covered by a rocket from t=8 to t=30 is given by a) Use Simpson s 1/3rd Rule to find the approximate value of x b) Find the true error, c) Find the absolute relative true error, 37
38 Solution a) 38
39 Solution (cont) b) The exact value of the above integral is True Error 39
40 Solution (cont) a)c) Absolute relative true error, 40
41 Multiple Segment Simpson s 1/3rd Rule 41
42 Multiple Segment Simpson s 1/3 rd Rule Just like in multiple segment Trapezoidal Rule, one can subdivide the interval [a, b] into n segments and apply Simpson s 1/3rd Rule repeatedly over every two segments. Note that n needs to be even. Divide interval [a, b] into equal segments, hence the segment width where 42
43 Multiple Segment Simpson s 1/3 rd Rule f(x)..... x x 0 x 2 x n-2 x n Apply Simpson s 1/3rd Rule over each interval, 43
44 Multiple Segment Simpson s 1/3 rd Rule Since 44
45 Multiple Segment Simpson s 1/3 rd Rule Then 45
46 Multiple Segment Simpson s 1/3 rd Rule 46
47 Example 2 Use 4-segment Simpson s 1/3rd Rule to approximate the distance covered by a rocket from t= 8 to t=30 as given by a) Use four segment Simpson s 1/3rd Rule to find the approximate value of x. b) Find the true error, for part (a). c) Find the absolute relative true error, for part (a). 47
48 Solution a) Using n segment Simpson s 1/3rd Rule, So 48
49 Solution (cont.) 49
50 Solution (cont.) cont. 50
51 Solution (cont.) b) In this case, the true error is c) The absolute relative true error 51
52 Solution (cont.) Table 1: Values of Simpson s 1/3rd Rule for Example 2 with multiple segments n Approximate Value E t Є t % % % % % 52
53 Error in the Multiple Segment Simpson s 1/3 rd Rule The true error in a single application of Simpson s 1/3rd Rule is given as In Multiple Segment Simpson s 1/3rd Rule, the error is the sum of the errors in each application of Simpson s 1/3rd Rule. The error in n segment Simpson s 1/3rd Rule is given by 53
54 Error in the Multiple Segment Simpson s 1/3 rd Rule... 54
55 Error in the Multiple Segment Simpson s 1/3 rd Rule Hence, the total error in Multiple Segment Simpson s 1/3rd Rule is 55
56 Error in the Multiple Segment Simpson s 1/3 rd Rule The term is an approximate average value of Hence where 56
57 Integration Topic: Gauss Quadrature Rule of Integration Major: General Engineering 57
58 Two-Point Gaussian Quadrature Rule 58
59 Basis of the Gaussian Quadrature Rule Previously, the Trapezoidal Rule was developed by the method of undetermined coefficients. The result of that development is summarized below. 59
60 Basis of the Gaussian Quadrature Rule The two-point Gauss Quadrature Rule is an extension of the Trapezoidal Rule approximation where the arguments of the function are not predetermined as a and b but as unknowns x 1 and x 2. In the two-point Gauss Quadrature Rule, the integral is approximated as 60
61 Basis of the Gaussian Quadrature Rule The four unknowns x 1, x 2, c 1 and c 2 are found by assuming that the formula gives exact results for integrating a general third order polynomial, Hence 61
62 Basis of the Gaussian Quadrature Rule It follows that Equating Equations the two previous two expressions yield 62
63 Basis of the Gaussian Quadrature Rule Since the constants a 0, a 1, a 2, a 3 are arbitrary 63
64 Basis of Gauss Quadrature The previous four simultaneous nonlinear Equations have only one acceptable solution, 64
65 Basis of Gauss Quadrature Hence Two-Point Gaussian Quadrature Rule 65
66 Higher Point Gaussian Quadrature Formulas 66
67 Higher Point Gaussian Quadrature Formulas is called the three-point Gauss Quadrature Rule. The coefficients c 1, c 2, and c 3, and the functional arguments x 1, x 2, and x 3 are calculated by assuming the formula gives exact expressions for integrating a fifth order polynomial General n-point rules would approximate the integral 67
68 Arguments and Weighing Factors for n-point Gauss Quadrature Formulas In handbooks, coefficients and arguments given for n-point Gauss Quadrature Rule are given for integrals Table 1: Weighting factors c and function arguments x used in Gauss Quadrature Formulas. Points Weighting Factors 2 c 1 = c 2 = Function Arguments x 1 = x 2 = as shown in Table 1. 3 c 1 = c 2 = c 3 = c 1 = c 2 = c 3 = c 4 = x 1 = x 2 = x 3 = x 1 = x 2 = x 3 = x 4 =
69 Arguments and Weighing Factors for n-point Gauss Quadrature Formulas Table 1 (cont.) : Weighting factors c and function arguments x used in Gauss Quadrature Formulas. 69 Points Weighting Factors 5 c 1 = c 2 = c 3 = c 4 = c 5 = c 1 = c 2 = c 3 = c 4 = c 5 = c 6 = Function Arguments x 1 = x 2 = x 3 = x 4 = x 5 = x 1 = x 2 = x 3 = x 4 = x 5 = x 6 =
