1. Finding zeros (also called roots) of a function
Overview:
Define the problem
Methods of solution
Graphical
Newton’s
Bisection
Secant

2. Define the problem
For a function f(x) of a single independent variable, find all x0
such that f(x0)=0.
(Some methods can be generalized to multiple independent
variables)
Other ways that the problem can be stated:
find all x where graphs of f1(x) and f2(x) intersect
find all extrema of f(x) on [a,b]

3. Graphical solutions Open MatLab
Under “new” open the macro for function definition
Type the following:
function [ ]=graphical(fh,xa,xb)
fplot(fh,[xa,xb])
grid on
end
fh is a “function handle” defined before graphical is called
xa and xb are the range of the independent variable in the output plot

4. Calling graphical in the command window
x^3-x^2-2*x+1;
>>graphical(myf,-1,1.5);
Read approximate zero of myf(x) from graph

5. Alternative way to define myf
>>myf=inline(‘x^3-x^2-2*x+1’);
>>graphical(myf,-1,1.5);
Every cubic has either 1 or 3 real zeros (why)
How many real zeros does this cubic have?
What are their approximate values?
Discover by extending the range of graphical

6. Expand the range of the x axis
>>myf=inline(‘x^3-x^2-2*x+1’);
>>graphical(myf,-1,1.5);
>>graphical(myf,-2,2.5);
approximate zeros are -1.25, 0.5, 1.8

7. Use graphical to find the intersections of ex and 3x2
In MatLab the function name for ex is exp(x)
In command window ?
You don’t know how many or where intersections occur
Start by casting a wide net
>>graphical(myf,-5,5)
Refine, if needed, to get better estimates of zeros

8. Solution after refining range of x axis:
approximate points of intersection of ex and 3x2 are -0.5, 1, and 3.75

9. Making a graph with MatLab’s plot function
>>myf=inline(‘exp(x)-3*x.^2’);
>>x=linspace(-1,4,50);
>>plot(x,myf(x));
>>grid on
Why x.^2 for x2?

10. Use graphical to estimate the zeros of
f(x) = ln(x+1) + tan(2x) on [-1,9]
Using MatLab documentation you could discover
Function name for ln is log
Function name for base 10 logarithm is log10

11. To make your graph more useful for estimating zeros, change range of y axis using
>>axis([xmin, xmax, ymin, ymax])

12. f(x) = ln(x+1) + tan(2x) with better range on the y axis
Where are the zeros?
f(x) X

13. Each branch contains a zero.
f(x) = ln(x+1) + tan(2x)
Each branch contains a zero.
Line connecting branches is plotting artifact
f(x) X

14. Newton’s Method

15. Non-linear problem reduced to sequence of linear problems
How do we get this result?
Non-linear problem reduced to sequence of linear problems

16. Generalize to xk and xk+1

17. Develop a MatLab code for Newton’s method
What is the basic idea
Put basic idea into a pseudocode
Translate pseudocode into MatLab syntax

18. Develop a pseudocode for Newton’s Method
xk+1 = xk – f(xk)/f’(xk)
Basic idea behind calculation;
calculate sequence x1, x2, … xk, xk+1,…
set x1 to the initial guess from graphical
stop when change xk-> xk+1 is small
assign root to xk+1
calculate f(xk+1)

19. pseudocode for Newton’s method xk+1 = xk – f(xk)/f’(xk)
Example of iterative code
function [r,fr]=newton(fh,dfh,x0)
Initialization
steps = 0
x <- x0
maxsteps
convergence requirement on fractional change (minre)
large initial value of fractional change (re)
while (re>minre) && (steps<maxsteps)
increment steps
save current estimate of root
calculate next estimate of root
re <- absolute value of fractional change
end while loop
r = best estimate of root
fr = fh(r)
end function

20. MatLab code for Newton’s method created in editor using the
function macro and saved as newton.m

21. Use Newton’s method to find all the values of x where the
graphs of ex and 3x2 intersect.
How many intersections and approximate values of x?
Call graphical
>>myf=inline(‘exp(x)-3*x.^2’);
>>graphical(myf,-1,4)

