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%% I. Matlab Introduction
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%% 1. Vector
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% Exercise 1
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x = [1, 2, 3, 4, 5];
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result1 = x(3) + x(2)
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% Exercise 2
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x_col = [1; 2; 3; 4; 5];
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x_transposed = x_col'
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% Exercise 3
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a_sum = sum(1:100)
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% Exercise 4
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syms x y
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f = x + y;
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f_val = subs(f, [x, y], [1, 2])
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% Exercise 5
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% a) sum from 1 to 10
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syms n
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s1 = symsum(n, n, 1, 10)
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% b) sum of n^2 from 1 to 5
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s2 = symsum(n^2, n, 1, 5)
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%% 2. Matrix
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% Exercise 1a
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% Define element by element
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A1 = [0 0 0; 0 0 0; 0 0 0];
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% Define using zeros
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A2 = zeros(3,3);
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% Exercise 1b
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A = zeros(3);
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A = A + diag([1 1 1])
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% Exercise 1c
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B = A'
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% Exercise 1d
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add = A + B
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sub = A - B
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mul_matrix = A * B
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mul_elem = A .* B
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% Remark: A*B is matrix multiplication, A.*B is element-wise
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% Exercise 1e
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A2x2 = A;
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A2x2(3,:) = [];
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A2x2(:,3) = [];
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A2x2
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B2x2 = B;
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B2x2(1,:) = [];
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B2x2(:,1) = [];
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B2x2
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% Exercise 1f
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A(1:3)
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A(1:3,1)
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A(:,1)
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%% 3. Function
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% Exercise 1
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function m = mean_vector(v)
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m = sum(v)/length(v);
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end
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% Application
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x = 1:3:100;
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m_val = mean_vector(x)
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% Exercise 2
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function roots = quadratic_solver(a, b, c)
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D = b^2 - 4*a*c;
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roots = [(-b + sqrt(D))/(2*a), (-b - sqrt(D))/(2*a)];
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end
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% Application
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roots_val = quadratic_solver(1, 2, -3)
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%% 4. If else structure
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% Exercise 1
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n = 5;
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if n > 1
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m = n + 1;
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else
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m = n - 1;
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end
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m
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n = -5;
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if n > 1
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m = n + 1;
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else
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m = n - 1;
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end
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m
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% Exercise 2
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x = 5;
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if x > 0 && x < 10
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y = 4*x;
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elseif x > 10 && x < 40
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y = 10*x;
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elseif x > 40 || x < 0
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y = x;
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end
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y
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x = -5;
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if x > 0 && x < 10
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y = 4*x;
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elseif x > 10 && x < 40
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y = 10*x;
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elseif x > 40 || x < 0
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y = x;
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end
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y
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x = 20;
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if x > 0 && x < 10
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y = 4*x;
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elseif x > 10 && x < 40
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y = 10*x;
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elseif x > 40 || x < 0
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y = x;
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end
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y
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% Exercise 3: Electricity bill
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kWh = 350;
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price = 0; % cant even initialize bru
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if kWh > 400
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price = (kWh-400)*2.587 + 100*2.503 + 100*2.242 + 100*1.786 + 50*1.533 + 50*1.484;
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elseif kWh > 300
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price = (kWh-300)*2.503 + 100*2.242 + 100*1.786 + 50*1.533 + 50*1.484;
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elseif kWh > 200
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price = (kWh-200)*2.242 + 100*1.786 + 50*1.533 + 50*1.484;
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elseif kWh > 100
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price = (kWh-100)*1.786 + 50*1.533 + 50*1.484;
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elseif kWh > 50
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price = (kWh-50)*1.533 + 50*1.484;
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else
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price = kWh * 1.484;
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end
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price
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%% 5. For loop
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% Exercise 1a: sum of squares
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sum_sq = 0;
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for i = 1:1000
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sum_sq = sum_sq + i^2;
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end
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sum_sq
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% Exercise 1b: 20!