70 Arguments and Weighing Factors for n-point Gauss Quadrature Formulas So if the table is given for integrals, how does one solve? The answer lies in that any integral with limits of can be converted into an integral with limits Let If If then then Such that: 70
71 Arguments and Weighing Factors for n-point Gauss Quadrature Formulas Then Hence Substituting our values of x, and dx into the integral gives us 71
72 Example 1 For an integral derive the one-point Gaussian Quadrature Rule. Solution The one-point Gaussian Quadrature Rule is 72
73 Solution Assuming the formula gives exact values for integrals and Since the other equation becomes 73
74 Solution (cont.) Therefore, one-point Gauss Quadrature Rule can be expressed as 74
75 Example 2 a) Use two-point Gauss Quadrature Rule to approximate the distance covered by a rocket from t=8 to t=30 as given by b) Find the true error, for part (a). c) Also, find the absolute relative true error, 75 for part (a).
76 Solution First, change the limits of integration from [8,30] to [-1,1] by previous relations as follows 76
77 Solution (cont) Next, get weighting factors and function argument values from Table 1. for the two point rule,. 77
78 Solution (cont.) Now we can use the Gauss Quadrature formula 78
79 Solution (cont) since 79
80 Solution (cont) b) The true error,, is c) The absolute relative true error,, is (Exact value = m) 80
81 81
82 82
83 83
Romberg Rule of Integration
Romberg Rule of ntegration Major: All Engineering Majors Authors: Autar Kaw, Charlie Barker Transforming Numerical Methods Education for STEM Undergraduates 1/10/2010 1 Romberg Rule of ntegration Basis
More informationConvergence of the Method
Gaussian Method - Integration Convergence of the Method Article Information Subject: This worksheet demonstrates the convergence of the Gaussian method of integration. Revised: 4 October 24 Authors: Nathan
More informationSection 6.6 Gaussian Quadrature
Section 6.6 Gaussian Quadrature Key Terms: Method of undetermined coefficients Nonlinear systems Gaussian quadrature Error Legendre polynomials Inner product Adapted from http://pathfinder.scar.utoronto.ca/~dyer/csca57/book_p/node44.html
More informationIn numerical analysis quadrature refers to the computation of definite integrals.
Numerical Quadrature In numerical analysis quadrature refers to the computation of definite integrals. f(x) a x i x i+1 x i+2 b x A traditional way to perform numerical integration is to take a piece of
More informationExtrapolation in Numerical Integration. Romberg Integration
Extrapolation in Numerical Integration Romberg Integration Matthew Battaglia Joshua Berge Sara Case Yoobin Ji Jimu Ryoo Noah Wichrowski Introduction Extrapolation: the process of estimating beyond the
More informationMA6452 STATISTICS AND NUMERICAL METHODS UNIT IV NUMERICAL DIFFERENTIATION AND INTEGRATION
MA6452 STATISTICS AND NUMERICAL METHODS UNIT IV NUMERICAL DIFFERENTIATION AND INTEGRATION By Ms. K. Vijayalakshmi Assistant Professor Department of Applied Mathematics SVCE NUMERICAL DIFFERENCE: 1.NEWTON
More informationNumerical Integra/on
Numerical Integra/on Applica/ons The Trapezoidal Rule is a technique to approximate the definite integral where For 1 st order: f(a) f(b) a b Error Es/mate of Trapezoidal Rule Truncation error: From Newton-Gregory
More informationBindel, Spring 2012 Intro to Scientific Computing (CS 3220) Week 12: Monday, Apr 16. f(x) dx,
Panel integration Week 12: Monday, Apr 16 Suppose we want to compute the integral b a f(x) dx In estimating a derivative, it makes sense to use a locally accurate approximation to the function around the
More informationPowerPoints organized by Dr. Michael R. Gustafson II, Duke University
Part 5 Chapter 17 Numerical Integration Formulas PowerPoints organized by Dr. Michael R. Gustafson II, Duke University All images copyright The McGraw-Hill Companies, Inc. Permission required for reproduction
More informationTABLE OF CONTENTS INTRODUCTION, APPROXIMATION & ERRORS 1. Chapter Introduction to numerical methods 1 Multiple-choice test 7 Problem set 9
TABLE OF CONTENTS INTRODUCTION, APPROXIMATION & ERRORS 1 Chapter 01.01 Introduction to numerical methods 1 Multiple-choice test 7 Problem set 9 Chapter 01.02 Measuring errors 11 True error 11 Relative
More informationRomberg Integration and Gaussian Quadrature
Romberg Integration and Gaussian Quadrature P. Sam Johnson October 17, 014 P. Sam Johnson (NITK) Romberg Integration and Gaussian Quadrature October 17, 014 1 / 19 Overview We discuss two methods for integration.
