euler’s and heun’s methods douglas wilhelm harder, m.math. lel department of electrical and...
TRANSCRIPT
![Page 1: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/1.jpg)
Euler’s and Heun’s Methods
Douglas Wilhelm Harder, M.Math. LELDepartment of Electrical and Computer Engineering
University of Waterloo
Waterloo, Ontario, Canada
ece.uwaterloo.ca
© 2012 by Douglas Wilhelm Harder. Some rights reserved.
![Page 2: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/2.jpg)
2
Outline
This topic discusses numerical differentiation:– Initial-value problems– Euler’s method– Heun’s method– Multi-step methods
Euler's and Heun's Methods
![Page 3: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/3.jpg)
3
Outcomes Based Learning Objectives
By the end of this laboratory, you will:– Understand how to approximate a solution to a 1st-order IVP
using Euler’s method– Understand the limitations of Euler’s method– Be able to apply the same ideas from the trapezoidal rule to
improve Euler’s method, i.e., Heun’s method
Euler's and Heun's Methods
![Page 4: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/4.jpg)
4
Initial-value Problems
Given the initial value problem
Invariably, initial-value problems deal with time:– We know the state y0 of a system at time t0
– We understand how the system evolves (through the ODE)– We want to approximate the state in the future
(1)
0 0
,y t f t y t
y t y
Euler's and Heun's Methods
![Page 5: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/5.jpg)
5
Ordinary Differential Equations
Your first question should be:
Can we always write a 1st-order ODE in the form:
?
For example, the ODE could be implicitly defined as:
Fortunately, the implicit function theorem says that, in almost all cases, “yes”– We may end up using a truncated approximation similar to
Taylor series
(1) ,y t f t y t
3(1) (1), , 1 sin 0F t y t y t t y t y t
Euler's and Heun's Methods
![Page 6: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/6.jpg)
6
Ordinary Differential Equations
What does the formula
mean?
Given any point (t*, y*), if a solution y(t) to the ODE passes through that point, the derivative of the solution must be:
(1) ,y t f t y t
(1) * * *,y t f t y
Euler's and Heun's Methods
![Page 7: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/7.jpg)
7
Ordinary Differential Equations
For example, the ODE
suggests, for example, at the point (1, 2), the slope is approximately
We could pick a few hundred points, determine the slopes at each of these lines, and plot that slope
(1) cosy t t y t y t t y t
1 2 2 1 cos 2 1.416146836
Euler's and Heun's Methods
![Page 8: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/8.jpg)
8
Ordinary Differential Equations
Doing this with the ODE
yields
(1) cosy t t y t y t t y t
Euler's and Heun's Methods
![Page 9: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/9.jpg)
9
Ordinary Differential Equations
The following are three solutions that satisfy these initial conditions
0 1
0 0
0 1
y
y
y
Euler's and Heun's Methods
![Page 10: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/10.jpg)
10
Ordinary Differential Equations
The ODE
was chosen because there is no explicit solution
The next example does have explicit solutions
(1) cosy t t y t y t t y t
Euler's and Heun's Methods
![Page 11: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/11.jpg)
11
Ordinary Differential Equations
Consider the ODE
This has the following field plot:
2 2(1) 1 1y t y t t
Euler's and Heun's Methods
![Page 12: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/12.jpg)
12
Ordinary Differential Equations
This clearly has y(t) = 1 as one solution; however, another solution is
2 2(1) 1 1y t y t t
3 2
3 2
3 3
3 3 3
t t ty t
t t t
Euler's and Heun's Methods
![Page 13: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/13.jpg)
13
