1. CS100A Lecture 21 Previous Lecture This Lecture
MatLab and its implementation, continued.
This Lecture
MatLab demonstration
CS 100
Lecture 21

2. MatLab
The notation j:k gives a row matrix consisting of the integers from j through k.
>> 1:8
ans =
The notation j:b:k gives a row matrix consisting of the integers from j through k in steps of b.
>> 1:2:10
To get a vector of n linearly spaced points between lo and hi, use linspace(lo,hi,n)
>> linspace(1,3,5)
To transpose a matrix m, write m’
>> (1:3)’ 1 2 3
CS 100
Lecture 21

3. MatLab Variables MatLab has variables and assignment:
>> evens = 2.*(1:8)
evens =
To suppress output, terminate line with a semicolon (not to be confused with vertical concatenation operator)
>> odds = evens - 1;
>>
To see the value of variable v, just enter expression v:
>> odds
odds =
CS 100
Lecture 21

4. MatLab Idiom
MatLab has for-loops, conditional statements, subscripting, etc., like Java. But the preferred programming idiom uses uniform operations on matrices and avoids explicit loops, if possible.
Essentially every operation and built in function takes a matrix as an argument.
CS 100
Lecture 21

5. sum, prod, cumsum, cumprod
Function sum adds row elements, and the function prod multiplies row elements:
>> sum(1:1000)
ans =
500500
>> prod(1:5)
120
Function cumsum computes a row of the partial sums and cumprod computes a row of the partial products:
>> cumprod(1:7)
>> cumsum(odds)
CS 100
Lecture 21

6. Compute Pi by Euler Formula
Leonard Euler ( ) derived the following infinite sum expansion:
2 / 6 =  1/j 2 (for j from 1 to )
>> pi = sqrt( 6 .* cumsum(1 ./ (1:10) .^ 2));
>> plot(pi)
To define a function, select New/m-file and type definition:
function e = euler(n)
% Return a vector of approximations to pi.
e = sqrt( 6 .* cumsum(1 ./ (1:n) .^ 2));
Select SaveAs and save to a file with the same name as the function.
To invoke:
>> pi = euler(100);
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Lecture 21

7. Help System Use on-line help system >> help euler
>> help function
... description of how to define functions ...
>> help euler
Return a vector of approximations to pi.
CS 100
Lecture 21

8. Compute Pi by Wallis Formula
John Wallis ( ) derived the following infinite sum expansion:
(2 * 2) * (4 * 4) * (6 * 6) * . . .
 / 2 =
(1 * 3) * (3 * 5) * (5 * 7) * . . .
Terms in Numerator
evens .^ 2
Terms in Denominator
odds
odds + 2
1*3 3*5 5*7 7*9 9*
i.e., odds .* (odds + 2)
Quotient
prod( (evens .^ 2) ./ (odds .* (odds+2)) )
Successive approximations to Pi
pi = 2 .* cumprod( (evens.^2) ./ (odds .* (odds+2)) )
CS 100
Lecture 21

9. Wallis Function Function Definition
function w = wallis(n)
% compute successive approximations to pi.
evens = 2 .* (1:n);
odds = evens - 1;
odds2 = odds .* (odds + 2);
w = 2 .* cumprod( (evens .^ 2) .* odds2 );
Contrasting Wallis and Euler approximations
>> plot(1:100, euler(100), 1:100, wallis(100))
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Lecture 21

10. Compute Pi by Throwing Darts
Throw random darts at a circle of radius 1 inscribed in a 2-by-2 square.
The fraction hitting the circle should be the ratio of the area of the circle to the area of the square:
f =  / 4
This is called a Monte Carlo method y 1 x 1
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Lecture 21

11. Darts
The expression rand(1,n) computes a row of n random numbers between 0 and 1.
>> x = rand(1,10);
>> y = rand(1,10);
Let d2 be the distance squared from the center of the circle.
>> d2 = (x .^ 2) + (y .^ 2);
Let in be a row of 0’s and 1’s signifying whether the dart is in (1) or not in (0) the circle. Note 1 is used for true and 0 for false.
>> in = d2 <= 1;
Let hits be the cumulative sum of darts in circle
>> hits = cumsum(in);
Let f be cumulative fraction of hits
>> f = hits ./ (1:n);
Let pi be successive approximations to pi
>> pi = 4 .* f;
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Lecture 21

12. Compute Pi by Throwing Needles
In 1777, Comte de Buffon published this method for computing :
N needles of length 1 are thrown at random positions and random angles on a plate rules by parallel lines distance 1 apart. The probability that a needles intersects one of the ruled lines is 2/. Thus, as N approaches infinity, the fraction of needles intersection a ruled line approaches 2/.
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Lecture 21

13. Subscripting Subscripts start at 1, not zero.
ans =
>> a(2,2) 5
A range of indices can be specified:
>> a(1:2, 2:3)
2 3
5 6
A colon indicates all subscripts in range:
>> a(:, 2:3)
8 9
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Lecture 21

14. Control Structures: Conditionals
if expression
list-of-statements
end
else
elseif expression
. . .
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15. Control Structures: Loops
while expression
list-of-statements
end
for variable = lo:hi
for variable = lo:by:hi
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Lecture 21