====== Math 445 HW1 Solutions ====== 1. Assign the value 0.00014 to the variable //x//, using compact scientific notation. <code> x = 1.4e-04; </code> 2. Set the variable //x// to a vector whose components are the even numbers between 14 and 36. <code> x = 14:2:36; </code> 3. Set the variable //x// to 200 evenly spaced points between 0 and 2. <code> x = linspace(0,2,200); </code> 4. Produce a plot of //y = x^2 - 2x + 3// for the //x// of the previous problem. Label the axes. <code> plot(x, x.^2 - 2*x + 3) xlabel('x') ylabel('y = x^2 - 2*x + 3)') </code> 5. Plot //tan(x)// versus //x// for 200 evenly spaced points between 0 and 1.57 ≈ pi/2, using a logarithmic scale on the //y// axis. <code> x = linspace(0, 1.57, 200); semilogy(x, tan(x)); xlabel('x') ylabel('y = tan(x)') </code> 6. Plot //sin(x)// and //cos(x)// for 200 evenly spaced points between -π and π, on the same plot, using red for //sin(x)// and blue for //cos(x)//. Add a legend that indicates which function is which color. <code> x = linspace(-pi, pi, 200); plot(x, sin(x), 'r-', x, cos(x), 'b-', ); xlabel('x') legend('sin(x)', 'cos(x)'); </code> 7. Produce a vector whose components are random integers between 0 and 10, inclusive. <code> x = randi([0,10], 13,1); % The problem didn't specify the length of the vector, choose 13 </code> 8. Produce a vector whose components are random real numbers between 0 and 10, inclusive. <code> x = 10*rand(13,1); % The problem didn't specify the length of the vector, choose 13 </code> 9. Show how you would solve the following system of equations with Matlab <code> x + 2y - z = 5 3x + y + 6z = 37 -3x + y + 2z = -11 </code> <code> A = [1 2 -1; 3 1 6; -3 1 2]; b = [5 37 -11]'; x = A\b </code> 10. Write a conditional expression that is true if scalar variables //x// and //y// are both nonzero and false otherwise. <code> (x ~= 0) && (y ~= 0) </code> 11. Set variable //A// to a 3 x 5 matrix of zeros. <code> A = zeros(3,5); </code> 12. Set variable //A// to a 4 x 7 matrix of random real numbers, using a guassian (normal) distribution. <code> A = randn(4,7); </code> 13. Write a conditional expression that is true if a matrix //A// is square and false otherwise. <code> size(A,1) == size(A,2) </code> 14. Write a conditional expression that is true if either //x// or //y// is an integer. <code> x == round(x) || y == round(y) </code> Write a Matlab function that 15. returns 1 (true) if its argument is divisible by 3 and 0 (false) if it's not. <code> function r = divisible_by_3(x) r = (x == 3*round(x/3)); end </code> or better, use the ''mod'' function <code> function r = divisible_by_3(x) r = (mod(x,3) == 0); end </code> 16. takes a vector //x// as input and returns 1 if the components of //x// are sorted in ascending order, 0 if not. <code> function s = is_sorted(x) s = 1; % start assuming x is sorted for n=1:length(s)-1; if s(n) > s(n+1) s = 0; break; end end end </code> 17. finds a zero of another function using the bisection search algorithm. <code> function xcenter = bisectsearch(f, xleft, xright) % function xcenter = bisectsearch(f, xleft, xright) % find a zero of f between xleft and xright using bisection search algorithm eps = 1e-13; % stopping condition: abs(f(xcenter)) < eps fleft = f(xleft); fright = f(xright); % should put in check that fleft and fright have opposite signs xcenter = 0.5*(xleft + xright); fcenter = f(xcenter); % repeat until f(xcenter) is close enough to zero while (abs(fcenter) > eps) if (sign(fleft) == sign(fcenter)) % replace the left stuff with the center stuff; fleft = fcenter; xleft = xcenter; else % replace the right stuff with the center stuff; fright = fcenter; xright = xcenter; end xcenter = 0.5*(xleft + xright); fcenter = f(xcenter); end end </code> 18. computes the product //y = Ax// of an //m x n// matrix //A// and an //n x 1// vector //x//, according the formula <latex> y_i = \sum_{j=1}^n A_{ij} x_j </latex> <code> function y = matvecmult(A,x); % returns y = A*x; [M,N] = size(A); % assume x is N x 1 y = zeros(M,1); for i = 1:M for j = 1:N y(i) = y(i) + A(i,j)*x(j); end end end </code>