More informationPHYS-4007/5007: Computational Physics Course Lecture Notes Appendix G
PHYS-4007/5007: Computational Physics Course Lecture Notes Appendix G Dr. Donald G. Luttermoser East Tennessee State University Version 7.0 Abstract These class notes are designed for use of the instructor
More informationNumerical Integra/on
Numerical Integra/on The Trapezoidal Rule is a technique to approximate the definite integral where For 1 st order: f(a) f(b) a b Error Es/mate of Trapezoidal Rule Truncation error: From Newton-Gregory
More information5 Numerical Integration & Dierentiation
5 Numerical Integration & Dierentiation Department of Mathematics & Statistics ASU Outline of Chapter 5 1 The Trapezoidal and Simpson Rules 2 Error Formulas 3 Gaussian Numerical Integration 4 Numerical
More informationIntegration, differentiation, and root finding. Phys 420/580 Lecture 7
Integration, differentiation, and root finding Phys 420/580 Lecture 7 Numerical integration Compute an approximation to the definite integral I = b Find area under the curve in the interval Trapezoid Rule:
More informationBACHELOR OF COMPUTER APPLICATIONS (BCA) (Revised) Term-End Examination December, 2015 BCS-054 : COMPUTER ORIENTED NUMERICAL TECHNIQUES
No. of Printed Pages : 5 BCS-054 BACHELOR OF COMPUTER APPLICATIONS (BCA) (Revised) Term-End Examination December, 2015 058b9 BCS-054 : COMPUTER ORIENTED NUMERICAL TECHNIQUES Time : 3 hours Maximum Marks
More informationAPPM/MATH Problem Set 6 Solutions
APPM/MATH 460 Problem Set 6 Solutions This assignment is due by 4pm on Wednesday, November 6th You may either turn it in to me in class or in the box outside my office door (ECOT ) Minimal credit will
More informationEquivalence Theorems and Their Applications
Equivalence Theorems and Their Applications Tan Bui-Thanh, Center for Computational Geosciences and Optimization Institute for Computational Engineering and Sciences (ICES) The University of Texas at Austin,
More informationNumerical Integration exact integration is not needed to achieve the optimal convergence rate of nite element solutions ([, 9, 11], and Chapter 7). In
Chapter 6 Numerical Integration 6.1 Introduction After transformation to a canonical element,typical integrals in the element stiness or mass matrices (cf. (5.5.8)) have the forms Q = T ( )N s Nt det(j
More informationNumerical Integration (Quadrature) Another application for our interpolation tools!
