Question

2.1 Summary In this part, you will create a figure, and use linear transformations (matrices) to move the figure around the screen. In the end, your figure should move up 8 steps, then turn and face left. Reference material for this part can be found in Linear Algebra, and its applications David Lay, Section 2.7 starting at the beginning of the section up to, but not including, 3D- Graphics. Also, this poster presentation does a pretty good job explaining the same concepts http://web.csulb.edu/~jchang9/m247/m247_sp12.Daniel Kris-James Walter.pdf 2.2 Create Figure in Plane Recreate the image below by creating a matrix whose first row is the x-coordinates of the vertices in clockwise order, and whose second row are the corresponding y-coordinates. Place the following function at the bottom of your script to plot the points in the matrix =plot-points (points) figure(1); fill (points(1, :), points (2, :), 'g'); set (gca,'color', [0, o, 0]); axis([-10, 10, -10, 10]); function [] You can then use this function in your script by using the command plot.points(P); Where P is the matrix created above. (It will look a bit different than shown) 1.e 1 4 门4 1 08 06 04 0.02 04 06 C1 2.3 Homogeneous Coordinates Convert the points stored in your matrix to homogeneous coordinates by adding a row of all ones to the bottom of your matrix of coordinates

Answer 1

Matlab code for conjugate gradient method clear all close all Running the program with given parameters All A, x0, b and epsilon values A-I4 3 0;3 4 -1;0-1 41i epsi-10-8; x0-0:0:0] fprintfExample code for Conjugate gradient \n' fprintf( The A matrix is In') disp (A) fprintf( The b matrix is \n) disp(b) x-x0 r-old-b-A*x; p-r old; epsi-epsi norm (r_old); cnt-0; %loop for conjugate gradient method while norm (r_old)>epsi cnt-cnt+l; alpha-(r_old' *r_old)/(p A*p); x x+alpha p; r_new-r_old-alpha*A*p; beta-(r new(r new-r old))(r old'*r old) p-r_new+beta*p; r old-r new; fprintf( The solution matrix using manual code is \n') disp(x) Running the program with conjugate gradient function [x,iter]-conjugategradient (A, x0, b) fprintf The solution matrix using con jugategradient function is \n') disp(x) fprint f ("Error in conjugate gradient method is %e.in.,norm(x-Ab)) Running program for iteration count r-4;

fprintf nSolution using conjugate gradient method for different n for i= 1:6 n-2* (rti); %creating tridiagonal matrix A A-full (gallery('tridiag' ,n,-1,2,-1)) considering exact solution value as 1 x-ones (n,1); creating b vector of length n %initial guess :0-zeros ( n , 1 ); %finding solution using conjugategradient solution [ x1 , iter ]=conjugategradient ( A, x0, b ) ; fprint f ( '\tFor n=2"%d the iteration count is 2d.\n', (r+1),iter) tprint f ( "\tError in conjugate gradient method is %e.\n\n', norm (x1- A\b)) end 8888옹옹옹옹옹옹88%옹88888888888888888888옹%88옹%88%%888888옹옹8888888888888 Function for Conjugate Gradient method function [x,iter1-conjugategradient (A, x0,b) epsi=10^-8 ; x-x0 r old-b-A*x; p-r_old; epsi-epsi norm(r old); cnt-0 while norm (r_old)>eps:i cnt=cnt+1; alpha-(r_old' *r_old)/(p A*p); x=x+alpha*p; r_new r_old-alpha*A*p; beta-(r ne(r new-r_old)) (r old'*r old); p-r_new+beta*p; r_ old-r_new; end iter cnt; end 용음음응용음음응용음음응용음음응용음음용용음음용용음음응용음음용용음응용용음응용욤음응용욤응응용욤응응용욤응응용욤응응욤욤응응욤욤응응 용%욤욤응용음음%%욤욤%용음 End of Code 용용음응용욤욤응용욤욤응용욤

Example code for Con jugate gradient The A matrix is The b matrix is The solution matrix using manual code is 0.75 -0.333333333333333 -0.0833333333333333 The solution matrix using conjugategradient function is 0. 75 -0.333333333333333 -0. 0833333333333333 Error in conjugate gradient method is 1.118863e-16. Solution using conjugate gradient method for different rn For n 25 the iterationcount is 16. Error in conjugate gradient method is 1.849413e-14 For n=2 ^6 the iteration count is 32. Error in conjugate gradient method is 3.575453e-14 For n-2 7 the iteration count is 64 Error in conjugate gradient method is 7.489604e-14. For n=2 ^8 the iteration count is 128 Error in conjugate gradient method is 1.186086e-13. For n-29 the iteration count is 256 Error in conjugate gradient method is 3.622280e-12. For n 2 10 the iteration count is 512 Error in conjugate gradient method is 1.203831e-11. Published with MATLAB R2018a

%%Matlab code for conjugate gradient method
clear all
close all

%Running the program with given parameters
%All A, x0, b and epsilon values

A=[4 3 0;3 4 -1;0 -1 4];
b=[2;1;0];
epsi=10^-8;
x0=[0;0;0];

fprintf('Example code for Conjugate gradient \n')
fprintf('The A matrix is \n')
disp(A)

fprintf('The b matrix is \n')
disp(b)

x=x0;
r_old=b-A*x;
p=r_old;
epsi=epsi*norm(r_old);
cnt=0;

%loop for conjugate gradient method
while norm(r_old)>epsi
  
    cnt=cnt+1;
    alpha=(r_old'*r_old)/(p'*A*p);
    x=x+alpha*p;
    r_new=r_old-alpha*A*p;
    beta=(r_new'*(r_new-r_old))/(r_old'*r_old);
    p=r_new+beta*p;
    r_old=r_new;
  
end

fprintf('The solution matrix using manual code is \n')
disp(x)

%Running the program with conjugate gradient function
[x,iter]=conjugategradient(A,x0,b);

fprintf('The solution matrix using conjugategradient function is \n')
disp(x)

fprintf('Error in conjugate gradient method is %e.\n',norm(x-A\b))

%Running program for iteration count
r=4;
fprintf('\nSolution using conjugate gradient method for different n.\n')

for i=1:6
  
    n=2^(r+i);
    %creating tridiagonal matrix A
    A=full(gallery('tridiag',n,-1,2,-1));
    %considering exact solution value as 1
    x=ones(n,1);
    %creating b vector of length n
    b=A*x;
    %initial guess
    x0=zeros(n,1);
  
    %finding solution using conjugategradient solution
    [x1,iter]=conjugategradient(A,x0,b);
    fprintf('\tFor n=2^%d the iteration count is %d.\n',(r+i),iter)
    fprintf('\tError in conjugate gradient method is %e.\n\n',norm(x1-A\b))
  
end
  
  
----------------------------------------------------------------------------------------------------------------
%%Function for Conjugate Gradient method
function [x,iter]=conjugategradient(A,x0,b)
    epsi=10^-8;
    x=x0;
    r_old=b-A*x;
    p=r_old;
    epsi=epsi*norm(r_old);
    cnt=0;
    while norm(r_old)>epsi

        cnt=cnt+1;
        alpha=(r_old'*r_old)/(p'*A*p);
        x=x+alpha*p;
        r_new=r_old-alpha*A*p;
        beta=(r_new'*(r_new-r_old))/(r_old'*r_old);
        p=r_new+beta*p;
        r_old=r_new;

    end
    iter=cnt;
end

       %%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%