Ordinary Differential Equations
This clearly has y(t) = 1 as one solution; however, another solution is
We can confirm this by substitution
3 2
3 2
3 3
3 3 3
t t ty t
t t t
2 2(1) 1 1y t y t t
Euler's and Heun's Methods
![Page 14: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/14.jpg)
14
Ordinary Differential Equations
Calculating the derivative:
Substituting the function into the equation
Everything cancels in the numerator except the one
23 2
23 2 3 2
9 13 3
3 3 3 3 3 3
td t t t
dt t t t t t t
23 22
3 2
2 23 2 3 2 3 2 3 22
23 2
3 31 1
3 3 3
3 3 2 3 3 3 3 3 3 3 31
3 3 3
t t tt
t t t
t t t t t t t t t t t tt
t t t
23 9
2 21 1y t t
Euler's and Heun's Methods
![Page 15: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/15.jpg)
15
Ordinary Differential Equations
Now, we see that y(1) = 1:
The slope at this point should be:
If we evaluate the calculated derivative at t = 1, we get:
3 2
3 2
1
2 6 6 11 2 6 6 11 31 1
2 6 6 17 2 6 6 17 3t
t t ty
t t t
2 2(1) 1 1,1 1 1 1 1 16y f
2
2
36 1 1 36 416
92 6 6 17
Euler's and Heun's Methods
![Page 16: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/16.jpg)
16
Euler’s Method
Now, suppose we have an initial condition:
y(t0) = y0
We want to approximate the solution at t0 + h; therefore, we can look at the Taylor series:
where
1 2 20 0 0
1
2y t h y t y t h y h
0 0,t t h
Euler's and Heun's Methods
![Page 17: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/17.jpg)
17
Euler’s Method
We can replace the initial condition
y(t0) = y0
into the Taylor series
Next, we also know what the derivative is from the ODE:
Thus,
1 2 20 0 0
1
2y t h y y t h y h
1 ,y t f t y t
2 20 0 0 0
1,
2y t h y f t y h y h
Euler's and Heun's Methods
![Page 18: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/18.jpg)
18
Euler’s Method
Thus, we have a formula for approximating the next point
together with an error term .
0 0 0 0,y t h y h f t y
2 21
2y h
Euler's and Heun's Methods
![Page 19: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/19.jpg)
19
Euler’s Method
Using our example:
we can implement both the right-hand side of the ODE and the solution:
2 2(1) 1 1
(0) 0
y t y t t
y
function [dy] = f2a(t, y) dy = (y - 1).^2 .* (t - 1).^2;end
function [y] = y2a( t ) y = (t.^3 - 3*t.^2 + 3*t)./(t.^3 - 3*t.^2 + 3*t + 3);end
Euler's and Heun's Methods
![Page 20: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/20.jpg)
20
Euler’s Method
Using our example:
we can therefore approximate y(0.1):>> approx = 0 + 0.1*f2a(0,0) actual = 0.100000000000000
>> actual = y2a(0.1) actual = 0.082849281565271
>> abs( actual - approx ) ans = 0.017150718434729
2 2(1) 1 1
(0) 0
y t y t t
y
Euler's and Heun's Methods
![Page 21: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/21.jpg)
21
Euler’s Method
Now, if we halve h, the error should drop by a factor of 4
We will therefore approximate y(0.05):>> approx = 0 + 0.05*f2a(0,0) approx = 0.050000000000000
>> actual = y2a(0.05) actual = 0.045384034047969
>> abs( actual - approx ) ans = 0.004615965952031
Previous error when h = 0.1: 0.017150718434729
Euler's and Heun's Methods
![Page 22: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/22.jpg)
22
Euler’s Method
Lets consider what we are doing:– The actual solution is in red– The two approximations are shown as circles
• We are following the same slope out from (0, 0)
Euler's and Heun's Methods
![Page 23: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/23.jpg)
23
Euler’s Method
The problem is, the second approximation does not approximate y(0.1)—it approximates the solution at the closer point t = 0.05– How can we proceed to approximate y(0.1)?
Euler's and Heun's Methods
![Page 24: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/24.jpg)
24
Euler’s Method
How about finding the slope at (0.05, 0.05) and following that out for another h = 0.05?