Numerical Integration (Quadrature) Another application for our interpolation tools! Integration: Area under a curve Curve = data or function Integrating data Finite number of data points spacing specified
More informationIntegration. Tuesday, December 3, 13
4 Integration 4.3 Riemann Sums and Definite Integrals Objectives n Understand the definition of a Riemann sum. n Evaluate a definite integral using properties of definite integrals. 3 Riemann Sums 4 Riemann
More information(A) Opening Problem Newton s Law of Cooling
Lesson 55 Numerical Solutions to Differential Equations Euler's Method IBHL - 1 (A) Opening Problem Newton s Law of Cooling! Newton s Law of Cooling states that the temperature of a body changes at a rate
More informationPractice Exam 2 (Solutions)
Math 5, Fall 7 Practice Exam (Solutions). Using the points f(x), f(x h) and f(x + h) we want to derive an approximation f (x) a f(x) + a f(x h) + a f(x + h). (a) Write the linear system that you must solve
More informationChapter 5: Numerical Integration and Differentiation
Chapter 5: Numerical Integration and Differentiation PART I: Numerical Integration Newton-Cotes Integration Formulas The idea of Newton-Cotes formulas is to replace a complicated function or tabulated
More informationNUMERICAL INTEGRATION. By : Dewi Rachmatin
NUMERICAL INTEGRATION By : Dewi Rachmatin The Trapezoidal Rule Theorem (Trapezoidal Rule) Consider y=f(x) over [x 0,x 1 ], where x 1 =x 0 +h. The trapezoidal rule is This is an numerical approximation
More informationMA2501 Numerical Methods Spring 2015
Norwegian University of Science and Technology Department of Mathematics MA5 Numerical Methods Spring 5 Solutions to exercise set 9 Find approximate values of the following integrals using the adaptive
More informationNumerical Analysis Solution of Algebraic Equation (non-linear equation) 1- Trial and Error. 2- Fixed point
Numerical Analysis Solution of Algebraic Equation (non-linear equation) 1- Trial and Error In this method we assume initial value of x, and substitute in the equation. Then modify x and continue till we
More informationPhysics 115/242 Romberg Integration
Physics 5/242 Romberg Integration Peter Young In this handout we will see how, starting from the trapezium rule, we can obtain much more accurate values for the integral by repeatedly eliminating the leading
More information8.3 Numerical Quadrature, Continued
8.3 Numerical Quadrature, Continued Ulrich Hoensch Friday, October 31, 008 Newton-Cotes Quadrature General Idea: to approximate the integral I (f ) of a function f : [a, b] R, use equally spaced nodes
More informationJim Lambers MAT 460/560 Fall Semester Practice Final Exam
Jim Lambers MAT 460/560 Fall Semester 2009-10 Practice Final Exam 1. Let f(x) = sin 2x + cos 2x. (a) Write down the 2nd Taylor polynomial P 2 (x) of f(x) centered around x 0 = 0. (b) Write down the corresponding
More informationCh. 03 Numerical Quadrature. Andrea Mignone Physics Department, University of Torino AA
Ch. 03 Numerical Quadrature Andrea Mignone Physics Department, University of Torino AA 2017-2018 Numerical Quadrature In numerical analysis quadrature refers to the computation of definite integrals. y
More informationLECTURE 16 GAUSS QUADRATURE In general for Newton-Cotes (equispaced interpolation points/ data points/ integration points/ nodes).
CE 025 - Lecture 6 LECTURE 6 GAUSS QUADRATURE In general for ewton-cotes (equispaced interpolation points/ data points/ integration points/ nodes). x E x S fx dx hw' o f o + w' f + + w' f + E 84 f 0 f
More informationApril 15 Math 2335 sec 001 Spring 2014
April 15 Math 2335 sec 001 Spring 2014 Trapezoid and Simpson s Rules I(f ) = b a f (x) dx Trapezoid Rule: If [a, b] is divided into n equally spaced subintervals of length h = (b a)/n, then I(f ) T n (f
More informationCOURSE Numerical integration of functions
COURSE 6 3. Numerical integration of functions The need: for evaluating definite integrals of functions that has no explicit antiderivatives or whose antiderivatives are not easy to obtain. Let f : [a,
More informationCOURSE Numerical integration of functions (continuation) 3.3. The Romberg s iterative generation method
COURSE 7 3. Numerical integration of functions (continuation) 3.3. The Romberg s iterative generation method The presence of derivatives in the remainder difficulties in applicability to practical problems
More informationAM205: Assignment 3 (due 5 PM, October 20)
AM25: Assignment 3 (due 5 PM, October 2) For this assignment, first complete problems 1, 2, 3, and 4, and then complete either problem 5 (on theory) or problem 6 (on an application). If you submit answers
More informationCS 450 Numerical Analysis. Chapter 8: Numerical Integration and Differentiation
Lecture slides based on the textbook Scientific Computing: An Introductory Survey by Michael T. Heath, copyright c 2018 by the Society for Industrial and Applied Mathematics. http://www.siam.org/books/cl80
More informationSimultaneous Linear Equations
Simultaneous Linear Equations PHYSICAL PROBLEMS Truss Problem Pressure vessel problem a a b c b Polynomial Regression We are to fit the data to the polynomial regression model Simultaneous Linear Equations
