1. Comp 208 Computers in Engineering Yi Lin Winter, 2007
Lecture 21 Integration
Comp 208 Computers in Engineering
Yi Lin
Winter, 2007
2019/5/13
Comp208 Computers in Engineering

2. Lecture 22 Learning goals
Integration
Rectangular midpoint rule
Trapezoidal rule
Simpson’s rule
Monte Carlo
2019/5/13
Comp208 Computers in Engineering

3. Comp208 Computers in Engineering
Integration
Recall the definition of the integral
As with differentiation, let’s experiment with small but finite deltas
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Comp208 Computers in Engineering

4. Meet our function for today
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Comp208 Computers in Engineering

5. ‘Dumb’ integrals, Left-point rule
Again, applying the definition directly
Obtain the left-point rectangular rule
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Comp208 Computers in Engineering

6. Left-point rectangular rule
Note how this scheme will always underestimate an increasing function and overestimate a decreasing function
This maps well to the definition but we can very easily make it much better so we won’t even bother to write code for this.
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Comp208 Computers in Engineering

7. Comp208 Computers in Engineering
Right-point rule
Using the right point as height, it will overestimate the ascending functions but underestimate the descending functions
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Comp208 Computers in Engineering

8. Comp208 Computers in Engineering
Midpoint rule
Each rectangular band which makes up the integral is evaluated at the midpoint
Hopefully the overestimation and underestimation effects cancel out on each panel
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Comp208 Computers in Engineering

9. Comp208 Computers in Engineering
Midpoint algorithm
double midpoint_int( DfD foo , double a , double b , int panels ) {
int i;
double sum = 0;
double dx = ( b - a ) / panels;
double x = a + dx / 2.0;
for( i = 0; i < panels; ++i ) {
sum += foo( x );
x += dx; }
return sum * dx;
2019/5/13
Comp208 Computers in Engineering

10. How good is the midpoint rule?
Absolutely accurate for constant and linear functions
How accurate for parabolas and other non-linear functions?
This is a pretty naïve way of doing things.
2019/5/13
Comp208 Computers in Engineering

11. Trying other geometries
The rectangle isn’t the only quadrilateral that we can use to evaluate an integral
What about a trapeze?
We could try even more sophisticated shapes
Fit a parabola through 3 points and integrate that
How can we make integration rules that guarantee higher accuracy?
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Comp208 Computers in Engineering

12. Comp208 Computers in Engineering
Trapezoidal rule
Approximate curves as sequences of straight lines instead of sequences of constants
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Comp208 Computers in Engineering

13. Comp208 Computers in Engineering
Trapezoidal integral
The integral formula comes from the Taylor series
The error term has a known order of magnitude which comes from the terms we truncate from the series to make the integration rule
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Comp208 Computers in Engineering

14. Comp208 Computers in Engineering
Summing trapezes
Area of a trapeze
Remember that all the trapezes share an edge
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Comp208 Computers in Engineering

15. Comp208 Computers in Engineering
Optimizing the sum
Because we are adding up all the inner edges twice but dividing all of our sums by two we can reduce the effort
Start sum with the outer edges divided by two
Then add all the inner edges undivided
2019/5/13
Comp208 Computers in Engineering

16. Trapezoidal integration
double trapezoidal_int( DfD foo , double a , double b , int panels ) {
int i;
double sum = ( foo( a ) + foo( b ) ) / 2.0;
double dx = ( b - a ) / panels;
double x = a + dx;
for( i = 0; i < ( panels - 1); ++i ) {
sum += foo( x );
x += dx; }
return sum * dx;
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Comp208 Computers in Engineering

17. Comp208 Computers in Engineering
How good is this?
If deltas are small enough then the straight line approximation can be good
How does it deal with errors? Is that a good way?
How does this method’s accuracy relate to how many points it uses (relate polynomials)
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Comp208 Computers in Engineering

18. Comp208 Computers in Engineering
Guarantees?
Midpoint and trapezoidal rules, amazingly enough, offer the same guarantees
100% accurate on constants and linear functions
BUT, trapezoidal deals with errors better, on average.
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Comp208 Computers in Engineering

19. Comp208 Computers in Engineering
Using a parabola
Simpson’s rule fits a parabola through 3 points and integrates it
Definition of what a panel is, is now somewhat up in the air… our old panels had 2 points but now we need three
We can subdivide our current panels…
…or we can use neighbour panels together.
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Comp208 Computers in Engineering

20. Comp208 Computers in Engineering
Using a parabola
As the trapezoidal rule for integration finds the area under the line connecting the endpoints of a panel, Simpson's rule finds the area under the parabola which passes through 3 points (the endpoints and the midpoint) on a curve.
Area under parabola: Δx Δx
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Comp208 Computers in Engineering

21. Comp208 Computers in Engineering
Simpson’s formula
Again, the formula comes from the taylor series:
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Comp208 Computers in Engineering

22. Comp208 Computers in Engineering
Simpson’s algorithm
double simpsons_int( DfD foo , double a , double b , int panels ) {
int i;
double sum = 0;
double dx = ( b - a ) / panels;
double x = a;
for( i = 0; i < panels; ++i ) {
sum += foo( x ) + 4*foo( x + dx/2.0 ) + foo( x + dx );
x += dx; }
return sum * dx / 6.0;
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Comp208 Computers in Engineering

23. Comp208 Computers in Engineering
How good is this?
Look at the error term… ∆x5!!
If ∆x is 0.01, then the precision is already beyond most engineering requirements…
Even more methods exist however…
Even methods to make methods of arbitrary precision
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Comp208 Computers in Engineering

24. Monte Carlo Integration
double monte_carlo_int(DfD f, double x0, double x1, int n) {
double sum = 0;
int i;
srand(time(NULL));
// Sum n random values of f(x) with x in [x0, x1].
for(i = 0; i < n; ++i)
sum += f( (rand() / RAND_MAX) * (x1 - x0) x );
// Divide the sum by n to get the average value of f(x) with x in [x0, x1].
// Multiply by (x1 - x0) to get the area.
return (sum / n) * (x1 - x0); }
2019/5/13
Comp208 Computers in Engineering