1. Comp 208 Computers in Engineering Yi Lin Winter, 2007
01/06/2019
Lecture 20 Root finding
Comp 208 Computers in Engineering
Yi Lin
Winter, 2007
6/1/201911/15/05
Comp208 Computers in Engineering

2. Comp208 Computers in Engineering
01/06/2019
01/06/2019
Lecture 19 Root finding
Objectives
Learn about 4 numerical methods
Bisection
Secant
False position
Newton Raphson
Learn new features of C
Pointer to function, pass as an argument to a function
typedef
Learn enough in-depth C to implement these
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3. Comp208 Computers in Engineering
01/06/2019
01/06/2019
Intro to Root Finding
A way to solve equations in one unknown that cannot be solved symbolically.
For example, suppose that we would like to solve the simple equation: x2 = 5
To solve this equation using the bisection method, we first manipulate it algebraically so that one side is zero. x2 - 5 = 0
Finding a solution to this equation is then equivalent to finding a root of the function
f(x) = x2 - 5
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4. Comp208 Computers in Engineering
01/06/2019
Bisection (con’t)
We next find two numbers, a positive guess and a negative guess, so that
f(positive guess) > 0 and f(negative guess) <0.
So long as the function whose root we are finding is continuous, there must be at least one root between the positive and negative guesses.
The bisection method locates such a root by repeatedly narrowing the distance between the two guesses.
middle = (positive guess + negative guess)/2
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Comp208 Computers in Engineering

5. Comp208 Computers in Engineering
01/06/2019
Bisection example X2
f(x2)
X3=(x1+x2)/2
f(x3)
X4=(x3+x1)/2
f(x4)
X5=(x4+x1)/2
f(x5)≈0 X1
f(x1)
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Comp208 Computers in Engineering

6. Non-recursive implementation
double bisection_rf(DfD f, double x0, double x1, double tol){
double middle = (x0 + x1) / 2.0;
while(fabs(f(middle)) > tol) {
if(f(middle) * f(x0) < 0)
x1 = middle;
else if ( f(middle) * f(x1) < 0 )
x0 = middle;
else
printf("Should not be here! f(x0) and f(x1) are the same sign!\n");
middle = (x0 + x1) / 2.0; }
return middle;
6/1/201911/15/05
Comp208 Computers in Engineering

7. Bisection-recursive implementation
01/06/2019
Bisection-recursive implementation
double bisection_rf(DfD f, double x0, double x1, double tol) {
double middle = (x0 + x1) / 2.0;
if((middle - x0) < tol)
return middle;
else if(f(middle) * f(x0) < 0.0)
// From the Intermediate Value Theorem, if there is a sign change then there is a root.
return bisection_rf(f, x0, middle, tol);
else
return bisection_rf(f, middle, x1, tol); }
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Comp208 Computers in Engineering

8. Bisection complete program
01/06/2019
Bisection complete program
#include <stdio.h>
#include <math.h>
typedef double (*DfD) (double);
double bisection_rf(DfD f, double x0, double x1, double tol) {
double middle = (x0 + x1) / 2.0;
if(fabs(f(middle)) < tol)
return middle;
else if(f(middle) * f(x0) < 0.0)
// From the Intermediate Value Theorem, if there is a sign change then there is a root.
return bisection_rf(f, x0, middle, tol);
else
return bisection_rf(f, middle, x1, tol); }
double func(double x){
return x*x - 5; }
int main(){
double root, tolerance;
tolerance = ;
root = bisection_rf(func, -4, 1, tolerance);
printf("root=%lf\n", root);
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Comp208 Computers in Engineering

9. Bisection counter-example
01/06/2019
Bisection counter-example
If f(x1) and f(x2) are both positive or both negative, unable to find the root! X1
f(x1)
X3=(x1+x2)/2
f(x3) x2
f(x2)
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Comp208 Computers in Engineering

10. Comp208 Computers in Engineering
01/06/2019
Secant method X1
f(x1) x2
f(x2)
f(x4)
f(x5)≈0
f(x3) x3
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Comp208 Computers in Engineering

11. Comp208 Computers in Engineering
01/06/2019
Secant method
Assume a function to be approximately linear in the region of interest.
Each improvement is taken as the point where the approximating line crosses the axis.
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Comp208 Computers in Engineering

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Secant Non-recursive
double secant(DfD f, double x1, double x2, double tol, double count){
double slope = (f(x1) - f(x2) )/(x1-x2);
double x = x2 - f(x2)/slope;
while(fabs(f(x)) > tol && count-->0){
x1 = x2;
x2 = x;
slope = (f(x1) - f(x2) )/(x1-x2);
x = x2 - f(x2)/slope; }
return x;
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Comp208 Computers in Engineering

13. Comp208 Computers in Engineering
01/06/2019
False position method
The secant method retains only the most recent estimate, so the root does not necessarily remain bracketed.
Combined with bisection in order to bracket the root.
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Comp208 Computers in Engineering

14. False position non-recursive
double falseposition(DfD f, double x1, double x2, double tol){
double slope = (f(x1) - f(x2) )/(x1-x2);
double x = x2 - f(x2)/slope;
while(fabs(f(x)) > tol){
if(f(x) * f(x1) < 0)
x2 = x;
else
x1 = x;
slope = (f(x1) - f(x2) )/(x1-x2);
x = x2 - f(x2)/slope; }
return x;
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Comp208 Computers in Engineering

15. Comp208 Computers in Engineering
01/06/2019
Newton-Raphson
Much like secant, but using the real derivative x1
f(x1) x4
f(x4)≈0
f(x2) x2 x3
f(x3)
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Comp208 Computers in Engineering

16. Comp208 Computers in Engineering
01/06/2019
Newton-Raphson
double newton_raphson_rf(DfD f, double x0, double tol) {
double h = 0.001;
double derivative;
do{
derivative = centered3_diff(f, x0, h);
x0 = x0 - f(x0)/derivative;
} while(fabs(f(x0)) > tol);
return x0; }
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Comp208 Computers in Engineering

17. How to calculate derivative
01/06/2019
How to calculate derivative
Forward 3 points differentiation
Backward 3 points differentiation
Centered 3 points differentiation
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Comp208 Computers in Engineering

18. Comp208 Computers in Engineering
01/06/2019
Forward 3 points
Mathematical formula:
double forward3_diff(double x, double h) {
return (-3*f(x) + 4*f(x+h) - f(x+2*h)) / (2 * h); }
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Comp208 Computers in Engineering

19. Comp208 Computers in Engineering
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Backward 3 points
Mathematical formula:
double backward3_diff(double x, double h) {
return (f(x - 2 * h) - 4 * f(x - h) + 3 * f(x)) / (2 * h); }
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Comp208 Computers in Engineering

20. Comp208 Computers in Engineering
01/06/2019
Centered 3 points
Mathematical formula:
double centered3_diff(double x, double h) {
return (-f(x - h) + f(x + h)) / (2 * h); }
6/1/201911/15/05
Comp208 Computers in Engineering