More informationSOLUTION OF EQUATION AND EIGENVALUE PROBLEMS PART A( 2 MARKS)
CHENDU COLLEGE OF ENGINEERING AND TECHNOLOGY (Approved by AICTE New Delhi, Affiliated to Anna University Chennai. Zamin Endathur Village, Madurntakam Taluk, Kancheepuram Dist.-603311.) MA6459 - NUMERICAL
More informationGENG2140, S2, 2012 Week 7: Curve fitting
GENG2140, S2, 2012 Week 7: Curve fitting Curve fitting is the process of constructing a curve, or mathematical function, f(x) that has the best fit to a series of data points Involves fitting lines and
More informationTo find an approximate value of the integral, the idea is to replace, on each subinterval
Module 6 : Definition of Integral Lecture 18 : Approximating Integral : Trapezoidal Rule [Section 181] Objectives In this section you will learn the following : Mid point and the Trapezoidal methods for
More informationContents. I Basic Methods 13
Preface xiii 1 Introduction 1 I Basic Methods 13 2 Convergent and Divergent Series 15 2.1 Introduction... 15 2.1.1 Power series: First steps... 15 2.1.2 Further practical aspects... 17 2.2 Differential
More information12.0 Properties of orthogonal polynomials
12.0 Properties of orthogonal polynomials In this section we study orthogonal polynomials to use them for the construction of quadrature formulas investigate projections on polynomial spaces and their
More informationMATHEMATICAL METHODS INTERPOLATION
MATHEMATICAL METHODS INTERPOLATION I YEAR BTech By Mr Y Prabhaker Reddy Asst Professor of Mathematics Guru Nanak Engineering College Ibrahimpatnam, Hyderabad SYLLABUS OF MATHEMATICAL METHODS (as per JNTU
More informationSlope Fields and Differential Equations. Copyright Cengage Learning. All rights reserved.
Slope Fields and Differential Equations Copyright Cengage Learning. All rights reserved. Objectives Review verifying solutions to differential equations. Review solving differential equations. Review using
More informationGAUSS-LAGUERRE AND GAUSS-HERMITE QUADRATURE ON 64, 96 AND 128 NODES
GAUSS-LAGUERRE AND GAUSS-HERMITE QUADRATURE ON 64, 96 AND 128 NODES RICHARD J. MATHAR Abstract. The manuscript provides tables of abscissae and weights for Gauss- Laguerre integration on 64, 96 and 128
More informationExam in TMA4215 December 7th 2012
Norwegian University of Science and Technology Department of Mathematical Sciences Page of 9 Contact during the exam: Elena Celledoni, tlf. 7359354, cell phone 48238584 Exam in TMA425 December 7th 22 Allowed
More informationThen, Lagrange s interpolation formula is. 2. What is the Lagrange s interpolation formula to find, if three sets of values. are given.
SRI RAMAKRISHNA INSTITUTE OF TECHNOLOGY, COIMBATORE- 10 DEPARTMENT OF SCIENCE AND HUMANITIES SUBJECT NUMERICAL METHODS & LINEAR PROGRAMMING ( SEMESTER IV ) IV- INTERPOLATION, NUMERICAL DIFFERENTIATION
More informationWarm up: Recall we can approximate R b
Warm up: Recall we can approximate R b a f(x) dx using rectangles as follows: i. Pick a number n and divide [a, b] into n equal intervals. Note that x =(b a)/n is the length of each of these intervals.
More informationX. Numerical Methods
X. Numerical Methods. Taylor Approximation Suppose that f is a function defined in a neighborhood of a point c, and suppose that f has derivatives of all orders near c. In section 5 of chapter 9 we introduced
More informationSYSTEMS OF NONLINEAR EQUATIONS
SYSTEMS OF NONLINEAR EQUATIONS Widely used in the mathematical modeling of real world phenomena. We introduce some numerical methods for their solution. For better intuition, we examine systems of two
More informationCalculus for the Life Sciences
Calculus for the Life Sciences Integration Joseph M. Mahaffy, jmahaffy@mail.sdsu.edu Department of Mathematics and Statistics Dynamical Systems Group Computational Sciences Research Center San Diego State
More informationIn manycomputationaleconomicapplications, one must compute thede nite n
Chapter 6 Numerical Integration In manycomputationaleconomicapplications, one must compute thede nite n integral of a real-valued function f de ned on some interval I of
More informationVirtual University of Pakistan
Virtual University of Pakistan File Version v.0.0 Prepared For: Final Term Note: Use Table Of Content to view the Topics, In PDF(Portable Document Format) format, you can check Bookmarks menu Disclaimer:
More informationReview. Numerical Methods Lecture 22. Prof. Jinbo Bi CSE, UConn
Review Taylor Series and Error Analysis Roots of Equations Linear Algebraic Equations Optimization Numerical Differentiation and Integration Ordinary Differential Equations Partial Differential Equations
More informationNumerical Integration and Numerical Differentiation. Hsiao-Lung Chan Dept Electrical Engineering Chang Gung University, Taiwan
Numerical Integration and Numerical Differentiation Hsiao-Lung Chan Dept Electrical Engineering Chang Gung University, Taiwan chanhl@mail.cgu.edu.tw Integration 2 Mean for discrete and continuous data
More informationNumerical methods. Examples with solution
Numerical methods Examples with solution CONTENTS Contents. Nonlinear Equations 3 The bisection method............................ 4 Newton s method.............................. 8. Linear Systems LU-factorization..............................
