CE 205 Numerical Methods. Some of the analysis methods you have used so far.. Algebra Calculus Differential Equations etc.

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1 CE 205 Numerical Methods Dr. Charisma Choudhury Lecture 1 March 30, 2009 Objective Some of the analysis methods you have used so far.. Algebra Calculus Differential Equations etc. Often not possible to determine analytical solution Complex calculation Sheer size of problem 1

2 Why Numerical Methods? Example: No closed form solution Length of one half of the curve y = sin(x): π 2 1+ cos ( ) 0 x dx Large matrices Computer is not so smart Need to break the problem to simple mathematical problems Add, subtract, multiply, divide, compare Practical considerations Example: Length of a board derived by solving equation: x 3 - x 2-3x + 3=0 How accurately can we measure for practical applications? Course Outline Numerical solution of algebraic and transcendental equations Solution of systems of linear equations Linear and non-linear curve-fitting by least square regression Finite differences Divided differences Interpolation Numerical differentiation and integration Numerical solution of differential equations 2

3 Course Outline Numerical solution of algebraic and transcendental equations Solution of systems of linear equations Linear and non-linear curve-fitting by least square regression Finite differences Divided differences Interpolation Numerical differentiation and integration Numerical solution of differential equations Lecture Plan Algebraic and transcendental equations Curve-fitting Differential equations CE applications and review 5 classes 3 classes 4 classes 1 class 3

4 References Any standard undergraduate textbook on Numerical Methods Some examples: Numerical Analysis: Goel & Mittal Applied Numerical Analysis: Gerald & Wheatley Numerical Methods for Engineers: Chapra & Canale Introductory Methods of Numerical Analysis: Sastry Grading Policy 3 quizzes Tentatively on 5 th, 9 th and 12 th class Or maybe one computer programming assignment? The last one is only a make-up quiz/assignment 4

5 Approach Often multiple methods for solving same problem Which is the most relevant method? Examine the problem What inputs do we have? What accuracy do we need? What is the computational burden? What is the rate of convergence? Errors in Numerical Methods Error in original data/measurement error Truncation error: e x =1+ x/1! + x 2 /2!+ Round-off error: 1/3, Calculations errors significant digits π etc. 5

6 Errors Abs error = true-apprx Relative= abs error /true 5000+/ /-0.1 True=10/3, apprx = Abs=1/3000 Rel=1/10000 Significant digits 4 Algebraic and Transcendental Equations Algebraic: y =ax+b Transcendental: y = a sinx+b cosx 6

7 Solution Methods Bisection/ Half-interval Search Method of false position/regula Falsi Secant Method Newton Raphson Iteration Method Many more Choice of Method Depends on.. Required accuracy Rate of convergence Inputs How does initial approximation affect the computation? Often combination of multiple methods is the optimum 7

8 Bisection/ Half-interval Search Background: Lessons from Graphical approach Solve: f(x)=0 Let y=f(x) Take a set of rectangular coordinates within a range (say x L and x U ) Plot Root(s): the point(s) where y crosses x Background Different variations (Examples shown in next page) Single root Multiple roots No root within specified range 8

9 1 root no roots 3 roots 2 roots Background Observations If f(x l ) and f(x U ) have opposite signs Odd number of roots in between If f(x l ) and f(x U ) have same signs Zero/even number of roots in between Exceptions Discontinuous function 9

10 Discontinuous Function Deductions If f(x) is continuous between x L and x U and f(x l ) * f(x U ) <0 (i.e. f(x l ) and f(x U ) have opposite signs) There is at least 1 real root f(x)=0 between x L and x U 10

11 Example Tangential functions can have any number of roots even irrespective of sign change/unchange Deductions If f(x) is continuous and strictly monotonic between x L and x U and f(x l ) * f(x U ) <0 (i.e. f(x l ) and f(x U ) have opposite signs) There is atmost 1 real root f(x)=0 between x L and x U 11

12 Basis of Bisection Method Narrow down interval [f(x L ) and f(x U ) ]to locate where the sign change occurs Divide into equal sub-intervals to search the point where the sign change occurs Steps Choose x L and x L such that f(x L )*f(x U )<0 Initial estimate of root x R =(x L + x U )/2 f(x R )*f(x L )<0, root is in lower interval, replace x U by x R f(x R )*f(x L )>0, root is in upper interval, replace x L by x R Terminate when f(x R )=f(x L ) or f(x U )-f(x L )<tolerance 12

13 Example f(x)=x 3 -x-1 x L =1,x U =2 x R =1.5 Solution: x=1.324 xl xr xu f(xl) f(xr) f(xu) DifferenceRemark ve +ve +ve replace upper ve -ve +ve replace lower ve +ve +ve replace upper ve -ve +ve replace lower ve +ve +ve replace upper ve +ve +ve replace upper ve -ve +ve replace lower ve -ve +ve replace lower ve +ve +ve OK at 0.01 tolerance Note: OK at 0.01 tolerance. Not ok at tolerance, need more iterations. 13

14 Practice Problems Find roots of the following in bisection method a. x π x e = Sin( ) 2 b. x xe = 2 14