Section 8.1 Problem 6

(a)

>> x=[.2 .3 .6 .9 1.1 1.3 1.4 1.6]'                              

x =

    0.2000
    0.3000
    0.6000
    0.9000
    1.1000
    1.3000
    1.4000
    1.6000

>> y=[.050446 .098426 .33277 .72660 1.0972 1.5697 1.8487 2.5015]'

y =

    0.0504
    0.0984
    0.3328
    0.7266
    1.0972
    1.5697
    1.8487
    2.5015

>> A=[x ones(8,1)]

A =

    0.2000    1.0000
    0.3000    1.0000
    0.6000    1.0000
    0.9000    1.0000
    1.1000    1.0000
    1.3000    1.0000
    1.4000    1.0000
    1.6000    1.0000


>> sol=inv(A'*A)*(A'*y)

sol =

    1.6655
   -0.5125

Now, you can create a function file (linear.m) that holds the linear function 
y = ax+b:

function v=linear(x)
    v=1.6655*x-0.5125;

And you can plot the discrete (given) data and the continuous linear fitting 
by doing:

>> x_cont=linspace(0.2,1.6,100);
>> y_cont=linear(x_cont);

>> plot(x,y,'x',x_cont,y_cont,'k')
>> axis([0 1.7 -.5 2.6])
>> title('exponential fitting (-) of given data (x)');

(d)

>> lny=log(y)

lny =

   -2.9869
   -2.3185
   -1.1003
   -0.3194
    0.0928
    0.4509
    0.6145
    0.9169

same A as in (a)

>> sol=inv(A'*A)*(A'*lny)

sol =

    2.7073
   -3.0855

>> a=sol(1)

a =

    2.7073

>> b=exp(sol(2))

b =

    0.0457

Now, you can create a function file (exponential.m) that holds the linear 
function y = be^(ax):

function v=exponential(x)
    v=0.0457*exp(2.7073*x);

And you can plot the discrete (given) data and the continuous exponential 
fitting by doing:

>> x_cont=linspace(0.2,1.6,100);
>> y_cont=exponential(x_cont);

>> plot(x,y,'x',x_cont,y_cont,'k')
>> axis([0 1.7 -.5 2.6])
>> title('exponential fitting (-) of given data (x)');

(e) CORRECTED

(1) polynomial fitting y=b*x^a

>> lnx=log(x)

lnx =

   -1.6094
   -1.2040
   -0.5108
   -0.1054
    0.0953
    0.2624
    0.3365
    0.4700

>> A=[lnx ones(8,1)]

A =

   -1.6094    1.0000
   -1.2040    1.0000
   -0.5108    1.0000
   -0.1054    1.0000
    0.0953    1.0000
    0.2624    1.0000
    0.3365    1.0000
    0.4700    1.0000

>> sol=inv(A'*A)*(A'*lny)

sol =

    1.8720
   -0.0511

>> b=exp(sol(2))

b =

    0.9502

Now, you can create a function file (pol.m) that holds the polynomial
function y = b*x^a:

function v=pol(x)
    v=.9502*x^1.8720;

And you can plot the discrete (given) data and the continuous polynomial 
fitting by doing:

>> x_cont=linspace(0.2,1.6,100);
>> for i=1:100, y_cont(i)=pol(x_cont(i)); end

>> plot(x,y,'x',x_cont,y_cont,'k')
>> axis([0 1.7 -.5 2.6])
>> title('polynomial fitting y = b*x^a (-) of given data (x)');

(2) Exponential fitting y=b*a^x:

Matrix A same as in parts (a) and (d).

>> sol=inv(A'*A)*(A'*lny)

sol =

    2.7073
   -3.0855

>> a=exp(sol(1))

a =

   14.9887

>> b=exp(sol(2))

b =

    0.0457

>> for i=1:100, y_cont(i)=expon(x_cont(i)); end
>> plot(x,y,'x',x_cont,y_cont,'k')

Now, you can create a function file (expon.m) that holds the exponential 
function y = ba^x:

function v=expon(x)
    v=.0457*(14.9887)^x;

And you can plot the discrete (given) data and the continuous exponential 
fitting by doing:

>> x_cont=linspace(0.2,1.6,100);
>> for i=1:100, y_cont(i)=expon(x_cont(i)); end

>> plot(x,y,'x',x_cont,y_cont,'k')
>> axis([0 1.7 -.5 2.6])
>> title('exponential fitting y = b*a^x (-) of given data (x)');