More informationFundamental Numerical Methods for Electrical Engineering
Stanislaw Rosloniec Fundamental Numerical Methods for Electrical Engineering 4y Springei Contents Introduction xi 1 Methods for Numerical Solution of Linear Equations 1 1.1 Direct Methods 5 1.1.1 The Gauss
More informationEngg. Math. II (Unit-IV) Numerical Analysis
Dr. Satish Shukla of 33 Engg. Math. II (Unit-IV) Numerical Analysis Syllabus. Interpolation and Curve Fitting: Introduction to Interpolation; Calculus of Finite Differences; Finite Difference and Divided
More informationSolving ODEs Euler Method & RK2/4
Solving ODEs Euler Method & RK2/4 www.mpia.de/homes/mordasini/uknum/ueb_ode.pdf Major: All Engineering Majors Authors: Autar Kaw, Charlie Barker Transforming Numerical Methods Education for STEM 02/11/10
More informationWeek 10 TENTATIVE SCHEDULE NUMERICAL APPROXIMATION LECTURE # 10
TENTATIVE SCHEDULE INTRODUCTION TO SCIENTIFIC & ENG. COMPUTING BIL 108E, CRN 44448 Week.9_ Polynomials Examples Week.10_ Application of curve fitting : Approximation & Inregration Dr. Feyzi HAZNEDAROĞLU
More informationOutline. 1 Numerical Integration. 2 Numerical Differentiation. 3 Richardson Extrapolation
Outline Numerical Integration Numerical Differentiation Numerical Integration Numerical Differentiation 3 Michael T. Heath Scientific Computing / 6 Main Ideas Quadrature based on polynomial interpolation:
More informationJean-Baptiste Joseph Fourier
Center of Atmospheric Sciences, UNAM September 30, 2016 Jean-Baptiste Joseph Jean-Baptiste Joseph (1768 1830) was a French mathematician and physicist born in Auxerre and best known for initiating the
More informationEAD 115. Numerical Solution of Engineering and Scientific Problems. David M. Rocke Department of Applied Science
EAD 115 Numerical Solution of Engineering and Scientific Problems David M. Rocke Department of Applied Science Multidimensional Unconstrained Optimization Suppose we have a function f() of more than one
More informationApplied Numerical Analysis
Applied Numerical Analysis Using MATLAB Second Edition Laurene V. Fausett Texas A&M University-Commerce PEARSON Prentice Hall Upper Saddle River, NJ 07458 Contents Preface xi 1 Foundations 1 1.1 Introductory
More informationNumerical Analysis. A Comprehensive Introduction. H. R. Schwarz University of Zürich Switzerland. with a contribution by
Numerical Analysis A Comprehensive Introduction H. R. Schwarz University of Zürich Switzerland with a contribution by J. Waldvogel Swiss Federal Institute of Technology, Zürich JOHN WILEY & SONS Chichester
More informationNUMERICAL ANALYSIS SYLLABUS MATHEMATICS PAPER IV (A)
NUMERICAL ANALYSIS SYLLABUS MATHEMATICS PAPER IV (A) Unit - 1 Errors & Their Accuracy Solutions of Algebraic and Transcendental Equations Bisection Method The method of false position The iteration method
More informationNumerical Methods I: Numerical Integration/Quadrature
1/20 Numerical Methods I: Numerical Integration/Quadrature Georg Stadler Courant Institute, NYU stadler@cims.nyu.edu November 30, 2017 umerical integration 2/20 We want to approximate the definite integral
More informationLogarithmic, Exponential, and Other Transcendental Functions. Copyright Cengage Learning. All rights reserved.
5 Logarithmic, Exponential, and Other Transcendental Functions Copyright Cengage Learning. All rights reserved. 5.5 Bases Other Than e and Applications Copyright Cengage Learning. All rights reserved.
More informationNumerical integration and differentiation. Unit IV. Numerical Integration and Differentiation. Plan of attack. Numerical integration.