Euler's and Heun's Methods
![Page 25: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/25.jpg)
25
Euler’s Method
How about finding the slope at (0.05, 0.05) and following that out for another h = 0.05?>> 0.05 + 0.05*f2a(0.05, 0.05)ans = 0.090725312500000
Euler's and Heun's Methods
![Page 26: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/26.jpg)
26
Euler’s Method
We could repeat this process again, and approximate the solution at t = 0.15?>> 0.090725312500000 + 0.05*f2a( 0.1, 0.090725312500000 )ans = 0.124209921021793
Euler's and Heun's Methods
![Page 27: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/27.jpg)
27
Euler’s Method
As you can see, the three points are shadowing the actual solution
Euler's and Heun's Methods
![Page 28: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/28.jpg)
28
Euler’s Method
Note that we require more work if we reduce h:– Dividing h by 2 requires twice the work, and– Dividing h by 10 requires ten times the work
to approximate the same final point
Euler's and Heun's Methods
![Page 29: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/29.jpg)
29
Euler’s Method
In addition, we are using an approximation to approximate the next approximation, and so on…– The error for approximating one point is O(h2)– In the laboratory, you will attempt to determine how this affects
the error
Euler's and Heun's Methods
![Page 30: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/30.jpg)
30
Euler’s Method
Thus, given an IVP
and suppose we want toapproximate y(tfinal)
We could simply use
h = tfinal – t0
and find y0 + h f(t0, y0)
Problem: we have no controlover the accuracy
(1)
0 0
,y t f t y t
y t y
Euler's and Heun's Methods
![Page 31: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/31.jpg)
31
Euler’s Method
Thus, given an IVP
and suppose we want toapproximate y(tfinal)
Instead, divide the interval [t0, tfinal] into n points and now repeat Euler’s method n – 1 times
(1)
0 0
,y t f t y t
y t y
Euler's and Heun's Methods
![Page 32: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/32.jpg)
32
Euler’s Method
For example, if we chose n = 11, we would find approximations at0 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 1.00000 ? ? ? ? ? ? ? ? ? ?
where y(0) = 0 and we want to approximate y(1)
Euler's and Heun's Methods
![Page 33: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/33.jpg)
33
Euler’s Method
Use the initial points to approximate y(0.1):0 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 1.00000 0.1000 ? ? ? ? ? ? ? ? ?
Euler's and Heun's Methods
![Page 34: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/34.jpg)
34
Euler’s Method
Use the next two points to approximate y(0.2):0 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 1.00000 0.1000 0.1656 ? ? ? ? ? ? ? ?
Euler's and Heun's Methods
![Page 35: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/35.jpg)
35
Use the next two points to approximate y(0.3):0 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 1.00000 0.1000 0.1656 0.2102 ? ? ? ? ? ? ?
Euler’s Method
Euler's and Heun's Methods
![Page 36: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/36.jpg)
36
Use these two points to approximate y(0.4):0 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 1.00000 0.1000 0.1656 0.2102 0.2407 ? ? ? ? ? ?
Euler’s Method
Euler's and Heun's Methods
![Page 37: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/37.jpg)
37
Use these two points to approximate y(0.5):0 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 1.00000 0.1000 0.1656 0.2102 0.2407 0.2615 ? ? ? ? ?
Euler’s Method
Euler's and Heun's Methods
![Page 38: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/38.jpg)
38
Use these two points to approximate y(0.6):0 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 1.00000 0.1000 0.1656 0.2102 0.2407 0.2615 0.2751 ? ? ? ?
Euler’s Method
Euler's and Heun's Methods
![Page 39: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/39.jpg)
39
Use these two points to approximate y(0.7):0 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 1.00000 0.1000 0.1656 0.2102 0.2407 0.2615 0.2751 0.2835 ? ? ?
Euler’s Method
Euler's and Heun's Methods
![Page 40: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/40.jpg)
40
Use these two points to approximate y(0.8):0 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 1.00000 0.1000 0.1656 0.2102 0.2407 0.2615 0.2751 0.2835 0.2882 ? ?
