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1 Mind the gap
function p = prime gap(n,d) % Returns the prime numbers separated by a gap d and smaller than n. % p is an array which contains pairs of primes in order.
x = primes(n); % get a list of primes smaller than n m = length(x) − 1; p = [];
for i = 1:m if (x(i+1)−x(i)) == d % check distance for consecutive pairs p = [p, x(i), x(i+1)]; end end
2 Frequency of letters
function n = letter freq(s) % Returns the number of occurences of letters in a strings. The index i % of the result array n, contains the occurence of the character % char('a'+i−1) The length of the output array should be equal to % 'z'−'a'+1 (=26).
n = zeros(1,'z'−'a'+1); % creates an array filled with zeros
s = lower(s); for c = s if c <= 'z' && c >= 'a', n(c−'a'+1) = n(c−'a'+1) + 1; end end
3 Polynomials
3.a Evaluation
function y = poly eval(p, x) % Evaluates the polynomial at x. The polynomial coefficients are % provided as an array p, in increasing order of the power of x.
y = 0;
for i = 1:length(p) y = y + xˆ(i−1) * p(i); end
3.b Multiple Points
function y = poly eval array(p, x) % Evaluate a polynomial at multiple values provided by array x. % The result, y, is another array. % % Example: If we want to evaluate p(x) = 3xˆ2 + 4x + 5 at x = [2 3] we % should call the function as follows: % % y = poly eval array([5 4 3], [2 3]) % result should be [25 44] %
n = length(x); % number of points to evaluate y = zeros(1,n); % result array for i = 1:n y(i) = poly eval(p, x(i)); end
3.c Multiplication
function p = poly multiply(p1, p2) % Multiply two polynomials represented with their coefficients and % returns the coefficients of the product polynomial.
n1 = length(p1); n2 = length(p2); n = n1 + n2 − 1;
p = zeros(1,n);
for i = 1:n for j = 1:n p(i+j−1) = p(i+j−1) + p1(i)*p2(j); end end
4 Astroid
% Fill in this script to accomplish the following tasks. % Task0: Write a couple of comment lines to describe the purpose of % the script
% In this script we will estimate the area of an astroid curve % using a numerical experiment. After generating uniformly % distributed random points inside a square region, we count % the number of points falling inside the astroid curve. The ratio % of number of points inside to the total number can be used to % compute the area.
PC = vC−vP;
% if P falls outside of the triangle, then the % sum of the areas of the triangles PAB, PAC, PBC % must be greater than that of ABC PAB = norm(cross(PA,PB))/2; PAC = norm(cross(PA,PC))/2; PBC = norm(cross(PB,PC))/2; ABC = norm(cross(AB,AC))/2;
if (PAB + PAC + PBC− ABC)/ABC> 1e−10, return; else hit = 1; return; end
5.b Probability Estimates
load('triangulation.mat');
N = 10000;
tic
HB = 0;
HW = 0;
for n = 1:N ray = rand isotropic(3); hit = 0;
c = 0; while ˜hit && c < 1080 vA = htria(c+1:c+3); vB = htria(c+4:c+6); vC = htria(c+7:c+9); hit = ray intersects triangle(ray, vA, vB, vC); c = c + 9; end
if hit, HW = HW + 1; else
c = 0; while ˜hit && c < 540 vA = ptria(c+1:c+3); vB = ptria(c+4:c+6); vC = ptria(c+7:c+9); hit = ray intersects triangle(ray, vA, vB, vC); c = c + 9; end if hit, HB = HB + 1; else error('uppps!'); end end
end
toc assert(HW+HB == N);
PW = HW/(HW+HB)
PB = HB/(HW+HB)
