1. Applications of Loops: The power of MATLAB Mathematics + Coding
Numerical Methods
Applications of Loops:
The power of MATLAB
Mathematics + Coding

2. Numerical Methods
Numerical methods are techniques used to find quantitative solutions to mathematical problems.
Many numerical methods are iterative solutions – this means that the technique is applied over and over, gradually getting closer (i.e. converging) on the final answer.
Iterative solutions are complete when the answer from the current iteration is not significantly different from the answer of the previous iteration.
“Significantly different” is determined by the program or user – for example, it might mean that there is less than 1% change or it might mean less than % change.

3. y(x) = x3 + 2x2 – 300/x Ex 1. Newton’s Method
Newton’s Method is used to find the roots of an equation.
“When the curve crosses the x-axis.”
“When f(x)=0.”
Prepare a MATLAB program that uses Newton’s Method to find the roots of the equation:
y(x) = x3 + 2x2 – 300/x

4. Newton’s Method f(x) = x3 + 2x2 – 300/x
What does it seem the answer is?

5. Newton’s Method
Closer look…
Actual Root!

6. Newton’s Method
X is the root we’re trying to find – the value where the function becomes 0
Xn is the initial guess (hopefully, a reasonably-close guess!)
Xn+1 is the next approximation using the tangent line
Credit to:

7. The Big Picture (1/4) 1st: guess! xn Guess an Xn
Hit the actual curve, find the slope of the tangent at that point (using f’(x))
Using this tangent, find the X intercept: Xn+1
1st: guess! xn

8. The Big Picture (1/4) 2nd: Go hit the actual curve
At this (xn , yn), calculate the slope of the curve using the derivatives:
m = f’(xn)

9. The Big Picture (1/4)
3rd: Using the slope m, the coordinates (Xn , Yn), calculate Xn+1 :
Remember: m was obtained from the derivative – only Xn+1 is unknown. What is Yn+1?
Guess an Xn
Hit the actual curve, find the slope of the tangent at that point (using f’(x))
Using this tangent, find the X intercept: Xn+1

10. The Big Picture (2/4)
TOO FAR
Guess again!

11. The Big Picture (2/4)
Iterate!

12. The Big Picture (2/4)
TOO FAR
Guess again!

13. The Big Picture (3/4) – Eventually…
CLOSE ENOUGH!
New estimate
Old estimate
Stop the loop, when there is not much difference between the guess and the new value!

14. Ex 1: algorithm, skeleton
% Provide an initial guess for the x-intercept
% Repeat until close enough: compare most recent x-intercept to previous x-intercept
% Find the slope of the tangent line at the point (requires derivative)
% Using the slope formula, find the x intercept of the tangent line

15. Newton’s Method
% Define initial difference as very large (force loop to start)
% Repeat as long as change in x is large enough
% Make the “new x” act as the current x
% Find slope of tangent line using f’(x)
% Compute y-value
% Use tangent slope to find the next value:
% two points on line are the current point (x, y)
% and the x-intercept (x_new, 0)

16. Xn+1 Xn Xn+1 Xn % Define initial difference as very large
% (force loop to start)
x_n = 1000;
x_np1 = 5; % initial guess
% Repeat as long as change in x is large enough
while (abs(x_np1 – x_n) > )
% x_np1 now becomes the new guess
x_n = x_np1;
% Find slope of tangent line f'(x) at x_n
m = 3*x_n^2 + 4*x_n + 300/(x_n^2);
% Compute y-value
y = x_n^3 + 2*x_n^ /x_n;
% Use tangent slope to find the next value:
% two points on line: the current point (x_n, y_n)
% and the x-intercept (x_np1, 0)
x_np1 = ((0 - y) / m) + x_n;
end
TOO FAR Xn
Xn+1 ? Xn
Xn+1

17. Newton’s Method
Add fprintf’s and see the results:
x y
Root of f(x) = x3 + 2x2 – 300/x is approximately

18. Real-World
Numerical Methods are used in CFD, where convergence can take weeks/months to complete (get “close enough”)...
while loops are generally used, since the number of iteration depends on how close the result should be, i.e. the precision.
Practice! Code this, and use a scientific calculator to see how close the results are.

19. Secant Method: Find the Derivative
Numerical Methods
Secant Method:
Find the Derivative

20. Derivatives, intro.
Another numerical method involves finding the value of the derivative at a point on a curve. Basically, this method uses secant lines as approximations to the tangent line. This method is used when the actual symbolic derivative is hard to find, the formula is simply not needed, or the function is prone to change as the main program does its job.