Euler’s Method
Euler's and Heun's Methods
![Page 41: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/41.jpg)
41
Use these two points to approximate y(0.9):0 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 1.00000 0.1000 0.1656 0.2102 0.2407 0.2615 0.2751 0.2835 0.2882 0.2902 ?
Euler’s Method
Euler's and Heun's Methods
![Page 42: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/42.jpg)
42
Finally, use these two to approximate y(1.0):0 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 1.00000 0.1000 0.1656 0.2102 0.2407 0.2615 0.2751 0.2835 0.2882 0.2902 0.2907
Our approximation is y(1.0) ≈ 0.290681404577720
Euler’s Method
Euler's and Heun's Methods
![Page 43: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/43.jpg)
43
Euler’s Method
You will implement Euler’s method:function [t_out, y_out] = euler( f, t_rng, y0, n )
wheref a function handle to the bivariate function f(t, y)
t_rng a row vector of two values [t0, tfinal]
y0 the initial condition
n the number of points that we will break the interval
[t0, tfinal] into
You will return two vectors:t_out a row vector of n equally spaced values from t0 to tfinal
y_out a row vector of n values where
y_out(1) equals y0
y_out(k) approximates y(t) at t_out(k) for k from 2 to n
Euler's and Heun's Methods
![Page 44: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/44.jpg)
44
Euler’s Method
This function will:1. Determine
2. Assign toa. tout a vector of n equally spaced points going from t0 to tfinal, and
b. yout a vector of n zeros where yout, 1 is assigned the initial value y0,
3. For k going from 1 to n – 1, repeat the following:a. Using f, calculate the slope K1 at the point tout,k and yout,k, and
b. Set .
final 0
1
t th
n
out, 1 out, 1k ky y h K
Euler's and Heun's Methods
![Page 45: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/45.jpg)
45
Euler’s Method
For example, consider our initial-value problem
Approximating the solution on [0, 1] with n = 11 points yields:
>> [t2a, y2a] = euler( @f2a, [0, 1], 0, 11 ) t2a = 0 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000
1.0000 y2a = 0 0.1000 0.1656 0.2102 0.2407 0.2615 0.2751 0.2835 0.2882 0.2902
0.2907
2 2(1) 1 1
0 0
y t ty t
y
Euler's and Heun's Methods
![Page 46: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/46.jpg)
46
Euler’s Method
The function ode45 is Matlab’s built-in ODE solver:[t2a, y2a] = euler( @f2a, [0, 1], 0, 11 );plot( t2a, y2a, 'or' ); hold on[t2a, y2a] = ode45( @f2a, [0, 1], 0 );plot( t2a, y2a, 'b' )
Euler's and Heun's Methods
![Page 47: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/47.jpg)
47
Euler’s Method
The function ode45 is Matlab’s built-in ODE solver:[t2a, y2a] = euler( @f2a, [0, 1], 0, 21 );plot( t2a, y2a, 'or' ); hold on[t2a, y2a] = ode45( @f2a, [0, 1], 0 );plot( t2a, y2a, 'b' )
Euler's and Heun's Methods
![Page 48: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/48.jpg)
48
Euler’s Method
For example, consider our initial-value problem
Approximating the solution on [0, 1] with n = 11 points yields:
>> [t2b, y2b] = euler( @f2b, [0, 1], 1, 11 ) t2b = 0 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000
1.0000 y2b = 1 1.0460 1.1000 1.1626 1.2343 1.3154 1.4059 1.5057 1.6144 1.7310
1.8543
(1) cos
0 1
y t t y t y t t y t
y
Euler's and Heun's Methods
![Page 49: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/49.jpg)
49
Euler’s Method
In this case, Euler’s method does not fare so well:hold on[t2b, y2b] = euler( @f2b, [0, 1], 1, 11 );plot( t2b, y2b, 'or' )[t2b, y2b] = ode45( @f2b, [0, 1], 1 );plot( t2b, y2b, 'b' )
Euler's and Heun's Methods
![Page 50: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/50.jpg)
50
Euler’s Method
We can increase the number of points by a factor of 10:hold on[t2b, y2b] = euler( @f2b, [0, 1], 1, 101 );plot( t2b, y2b, '.r' )[t2b, y2b] = ode45( @f2b, [0, 1], 1 );plot( t2b, y2b, 'b' )
Euler's and Heun's Methods
![Page 51: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/51.jpg)