Unit IV Numerical Integration and Differentiation Numerical integration and differentiation quadrature classical formulas for equally spaced nodes improper integrals Gaussian quadrature and orthogonal
More informationCalculus: Area. Mathematics 15: Lecture 22. Dan Sloughter. Furman University. November 12, 2006
Calculus: Area Mathematics 15: Lecture 22 Dan Sloughter Furman University November 12, 2006 Dan Sloughter (Furman University) Calculus: Area November 12, 2006 1 / 7 Area Note: formulas for the areas of
More informationNUMERICAL METHODS. lor CHEMICAL ENGINEERS. Using Excel', VBA, and MATLAB* VICTOR J. LAW. CRC Press. Taylor & Francis Group
NUMERICAL METHODS lor CHEMICAL ENGINEERS Using Excel', VBA, and MATLAB* VICTOR J. LAW CRC Press Taylor & Francis Group Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Croup,
More informationECEN 615 Methods of Electric Power Systems Analysis Lecture 18: Least Squares, State Estimation
ECEN 615 Methods of Electric Power Systems Analysis Lecture 18: Least Squares, State Estimation Prof. om Overbye Dept. of Electrical and Computer Engineering exas A&M University overbye@tamu.edu Announcements
More informationDynamic Programming. Macro 3: Lecture 2. Mark Huggett 2. 2 Georgetown. September, 2016
Macro 3: Lecture 2 Mark Huggett 2 2 Georgetown September, 2016 Three Maps: Review of Lecture 1 X = R 1 + and X grid = {x 1,..., x n } X where x i+1 > x i 1. T (v)(x) = max u(f (x) y) + βv(y) s.t. y Γ 1
More informationMathematics for Engineers. Numerical mathematics
Mathematics for Engineers Numerical mathematics Integers Determine the largest representable integer with the intmax command. intmax ans = int32 2147483647 2147483647+1 ans = 2.1475e+09 Remark The set
More informationDay 2 Notes: Riemann Sums In calculus, the result of f ( x)
AP Calculus Unit 6 Basic Integration & Applications Day 2 Notes: Riemann Sums In calculus, the result of f ( x) dx is a function that represents the anti-derivative of the function f(x). This is also sometimes
More information1.8. Integration using Tables and CAS
1.. INTEGRATION USING TABLES AND CAS 39 1.. Integration using Tables and CAS The use of tables of integrals and Computer Algebra Systems allow us to find integrals very quickly without having to perform
More informationPARTIAL DIFFERENTIAL EQUATIONS
MATHEMATICAL METHODS PARTIAL DIFFERENTIAL EQUATIONS I YEAR B.Tech By Mr. Y. Prabhaker Reddy Asst. Professor of Mathematics Guru Nanak Engineering College Ibrahimpatnam, Hyderabad. SYLLABUS OF MATHEMATICAL
More informationPreface. 2 Linear Equations and Eigenvalue Problem 22
Contents Preface xv 1 Errors in Computation 1 1.1 Introduction 1 1.2 Floating Point Representation of Number 1 1.3 Binary Numbers 2 1.3.1 Binary number representation in computer 3 1.4 Significant Digits
More informationNumerical Integration
Numerical Integration Sanzheng Qiao Department of Computing and Software McMaster University February, 2014 Outline 1 Introduction 2 Rectangle Rule 3 Trapezoid Rule 4 Error Estimates 5 Simpson s Rule 6
More informationNumerical Analysis. Introduction to. Rostam K. Saeed Karwan H.F. Jwamer Faraidun K. Hamasalh
Iraq Kurdistan Region Ministry of Higher Education and Scientific Research University of Sulaimani Faculty of Science and Science Education School of Science Education-Mathematics Department Introduction
More information19.4 Spline Interpolation
c9-b.qxd 9/6/5 6:4 PM Page 8 8 CHAP. 9 Numerics in General 9.4 Spline Interpolation Given data (function values, points in the xy-plane) (x, ƒ ), (x, ƒ ),, (x n, ƒ n ) can be interpolated by a polynomial
More informationIntegration. Antiderivatives and Indefinite Integration 3/9/2015. Copyright Cengage Learning. All rights reserved.
