2

%Problem 1
A1 = [];
A2 = [];
A3 = [];
tol = 1e16;
for i = 2:50
    hilbert = hilb(i);
    if cond(hilbert, inf) < tol
        A1 = [A1; cond(hilbert, inf)];
    end
    if cond(hilbert, inf) > tol
        break
    end
end
i = i-2;
save A1.dat A1 '-ascii';
temp = [];


tempArray1 = [];
tempArray2 = [];
for n = 2:i
    %temp(1:n) = 1;
    %temp = temp.';
    temp = ones([n, 1]);
    hilbert1 = hilb(n);
    z1 = temp - (hilbert1.'*(hilbert1*temp));
    tempArray1 = [tempArray1; norm(z1, inf)];

    z2 = temp - (hilbert1 * (hilbert1.' * temp));
    tempArray2 = [tempArray2; norm(z2, inf)];
end
A2 = [tempArray1 tempArray2];
save A2.dat A2 '-ascii';

temp2 = [];


tempArray12 = [];
tempArray22 = [];
for n = 2:i
    temp2 = ones([n, 1]);
    hilbert2 = hilb(n);
    invhilb2 = invhilb(n);
    z12 = temp2 - (invhilb2*(hilbert2*temp2));
    tempArray12 = [tempArray12; norm(z12, inf)];

    z22 = temp2 - (hilbert2 * (invhilb2 * temp2));
    tempArray22 = [tempArray22; norm(z22, inf)];
end

A3 = [tempArray12 tempArray22];
save A3.dat A3 '-ascii';

%Problem 2
clear all

A =  [ 4 -1 -1 0 0 0 0 0 0 0;
      -1 5 -1 -1 -1 0 0 0 0 0;
      -1 -1 5 0 -1 -1 0 0 0 0;
      0 -1 0 5 -1 0 -1 -1 0 0;
      0 -1 -1 -1 6 -1 0 -1 -1 0;
      0 0 -1 0 -1 5 0 0 -1 -1;
      0 0 0 -1 0 0 4 -1 0 0;
      0 0 0 -1 -1 0 -1 5 -1 0;
      0 0 0 0 -1 -1 0 -1 5 -1;
      0 0 0 0 0 -1 0 0 -1 4;];

  b = [0; 0; 0; 0; 0; 0; 1; 1; 1; 1];
  x = A\b;
save A4.dat x '-ascii';

tol = 1e-4;
maxIter = 100;
%Split up matrix
D = diag(diag(A));
L = tril(A) - D;
U = triu(A) - D;

guess = [1; 0; 0; 0; 0; 0; 0; 0; 0; 0];
%Jacobi
for i = 1:maxIter
    guess(:, i+1) = D\(b-(L+U)*guess(:,i));
    if norm(guess(:,i+1)-guess(:,i), inf) < tol
        break
    end
end
save A5.dat guess '-ascii'
%Gauss-Seidel
guess1 = [1; 0; 0; 0; 0; 0; 0; 0; 0; 0];
for i = 1:maxIter
    guess1(:,i+1) = (L+D)\(b-U*guess1(:,i));
    if norm(guess1(:,i+1) - guess1(:,i),inf) < tol
        break
    end
end
save A6 guess1 '-ascii'
[X,FLAG,RELRES,ITER] = bicg(A,b);
A7 = [X;ITER];
save A7.dat A7 '-ascii'

M = [0 1/4 1/4 0 0 0 0 0 0 0;
    1/2 0 1/4 1/4 1/6 0 0 0 0 0;
    1/2 1/4 0 0 1/6 1/4 0 0 0 0;
    0 1/4 0 0 1/6 0 1/2 1/4 0 0;
    0 1/4 1/4 1/4 0 1/4 0 1/4 1/4 0;
    0 0 1/4 0 1/6 0 0 0 1/4 1/2;
    0 0 0 1/4 0 0 0 1/4 0 0;
    0 0 0 1/4 1/6 0 1/2 0 1/4 0;
    0 0 0 0 1/6 1/4 0 1/4 0 1/2;
    0 0 0 0 0 1/4 0 0 1/4 0];
sum(M,1);
[V,D] = eig(M);
diagD = diag(D);
save A8.dat diagD '-ascii'

%Problem 3
clear all
N = [  7.24;  9.64; 12.87; 17.07; 23.19;
      31.44; 38.56; 50.19; 62.98; 76.21;
      92.23;106.02;123.20;132.16;151.33;
     179.32;203.30;226.54;248.71;281.42;
     307.75];
 t = [-90; -80; -70; -60; -50; -40; -30; -20; -10;
     0; 10; 20; 30; 40; 50; 60; 70; 80; 90; 100; 110];
 t = t.';
 N = N.';
 y = @(x) log((500/x)-1);
 ytemp = [21,1];
 A9 = polyfit(t,N,1);
 A9(1,2) = log(A9(1,2));
 save A9.dat A9 '-ascii'
temp = [-90 1;-80 1;-70 1;-60 1;-50 1;-40 1;
        -30 1;-20 1;-10 1;0 1;10 1;20 1;30 1;
         40 1;50 1;60 1;70 1;80 1;90 1;100 1;
        110 1];
years = linspace(-90,110,21);
%what you had before was down below
%in the for loop
%ytemp(i,1) = y(i);
for i = 1:21
   ytemp(i,1) = y(years(1,i));
end
ytemp
ytemp = ytemp(1:21,1);
%ans = temp\ytemp;
[X,FLAG,RELRES,ITER] = lsqr(temp,ytemp);
X = X.';
lsqrans = A9-X;
lsqrans = [lsqrans ITER];
save A10.dat lsqrans '-ascii';

predTemp = linspace(120,300,20); %list of years
predTemp = predTemp.'; %transpose since it was naughty before
eqTemp = [20,1]; %hold values of y(predTemp)
for i = 1:20
   eqTemp(i,1) = y(predTemp(i)); %plug into anon eq
end

nEq = @(m) 500/(1+exp(m)); %original funct
for i = 1:20
    eqTemp(i,1) = nEq(eqTemp(i,1)); %plugging back in
end
eqTemp = eqTemp(1:20,1).';
save A11.dat eqTemp '-ascii';