21. Derivatives, big picture
What is the slope at this point?

22. Derivatives (1/5)
Choose a and b to be equidistant from that point, calculate slope 1
Slope 1
range X
range/2

23. Derivatives (2/5) Reduce the range for the second iteration. Slope 1 x
loX
hiX
Zoom in on x

24. Derivatives (3/5)
Slope 1 x
loX
hiX
Find new secant line.

25. Derivatives (3/5) Slope 1 Slope 2 hiY loY x loX hiX
Calculate new slope

26. Derivatives (3/5) Is slope 2 “significantly different” from slope 1?x
loX
hiX
Assume they are different… let’s iterate!...

27. Derivatives (4/5)
Slope 1
Slope 2
Slope3 x
loX
hiX
Zoom in again..

28. Derivatives (4/5) Is Slope2 very different from Slope3?
Iterate while yes! x
loX
hiX
Zoom in again..

29. (5/5) Eventually…
Stop the loop when the slopes are no longer “significantly different”
Remember: “significantly different” means
maybe a 1% difference?
maybe a 0.1% difference?
maybe a % difference?

30. Derivatives - Algorithm
% Define initial difference as very large % (force loop to start) % Specify starting range % Repeat as long as slope has changed “a lot” % Cut range in half % Compute the new low x value % Compute the new high x value % Compute the new low y value % Compute the new high y value % Compute the slope of the secant line % end of loop
Let’s do this for f(x) = sin(x) + 3/sqrt(x)

31. Code % Define initial difference as very large % (force loop to start)
m = 1;
oldm = 2;
% Specify starting range and desired point
range = 10;
x=5;
ctr=0;
% Repeat as long as slope has changed a lot
while (ctr<3 || (abs((m-oldm)/oldm) > ))
% Cut range in half
range = range / 2;
% Compute the new low x value
lox = x – range ;
% Compute the new high x value
hix = x + range ;
% Compute the new low y value
loy = sin(lox) + 3/sqrt(lox);
% Compute the new high y value
hiy = sin(hix) + 3/sqrt(hix);
% Save the old slope
oldm = m;
% Compute the slope of the secant line
m = (hiy – loy) / (hix – lox);
% increment counter
ctr = ctr + 1;
end % end of loop
Code
(or have just started)

32. Derivatives - the end
Add some fprintf’s and…
LOW x HIGH x Slope Difference in slope
1: , Inf Inf
2: , Inf
3: ,
4: ,
5: ,
6: ,
7: ,
8: ,
9: ,
10: ,
11: ,
12: ,
13: ,
14: ,
15: ,
16: ,
17: ,
Slope of f(x) = sin(x) + 3/sqrt(x) at x=5 is approximately

33. Numerical Methods
Bisection Method A general purpose mathematical solution finder

34. More Examples ES206 Fluid Dynamics
Solve for f
The Colebrook Equation

35. ES206 – Fluid Dynamics “WOW – that’s hard to solve!”
How about this instead:
“By hand, find the square root of 127.3”
Both problems use the same technique to solve: the Bisection Method.

36. Bisection Method
The Bisection Method is a very common technique for solving mathematical problems numerically.
Most people have done this even if they didn’t know the name.
The technique involves taking a guess for the desired variable, then applying that value to see how close the result is to a known value.
Based upon the result of the previous step, a better “guess” is made and the test repeated.

37. Bisection Method Typical human method: Find the square root of 127.3
First guess: 11 (since we know that 112 is 121)
11*11 = 121  too low (looking for 127.3)
Adjust: 12 is the new guess
12*12 = 144  too high
Adjust: (“bisect” the difference)
11.5*11.5 =  too high
Adjust: 11.25
11.25 * =  too low
etc.

38. Bisection Method Let’s use the programmer’s method – an algorithm:
Define highest possible answer (HI) and lowest possible answer (LO)
NEW is the most recent estimate: halfway between HI and LO. OLD is defined to be very far from NEW.
As long as NEW is far enough from OLD:
If NEW is too high, HI gets the value of NEW; if it’s too low, LO gets the value of NEW.
OLD gets the value that was in NEW
NEW gets the value that lies halfway between HI and LO.

39. Bisection Method
For the Colebrook Equation, we know that this equation must hold:
Re-arrange and you have a roots problem. Guess for f and if the equation is not satisfied, adjust f and try again.
This is the same as solving for = X2 . We made it X = 0 and found the roots.

40. Wrapping Up Very small amount of code to solve a mathematical problem
MATLAB can iterate 5 times, or 1,000 times, or a million!
The more precision needed, the more iterations, the more time!
Try to code it. See it you can find the slope of any equation.

41. Wrapping Up We learned about three numerical methods
1. Newton’s Method for finding the roots of an equation
Requires the symbolic derivative
2. Secant line approximation of a tangent point
Finds the numerical value of the derivative at a point
3. Bisection Method
Finds numerical solutions to complex mathematical problems
Very general – essentially “Guess, Test, Adjust”