Integration Copyright Cengage Learning. All rights reserved. Antiderivatives and Indefinite Integration Copyright Cengage Learning. All rights reserved. 1 Objectives Write the general solution of a differential
More informationCurriculum Map: Mathematics
Curriculum Map: Mathematics Course: Calculus Grade(s): 11/12 Unit 1: Prerequisites for Calculus This initial chapter, A Prerequisites for Calculus, is just that-a review chapter. This chapter will provide
More informationDifferentiation of Discrete Functions
Differentiation of Discrete Functions Differentiation of Discrete Functions Introduction Ana Catalina Torres, Autar Kaw University of South Florida United States of America kaw@eng.usf.edu This worksheet
More informationReview for Exam 2 Ben Wang and Mark Styczynski
Review for Exam Ben Wang and Mark Styczynski This is a rough approximation of what we went over in the review session. This is actually more detailed in portions than what we went over. Also, please note
More informationMA1023-Methods of Mathematics-15S2 Tutorial 1
Tutorial 1 the week starting from 19/09/2016. Q1. Consider the function = 1. Write down the nth degree Taylor Polynomial near > 0. 2. Show that the remainder satisfies, < if > > 0 if > > 0 3. Show that
More informationStat 451 Lecture Notes Numerical Integration
Stat 451 Lecture Notes 03 12 Numerical Integration Ryan Martin UIC www.math.uic.edu/~rgmartin 1 Based on Chapter 5 in Givens & Hoeting, and Chapters 4 & 18 of Lange 2 Updated: February 11, 2016 1 / 29
More informationDynamic Trapezoidal Rule
Dynamic Trapezoidal Rule IRVAN JAHJA / 3509099 Program Studi Teknik Informatika Sekolah Teknik Elektro dan Informatika Institut Teknologi Bandung, Jl. Ganesha 0 Bandung 403, Indonesia 3509099@std.stei.itb.ac.id
More informationONE - DIMENSIONAL INTEGRATION
Chapter 4 ONE - DIMENSIONA INTEGRATION 4. Introduction Since the finite element method is based on integral relations it is logical to expect that one should strive to carry out the integrations as efficiently
More informationIntegration Using Tables and Summary of Techniques
Integration Using Tables and Summary of Techniques Philippe B. Laval KSU Today Philippe B. Laval (KSU) Summary Today 1 / 13 Introduction We wrap up integration techniques by discussing the following topics:
More informationMATHEMATICAL FORMULAS AND INTEGRALS
MATHEMATICAL FORMULAS AND INTEGRALS ALAN JEFFREY Department of Engineering Mathematics University of Newcastle upon Tyne Newcastle upon Tyne United Kingdom Academic Press San Diego New York Boston London
More informationCentral Difference Approximation of the First Derivative
Differentiation of Continuous Functions Central Difference Approximation of the First Derivative Introduction Ana Catalina Torres, Autar Kaw University of South Florida United States of America kaw@eng.usf.edu
More informationMATHEMATICAL FORMULAS AND INTEGRALS
HANDBOOK OF MATHEMATICAL FORMULAS AND INTEGRALS Second Edition ALAN JEFFREY Department of Engineering Mathematics University of Newcastle upon Tyne Newcastle upon Tyne United Kingdom ACADEMIC PRESS A Harcourt
More informationDo not turn over until you are told to do so by the Invigilator.
UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 216 17 INTRODUCTION TO NUMERICAL ANALYSIS MTHE612B Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other questions.
More informationMAC Rev.S Learning Objectives. Learning Objectives (Cont.)
MAC 1140 Module 6 Nonlinear Functions and Equations II Learning Objectives Upon completing this module, you should be able to 1. identify a rational function and state its domain.. find and interpret vertical
More informationIntegration. Copyright Cengage Learning. All rights reserved.
4 Integration Copyright Cengage Learning. All rights reserved. 1 4.3 Riemann Sums and Definite Integrals Copyright Cengage Learning. All rights reserved. 2 Objectives Understand the definition of a Riemann
More informationName: Date: Block: Algebra 1 Function/Solving for a Variable STUDY GUIDE
Algebra Functions Test STUDY GUIDE Name: Date: Block: SOLs: A.7, A. Algebra Function/Solving for a Variable STUDY GUIDE Know how to Plot points on a Cartesian coordinate plane. Find the (x, y) coordinates
More informationCase Study Number 2: Pressure of Strip Load on Retaining Wall - Verifying Jarquio s Solution
nesse@mindspring.com pg. 1 of 8 Case Study Number : Pressure of Strip Load on Retaining Wall - Verifying Jarquio s Solution Introduction At a retaining wall site along new construction on Highway 1 in
More information2.6. Solve Factorable Polynomial Inequalities Algebraically. Solve Linear Inequalities. Solution
.6 Solve Factorable Polynomial Inequalities Algebraically A rectangular in-ground swimming pool is to be installed. The engineer overseeing the construction project estimates that at least 148 m of earth
More information