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fact = 1;
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for i = 1:20
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fact = fact * i;
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end
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fact
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% Exercise 1c: Series
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sum_alt = 0;
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for i = 1:2:1003
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sum_alt = sum_alt + ((-1)^((i-1)/2)) * (1/i);
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end
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sum_alt
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% Exercise 1d: π approximation
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sum_pi = 0;
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for k = 1:1000
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sum_pi = sum_pi + 1/((2*k-1)^2);
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end
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pi_squared_over_8 = sum_pi
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approx_pi = sqrt(8 * sum_pi)
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% Exercise 2a: Combination function
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function c = combination(n, k)
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c = factorial(n)/(factorial(k)*factorial(n-k));
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end
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comb_val = combination(5, 3)
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% Exercise 2b: Permutation function
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function p = permutation(n, k)
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p = factorial(n)/factorial(n-k);
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end
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perm_val = permutation(5, 3)
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%% 6. While loop
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% Exercise 1a: Count numbers from 1 to N
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N = 5;
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i = 1;
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while i <= N
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disp(i)
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i = i + 1;
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end
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% Exercise 1b: Count odd numbers from 1 to N
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i = 1;
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while i <= N
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if mod(i,2) == 1
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disp(i)
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end
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i = i + 1;
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end
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% Exercise 2: Sum sqrt(2i-1) until > 20
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i = 1;
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S = 0;
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while S <= 20
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S = S + sqrt(2*i - 1);
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i = i + 1;
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end
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S
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i
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% Exercise 3: Doubling investment
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investment = 1;
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years = 0;
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while investment < 2
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investment = investment * 1.1;
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years = years + 1;
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end
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years
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%% 7. Graphics
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% Exercise 1a: Linear plot
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x = linspace(-5,5,100);
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y1 = x + 1;
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plot(x, y1)
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% Exercise 1b: Line style and color
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plot(x, y1, '--r')
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% Exercise 1c: Plot second function
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hold on
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y2 = -x + 2;
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plot(x, y2, ':b')
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hold off
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% Exercise 1d: Add title and labels
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title('Linear Functions')
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xlabel('x')
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ylabel('y')
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legend('y = x+1', 'y = -x+2')
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+127
@@ -0,0 +1,127 @@
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%% I. MATLAB FUNCTION
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%% 1. Solve / fzero
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% (a) Exact solution: 67x + 32 = 0
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syms x
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sol_exact = solve(67*x + 32 == 0, x)
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% (b) Numerical solution to 15 places
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format long
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f = @(x) 67*x + 32;
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sol_numeric = fzero(f, -1)
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% (c) x^3 + p*x + q = 0
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syms x p q
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sol_cubic = solve(x^3 + p*x + q == 0, x)
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% (d) e^x = 8x - 4
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syms x
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eqn = exp(x) == 8*x - 4;
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% Plotting to identify real solutions
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fplot(@(x) exp(x), [-2, 2]); hold on;
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fplot(@(x) 8*x - 4, [-2, 2]);
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legend('e^x','8x-4');
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grid on;
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title('Graph of e^x and 8x-4 to visualize intersection points');
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% Solving numerically
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f = @(x) exp(x) - (8*x - 4);
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sol1 = fzero(f, 0) % Guess from plot
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%% 2. Plot
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% (a) y = x^3 - x for x in [-4, 4]
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x = linspace(-4, 4, 1000);
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y = x.^3 - x;
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figure
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plot(x, y); grid on; title('y = x^3 - x');
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% (b) y = tan(x/2) for x in [-pi, pi], clip y in [-10, 10]
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x = linspace(-pi, pi, 1000);
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y = tan(x/2);
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figure
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plot(x, y);
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axis([-pi pi -10 10]);
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title('y = tan(x/2)');
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% (c) y1 = e^(-x/2), y2 = x^4 - x^2 for x in [-1.5, 1.5]
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x = linspace(-1.5, 1.5, 1000);
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y1 = exp(-x/2);
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y2 = x.^4 - x.^2;
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figure
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plot(x, y1, 'b', x, y2, 'r--');
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legend('e^{-x/2}', 'x^4 - x^2');
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title('Overlay of two functions');
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%% II. MATLAB NUMERICAL METHODS
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%% 1. Bisection Method
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% Function file already given above as bisection.m
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% Exercise 1: sqrt(2), f(x) = x^2 - 2
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f = @(x) x^2 - 2;
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[root1, fx1, ea1, iter1] = bisection(f, 1, 2, 0.05)
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% Exercise 2: ln(x^2) = 0.7 --> f(x) = log(x^2) - 0.7
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f = @(x) log(x^2) - 0.7;
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[root2, fx2, ea2, iter2] = bisection(f, 0.5, 2, 100, 3)
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% Exercise 3: x^2 + 2x - 3 = 0
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f = @(x) x^2 + 2*x - 3;
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[root3, fx3, ea3, iter3] = bisection(f, 0, 2)
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%% 2. Newton-Raphson Method
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% f.m file
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% function y = f(x)
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% y = x^3 + x - 3;
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% df.m file
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% function y = df(x)
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% y = 3*x^2 + 1;
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% Exercise 1: Solve x^3 + x - 3 = 0
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f = @(x) x^3 + x - 3;
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df = @(x) 3*x^2 + 1;
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x0 = 1; tol = 1e-8; max_iter = 20;
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x = x0;
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for i = 1:max_iter
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x_new = x - f(x)/df(x);
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err = abs((x_new - x)/x_new);
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x = x_new;
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fprintf('Iter %d: x = %.15f, f(x) = %.15f\n', i, x, f(x));
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if err < tol
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break
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end
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end
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% Exercise 2: f(x) = x^10 - 1, x0 = 0.5
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f = @(x) x^10 - 1;
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df = @(x) 10*x^9;
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x = 0.5;
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X_history = x;
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for i = 1:20
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x_new = x - f(x)/df(x);
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X_history(end+1) = x_new;
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if abs((x_new - x)/x_new) < 1e-8
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break
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end
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x = x_new;
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end
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fprintf('Final approx root: %.15f\n', x);
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% Plot convergence
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figure;
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plot(0:length(X_history)-1, X_history, '-o');
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xlabel('Iteration');
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ylabel('x');
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title('Newton-Raphson convergence for x^{10} - 1');
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% Exercise 3: Solve x^2 + 2x - 3 = 0
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f = @(x) x^2 + 2*x - 3;
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df = @(x) 2*x + 2;
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x = 1;
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for i = 1:20
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x_new = x - f(x)/df(x);
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fprintf('Iter %d: x = %.10f, f(x) = %.10f\n', i, x_new, f(x_new));
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if abs((x_new - x)/x_new) < 1e-8
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break
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end
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x = x_new;
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end
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