## Contents

```%
%             alias.m
%
%
```

## Code used to generate the figures and ranks for Case ii: n = 101, p = 1000 for Example 6.11.

`This can be easily modified for Case i: p = n = 5 or truncated for Case iii.`
```  clear all
close all
```

Input the response and parameter dimensions. When using the Random Range Finding algorithm, we will use ell = 75 samples.

```  n = 101;
p = 1000;
ell = 75;
tol = 1e-15;
h = 1/(n-1);
t = 0:h:1;
```

Construct the matrix A and compute the singular values and QR matrices. Construct the random matrix Y and its QR decomposition.

```  for i = 1:p
A(:,i) = sin(2*pi*i*t);
end

[Q,R,P] = qr(A');
[U,S,V] = svd(A);
rankA = rank(A,tol)

params = rand(p,ell);
Y = A*params;

[Qs,Rs,Ps] = qr(Y);
B = Qs'*A;
[Us,Ss,Vs] = svd(B);
```
```rankA =

100

```

Plot the columns of A shown in Figure 6.4 and the singular values and errors shown in Figure 6.5.

```  figure(1)
plot(t,A(:,1),'-k',t,A(:,101),'--r',t,A(:,201),'-.b',t,A(:,901),':k','linewidth',3)
hold on
plot(t,0*t,'k-','linewidth',3)
hold off
set(gca,'Fontsize',[22]);
xlabel('Time (s)')
ylabel('Column Entries of A')
legend('A(:,1)','A(:,101)','A(:,201)','A(:,901)','Location','NorthEast')

figure(2)
plot(t,A(:,1),'--b',t,A(:,51),'-r','linewidth',3)
hold on
plot(t,0*t,'k-','linewidth',3)
hold off
set(gca,'Fontsize',[22]);
xlabel('Time (s)')
ylabel('Column Entries of A')
legend('A(:,1)','A(:,51)','Location','South')

figure(3)
plot(diag(S),'x','linewidth',3)
axis([0 101 0 35])
hold on
plot(diag(Ss),'o','linewidth',3)
hold off
set(gca,'Fontsize',[22]);
legend('Deterministic Algorithm','Random Algorithm','Location','East')
ylabel('Singular Values')

Sdiff = abs(diag(S) - diag(Ss));
figure(4)
semilogy(Sdiff,'x','linewidth',3)
axis([0 101 1e-16 1e-12])
set(gca,'Fontsize',[22]);
ylabel('Absolute Difference in Singular Values')
```