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I'm solving a problem (page 16 is in English) in numerical analysis and this is the solution:

>> z=[0:500:4000 5000:1000:12000];
>> data=[5050 4980 4930 4890 4870 4865 4860 4860 4865 4875 4885 4905 4920 4935 4950 4970 4990];
>> fun=@(p)(4800 + p(1))*ones(size(z)) +p(2)/1000*z+p(3)*exp(p(4)/1000*z)-data;
>> x0=[0 0 0 -1];
>> opt = optimset('MaxFunEvals',1000);
>> p=lsqnonlin(fun,x0,[],[],opt);

Local minimum possible.

lsqnonlin stopped because the final change in the sum of squares relative to 
its initial value is less than the default value of the function tolerance.

<stopping criteria details>

>> fitf=@(t)(4800 + p(1))*ones(size(t)) + p(2)/1000*t+ p(3)*exp(p(4)/1000*t);
>> tt=linspace(0,12000,1000);
>> plot(z,data,'r-',tt,fitf(tt),'b-');
>> p

p =

  -20.2090   17.3368  272.9057   -0.7528

>> 

On the plot it looks alright:

enter image description here

But when I asked about it the coefficients to the polynomial in the answer were some other values:

ode45 usage in this case?

In my program run, the coeff. appear to be p4=-0.7528 and the other are p1=-20.2090, p2=17.3368, p3=272.9057.

So c(z) becomes

c(z)=4800-20.2090+(17.3368)z/1000+(272.9057)exp(0.7528z/1000)

and

c’(z)=17.3368/1000-0.2054exp(-z/1000)

And then I can make the substitution z’=u and z’’=u’ so the matlab code to continue with will be:

function dZ=sys(x,Z)
    c=@(z)4800 -20.2090 + (17.3368)*z/1000+ (272.9057)*exp(0.7528z/1000); % c(z)
    c=c(2000);

% Z(1):=z
% Z(2):=u
    dZ=zeros(2,1);    % a column vector
    dZ(1)=Z(2);
    dZ(2)=(c/cosd(7.8))^2*((1287*exp(Z(1)/1000))/5000 + 729/50000)/...
        (4800 + 5.66 + (14.58)*Z(1)/1000+ (257.4)*exp(-Z(1)/1000))^3;
    end

Do you agree?

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Both your coefficients and the coefficients in the answer are correct but yours are better.

The information of fit for the coefficients in the answer:

Chi-squared: 813.032070619
R-squared: 0.983460126607
R-squared adjusted: 0.981097287551
Root Mean Squared Error (RMSE): 6.91559223196
p1 = 5.6596316127433939E+00
     std err squared: 2.17628E+01
     t-stat: 1.21319E+00
     p-stat: 2.45129E-01
     95% confidence intervals: [-4.34594E+00, 1.56652E+01]
p2 = 1.4581975826123806E+01
     std err squared: 4.22221E-01
     t-stat: 2.24412E+01
     p-stat: 2.24487E-12
     95% confidence intervals: [1.31883E+01, 1.59756E+01]
p3 = 2.5743180704436827E+02
     std err squared: 8.25256E+01
     t-stat: 2.83379E+01
     p-stat: 9.14824E-14
     95% confidence intervals: [2.37948E+02, 2.76916E+02] 

The information of fit for your coefficients:

Chi-squared: 123.211540448
R-squared: 0.997493452776
R-squared adjusted: 0.996915018801
Root Mean Squared Error (RMSE): 2.69216226729
p1 = -2.0209016655555111E+01
     std err squared: 1.78565E+01
     t-stat: -4.78242E+00
     p-stat: 3.57957E-04
     95% confidence intervals: [-2.93381E+01, -1.10800E+01]
p2 = 1.7336770761968697E+01
     std err squared: 2.24807E-01
     t-stat: 3.65648E+01
     p-stat: 1.70974E-14
     95% confidence intervals: [1.63125E+01, 1.83611E+01]
p3 = 2.7290573753852300E+02
     std err squared: 1.98905E+01
     t-stat: 6.11913E+01
     p-stat: 0.00000E+00
     95% confidence intervals: [2.63271E+02, 2.82541E+02]
p4 = 7.5277833016177342E-01
     std err squared: 6.80991E-04
     t-stat: 2.88467E+01
     p-stat: 3.59046E-13
     95% confidence intervals: [6.96402E-01, 8.09155E-01] 

These fits calculated by zunzun

Additionally i think you missed a minus before p4 in c(z) and you should change the coefficients in dZ(2) with your coeffs.

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  • $\begingroup$ Thank you for the answer. Can there be even more solution? Why can there be several solutions? How do I justify which coefficients I use? I will work through the program and run it once more to be sure that the calculations are correct. $\endgroup$
    – Niklas R.
    Sep 18 '13 at 14:53
  • 2
    $\begingroup$ Actually the two coefficient sets (in the answer and yours) are not for the same problem namely not solutions of same fitting problem. Because we actually ignore $p4$ by taking $p4=1$ and then find the other coefficients. But you, you find four coefficients. Although i am not expert in statistics as i know when you fit a curve or a line to a given data set you check the value of "R-squared" to measure the goodness of the fit. This value is between $0$ and $1$. As $R^2$ approach to $1$ your fit gets better. As you see the coefficients you have are better for the fit. $\endgroup$
    – Ömer
    Sep 18 '13 at 15:58

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