51
Error Analysis
Now, we saw the error for Euler’s method was O(h2)– However, except with the first point, we are using an
approximation to find an approximation
– Thus, repeatedly applying Eulerresults in an error of O(h)
Euler's and Heun's Methods
1
2 2
1
1
2
n
kk
E y h
1,k k kt t
1
2
12
n
kk
hy h
final
0
12 2
1
tn
kk t
y h y d
final
0
2
2
t
t
hy d
![Page 52: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/52.jpg)
52
Improving on Euler’s Method
In the lab, you will find that, for Euler’s method:– Reducing the error by half requires twice as much effort and
memory– Reducing the error by a factor of 10 requires ten times the time
and memory
This is exceptionally inefficient and we will therefore take this lab and the next lab to see how we can improve on Euler’s method
Euler's and Heun's Methods
![Page 53: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/53.jpg)
53
Improving on Euler’s Method
Suppose you are approximating the integral of a function over an interval:
b
a
g x dx
Euler's and Heun's Methods
![Page 54: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/54.jpg)
54
Improving on Euler’s Method
One of the worst approximations would be to simply use the value of the function at one end-point:
b
a
g x dx g a b a
Euler's and Heun's Methods
![Page 55: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/55.jpg)
55
Improving on Euler’s Method
At the very least, it would be better to approximate the integral by taking the average of the two end-points:
This is the trapezoidal rule of integration
2
b
a
g a g bg x dx b a
Euler's and Heun's Methods
![Page 56: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/56.jpg)
56
Improving on Euler’s Method
When we are essentially integrating using information only at the initial value:
1 ,y t f t y t
Euler's and Heun's Methods
0 0
0 0
1 ,t h t h
t t
y t dt f t y t dt
0
0
0 0 ,t h
t
y t h y t f t y t dt
0
0
0 0 ,t h
t
y t h y t f t y t dt
0 0 0,y t h f t y
![Page 57: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/57.jpg)
57
Improving on Euler’s Method
The problem is, we would have to know the slope at t0 + h in order to approximate mimic the trapezoidal rule
Note, however, that Euler’s method gives usan approximationof y(t0 + h)
y(t0 + h) ≈ y0 + hK1
Therefore, we can approximate thethe slope at t0 + h with
1 0 0,K f t y
2 0 0 1,K f t h y h K
Euler's and Heun's Methods
![Page 58: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/58.jpg)
58
Improving on Euler’s Method
Thus, we have one slope and one approximation of a slope:
Applying the same principle as thetrapezoidal rule, we would thenapproximate
1 0 0
2 0 0 1
,
,
K f t y
K f t h y h K
1 20 0 2
K Ky t h y h
Euler's and Heun's Methods
![Page 59: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/59.jpg)
59
Heun’s Method
Graphically, Euler’s method follows the initial slope out a distance h– We calculate only one slope:
K1
1 0 0,K f t y
Euler's and Heun's Methods
![Page 60: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/60.jpg)
60
Heun’s Method
Heun’s method states that we determine the slope at the second point, too
K1
K2
2 0 0 1,K f t h y h K
Euler's and Heun's Methods
![Page 61: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/61.jpg)
61
Heun’s Method
Take the average of the two slopes and follow that new slope out a distance h:
K1 1 2
2
K K
K2
1 20 2
K Ky h
Euler's and Heun's Methods
![Page 62: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/62.jpg)
62
Heun’s Method
Thus, you will write a second function, heun(), that has the same signature as euler(), where you will1. Determine
2. Assign toa. tout a vector of n equally spaced points going from t0 to tfinal, and
b. yout a vector of n zeros where yout, 1 is assigned the initial value y0,
3. For k going from 1 to n – 1, repeat the following:a. Using f, calculate the slope K1 at the point tout,k and yout,k,
b. Use K1 to find K2, and
c. Set .