22. Solution after refining range of x axis:
approximate points of intersection of ex and 3x2 are -0.5, 1, and 3.75

23. Write a script to ind all intersections of ex and 3x2 by Newton’s method.

24. Write a script to ind all intersections of ex and 3x2 by Newton’s method.
In the editor
exp(x)-3*x^2;
exp(x)-6*x;
[r1,fr1]=newton(myf,mydf,0.5);
disp([r1,fr1])
[r2,fr2]=newton(myf,mydf,1);
disp([r2,fr2])
[r3,fr3]=newton(myf,mydf,3.7);
disp([r3,fr3])
Copy and paste into the command window

25. Script to refine multiple intersections of ex and 3x2 by Newton’s method
Writing a script in the editor, whether you
save it or not, has the advantage that:
Typos don’t lead to a cluttered command
window and
Editor copy and paste allows less typing
This screen shot shows copy and paste
into the command window
The script in the editor is called “Untitled”
If you want to keep it, use “save as”

26. Assignment 1: due 1/17/19
Estimate all of the zero of x3-x2-2x+1 graphically.
Write a MatLab code for Newton’s method.
Use your code to refine the graphical estimates.
Hand in copies of your graph and the command window where the functions were called and zeros returned.

27. Derive expected convergence rate of Newton’s method

28. The convergence of Newton’s method is usually quadratic.
Error in the n+1 estimate is proportional to the square of error in nth estimate.
Makes convergence fast for any “good” initial estimate.
Proportionality constant is different for different initial guesses.
Makes convergence slightly dependent on the initial guess.
How do I change function newton to convergence data?

29. Extended pseudocode for Newton’s method
function [r,fr,logre]=newton(fh,dfh,x0)
Initialization
steps = 0
x1 <- x0
maxsteps
convergence requirement on fractional change (minre)
assign a large initial value of fractional change (re)
while (re>minre) && (steps<maxsteps)
increment steps
save current estimate of root
calculate next estimate of root
re <- absolute value of fractional change
logre(steps)=log base 10(re)
end while loop
r = best estimate of root
fr = fh(r)
end function

30. Modify Newton’s method to get convergence data

31. Get convergence data to compare rates of convergence
for different initial guesses
Note alternative to “inline”
Note suppressed output of logre

32. Write script to get convergence data for same root with different initial guesses
How do we ensure convergence to the same root?
Make a semi-log plot convergence data
How do we make a semi-log plot?
Put results for both guesses on the same set of axes
How do we put 2 graphs on the same axes?

33. Find intersections of ex and 3x2 by Newton’s method.
For initial guesses 0.5 and 1.5, Newton’s method should converge to the zero near 1, since max and min separate these initial guesses from other zeros.

34. Since the convergence data is saved as log10(myrel),
we can use a linear plot.
Log10(myrel) = -5 means convergence to 10-5
Note alternative to “inline”
Note suppressed output of logre

35. Pseudocode for script to compare convergence Newton’s method for different intial guesses
define function handles for f(x) and f’(x)
Call newtonzeros with initial guess1
Save convergence data as array1
Call newtonzeros with initial guess2
Save convergence data as array2
Call plot(array1) Note: just one argument
hold on
Call plot(array2)
hold off

36. Script to compare convergence of Newton’s method

37. Convergence of Newton’s method with different initial guesses
Note: only axes and curves returned from MatLab
X0=0.5
X0=1.5
Y axis is log10(re)
-5 corresponds to
re = 10-5
Convergence to root of ex – 3x2
in [0.5,1.5] with different x0
X axis is number of iterations

38. Assignment 2, Due 1/22/19
f(x) = ex - 3x2 has a zero in the interval [-1, 0].
Modify your Newton’s method code to return convergence data as log10(re).
Use plot to compare the rates of convergence to the root with initial guesses
0 and -1.
Verify that both initial guesses converge to the same zero. Hand in a copy
of command window where Newton’s method was called Hand in your plot
with labels (by hand is OK) on axes and curves to show which curve goes
with which initial guess.