final 0
1
t th
n
1 2out, 1 out, 2k k
K Ky y h
Euler's and Heun's Methods
![Page 63: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/63.jpg)
63
Heun’s Method
For example, consider our initial-value problem
Approximating the solution on [0, 1] with n = 11 points yields:
[t2a, y2a] = heun( @f2a, [0, 1], 0, 11 ) t2a = 0 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000
1.0000 y2a = 0 0.0828 0.1399 0.1798 0.2074 0.2262 0.2382 0.2454 0.2491 0.2505
0.2508
2 2(1) 1 1
0 0
y t ty t
y
Euler's and Heun's Methods
![Page 64: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/64.jpg)
64
Heun’s Method
The function ode45 is Matlab’s built-in ODE solver:[t2a, y2a] = heun( @f2a, [0, 1], 0, 11 );plot( t2a, y2a, 'or' ); hold on[t2a, y2a] = ode45( @f2a, [0, 1], 0 );plot( t2a, y2a, 'b' )
Euler's and Heun's Methods
![Page 65: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/65.jpg)
65
Heun’s Method
For example, consider our initial-value problem
Approximating the solution on [0, 1] with n = 11 points yields:
>> [t2b, y2b] = heun( @f2b, [0, 1], 1, 11 ) t2b = 0 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000
1.0000 y2b = 1 1.0500 1.1091 1.1779 1.2569 1.3463 1.4462 1.5562 1.6756 1.8029
1.9362
(1) cos
0 1
y t t y t y t t y t
y
Euler's and Heun's Methods
![Page 66: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/66.jpg)
66
Heun’s Method
Heun’s method is significant better than Euler:[t2b, y2b] = heun( @f2b, [0, 1], 1, 11 );plot( t2b, y2b, 'or' ); hold on[t2b, y2b] = ode45( @f2b, [0, 1], 1 );plot( t2b, y2b, 'b' )
Euler’s Method
Euler's and Heun's Methods
![Page 67: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/67.jpg)
67
Heun’s Method
Comparing the accuracy of– Euler’s method (11 and 41 points in magenta) , and– Heun’s method (11 points in red)
We see that Heun is significantly better
Euler's and Heun's Methods
![Page 68: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/68.jpg)
68
Heun’s Method
The absolute errors are also revealing:– A reduction by a factor of three
0.02390.00705
Euler's and Heun's Methods
![Page 69: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/69.jpg)
69
0.02390.00705
Heun’s Method
To be fair, we should count function evaluations:– Euler’s method with n points has n – 1 function evaluations– Heun’s method with n points has 2(n – 1) function evaluations
Still, Heun’s method comes out ahead...
Euler's and Heun's Methods
![Page 70: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/70.jpg)
70
Error Analysis
Without proof, the error for Heun’s method is O(h3)– However, again, except with the first point, we are using an
approximation to find an approximation– As with Euler’s method, repeatedly applying Heun’s method will
results in an error of O(h2)
Euler's and Heun's Methods
![Page 71: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/71.jpg)
71
Summary
We have looked at Euler’s and Heun’s methods for approximating1st-order IVPs:– Euler’s method is a direct application of Taylor’s series– Heun’s method uses the ideas from the trapezoidal rule to
improve on Euler’s method– Heun’s method requires twice as many function evaluations as
does Euler’s method and yet it is significantly more accurate
Euler's and Heun's Methods
![Page 72: Euler’s and Heun’s Methods Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario,](https://reader035.vdocuments.us/reader035/viewer/2022062417/551aa1d75503466b3a8b5766/html5/thumbnails/72.jpg)
72
References
[1] Glyn James, Modern Engineering Mathematics, 4th Ed., Prentice Hall, 2007.
[2] Glyn James, Advanced Modern Engineering Mathematics, 4th Ed., Prentice Hall, 2011.
Euler's and Heun's Methods