39. Usually the convergence of Newton’s method is quadratic. Error in the
n+1 estimate of the root is proportional to the square of the error in the
nth estimate of the root.
Since xk+1 = xk – f(xk)/f ’(xk) involves f ’(x) in the denominator, convergence
is not as fast when f ’(r)=0. Called a “multiple” root with multiplicity 2.

40. Ever other root has multiplicity at least 2
f(x) = cos(x) – cos(3x)
Ever other root has multiplicity at least 2 X

41. Convergence problems with Newton’s method
All can be avoided by a good initial guess

42. Bisection method:
Does not require a derivative of f(x)
Cannot have the convergence problems shown on previous slide
Disadvantage: slow convergence

43. Bisection method for continuous functions
Implies at least one zero
between a and b
c=(a+b)/2
Exactly one of the following will be true
If f(c)=0 then c is a root.
If f(c)f(a)<0 then a root is in [a,c].
If f(c)f(b)<0 then a root is in [c,b].
Which is the case illustrated here?
Requires starting values a and b with f(a)f(b)<0
What is wrong with the choice of a and b illustrated here?

44. Starting values for bisection method
Let c = (a+b)/2
f(a)f(b)<0, but a and b are not good starting values.
Only one root should be between starting values.
Use graphical solution to choose good starting values.

45. Basic idea of bisection method
In each cycle of iteration with a and b as end points,
c = (a+b)/2 is the next estimate of the zero
If f(c) = 0 stop. c is the zero between a and b.
Decrease the interval around the zero by half
c becomes one end of the new interval
If f(a)f(c) < 0, a is the other end: b <- c
otherwise b is the other end: a <- c
converged?
no: recycle
yes: stop

46. Basic pseudocode for bisection method
function [r,fr]=bisection(fh,xa,xb)
Initialize steps, maxsteps, re and minre (as in Newton)
xc=(xa+xb)/2;
while (re>minre) && (steps<maxsteps)
increment steps
save current estimate of root (oldxc)
reassign either xa or xb to xc
calculate a new xc=(xa+xb)/2
if re <- absolute value of fractional change in xc
end while loop
r = best estimate of root
fr = fh(r)
end function
No check for a sign change of function between initial xa and xb
Does not allow for f(xc)=0

47. Extended pseudocode for bisection method
function [r,fr]=bisection(fh,xa,xb)
If f(xa)f(xb)>0, write message and return
Initialize steps and re (as in Newton)
xc=(xa+xb)/2; if f(xc)=0 write message and return
while (re>minre) && (steps<maxsteps)
increment steps
save current estimate of root (oldxc)
reassign either xa or xb to xc,
calculate a new xc (probably don’t need to check xc=0)
if re <- absolute value of fractional change in xc
end while loop
r = best estimate of root
fr = fh(r)
end function
Very unlikely that f(xc) is identically zero, unless starting values
have are chosen to make that true

48. x=0 is a root. Don’t need a root finder
f(x) = ln(x+1) + tan(2x)
x=0 is a root. Don’t need a root finder
If bisection call with xb = -xa, initial xc is a root
Code should stop
f(x) X

49. Implement logic of bisection method in MatLab
Relational operators in MatLab
< less than
<= less than or equal
== equal
-= not equal
> greater than
>= greater than or equal

50. Matlab code for bisection method
disp([‘landed on a zero’])

51. Test your bisection function
f(x)=x5 + x3 +3 does not have a zero on [-1,1]
Call your bisection function with xa=-1 and xb=1
Does it generates warning message?
f(x) has a zero between -1.5 and -1
Did your bisection function find a zero at ?

52. Assignment 3, Due 1/24/19
Write bisection function that finds root and saves convergence data
Use code to find the zero f(x) = ex - 3x2 in the interval [-1, 0].
On a semi-log plot, compare the rate of convergence of the bisection
method with starting interval [-1,0] to that of Newton’s method with initial
guesses -1 and 0.
Verify that Bisection and Newton’s method converge to the same zero.
Hand in a copy of the command window where functions were called.
Hand in a plot of the convergence data for bisection and Newton’s method
with axes and curves labeled.

53. -1 and 0 are good starting values for bisection method to
converge to root near -0.5

54. Bisection method converges slower than Newton

56. Linear on semi_log_y plot

57. Exactly the result shown by this plot

58. Bisection method does not require the derivative of a function to find
its zeros.
However, convergence is slow.
Secant method does not require a derivative and converges almost
as fast as Newton’s method.

59. Newton’s method without derivatives
~ xn – f(xn)/f’(xn)
Newton’s method without derivatives

60. Basic pseudocode for secant method is almost the same as for Newton’s method
function [r,fr]=secant(fh,xn,xnm1)
Initialize: steps, max steps, re, minre
x = xn
while (re>minre) && (steps<maxsteps)
increment steps
save current estimate of root
calculate next estimate of root using numerical derivative
re <- absolute value of fractional change
end while loop
r = best estimate of root
fr = fh(r)
end function
xn should be the better of the 2 initial estimates of the root
How do we confirm that this true?
In some steps, estimate of the root may not improve
How do we deal with this problem?

61. Convergence may not be uniform
Always use best estimate of root as xn in each cycle
If starting values are 1 and -1,
what do we expect to happen?

62. Initial xn+1 is not as good as xn, (how do we know this?)
Use best estimate of root as xn (what value is this?)
Use bad first estimate as xn-1
Did the second attempt work?

63. Extended pseudocode for secant method
function [r,fr]=secant(fh,xnm1,xn)
Test if |fn|<|fnm1|; otherwise interchange xn and xnm1
Initialize steps maxsteps, minre and re (as in Newton)
x=xn
while (re>minre) && (steps<maxsteps)
increment steps
approximate derivative and calculate xnp1
if |fnp1|<|fn| xn=xnp1 and xnm1=xn
if |fnp1|>|fn| keep xn and xnm1=xnp1
calculate re = |(xn-xnm1)/xn|
end while loop
r = best estimate of root
fr = fh(r)
end function

65. Assignment 4, Due 1/24/19
Write secant function that finds root and saves convergence data
Use code to find zero f(x) = ex - 3x2 in the interval [-1,0].
On a semi-log plot, compare the rate of convergence of the secant
method to that of the bisection method with the same starting values [-1,0]
and to Newton’s method with initial guesses -1 and 0.
Hand in a copy of the command window where you verified that all methods converged
to same zero.
Hand in a plot of the convergence data for secant, bisection, and Newton’s method
with axes and curves labeled.
Write a script for this problem when root is between 0.5 and 1.5.

68. Assignment 4, Due 1/24/19
Write secant function that finds root and saves convergence data
Use code to find zero f(x) = ex - 3x2 in the interval [-1, 0].
On a semi-log plot, compare the rate of convergence of the secant
method to that of the bisection method with the same starting values
of -1 and 0. Compare these results to Newton’s method with initial
guesses 0 and -1.
Hand in a plot with labels (by hand is OK) on axes and the curves
to show which curve goes with which initial guess and which method.
Hand in a copy of the command window where you confirmed that all
methods converge to the same root.

69. Move due date for HW4 to 1/29/19
Return graded HW1-HW4 by 1/31/19
Quiz on finding zeros of a function 2/5/19

71. Note similarity to
Newton’s method (slide 20)
en+1 = [ ] en2

72. Suggested problems from text (6th edition)
Chapter 3.1 p85
Problems 1, 4, 9, 10
Computer problems 1, 2, 4, 7
Chapter 3.2 p101
Problems 4, 15, 17, 19
Computer problems 1, 2, 4, 8, 9, 14
Chapter 3.3 p119
Problems 2, 3, 5
Computer problems 1, 3, 5, 6, 7