How to change my spline code to natural spline?

I have written the following code to recieve a function from the user and a set of points to be interpolated by natural cubic spline.

funstr = input('Please enter your function: ', 's');
f = str2func( ['@(x) ' funstr ] );
x = input('Please enter the points separated with space in row vector form: ');
y = f(x);
m = min(x);
M = max(x);
xx = linspace(m-10,M+10,100);
values = csapi(x,y,xx);
plot(x,y,'b');
hold on
plot(xx, values - xx,'r')

How can I modify it so that it uses Natural cubic spline rather than spline? Also is there a way to access the coefficients of each spline polynomial s(x) for each previously given node point in some [a,b] interval?

P.S: I have already searched every spline related quetsion on the site and they were not helpful.

[John D'Errico]

No. It is not that they were not helpful. Merely that you did not understand what was answered, or you did not find the questions that would have helped you. This said, since I have repeatedly answered similar questions, and explained the same things many times before. So the odds are you did not look very hard.

[Walter Roberson]

What difficulty did you encounter when you used the advice from

https://www.mathworks.com/matlabcentral/answers/387177-plot-natural-cubic-spline#answer_309106

[John D'Errico]

Good. I really did not want to write the whole thing again. :)

Perhaps @Mali Yaser was hoping to find a way to force csapi to produce a natural cubic spline. This will not happen. Or perhaps @Mali Yaser did not understand that the "variational" end conditions are the same thing as what some others have called the natural end conditions. Or that the default end conditions for the "second" option for csape are in fact the same as what variational does.

As far as extracting the polynomial segments themselves, I know I've written about how to do that many times before too.

[Mali Yaser]

Maybe my lack of perspicacity could be measured up with your pompousness and sarcasm.

[John D'Errico]

And that is a problem for who?

Note that had you simply asked how to do that, I would have told you to look at csape, or perhaps at one of my many answers on exactly that question. But telling us that nothing on the site, and you have searched the entire site, was helpful? That merely said you were telling a falsehood, since you could not have done any kind of search. The response you got was in kind.

[Mali Yaser]

And because I have by your estimation stated a falsehood, you must defend yourself by making another falsehood, that I "could not have done any kind of search". Were you present by my side when I was searching for the last week for an answer to this in this site? Has it occured to your hurt ego that I might be speaking of helpful as useful for me i.e. what I comprehend and can apply and that your presupposed offence could simply be a misunderstanding produced by language difference? Has the possibility of diverse coginitive capacities among humans presented itself to you?

[Walter Roberson]

@Mali Yaser

What difficulty did you encounter with https://www.mathworks.com/matlabcentral/answers/387177-plot-natural-cubic-spline#answer_309106 ? Which specifically indicates the call and option to use, and specifically says "A natural cubic spline is the one that has zero second derivatives at the ends." ?

When you tested using csape() in response to that posting, did you encounter a mismatch between the results csape() produced and the results you expected?

[Mali Yaser]

I could not understand 'variational' at first but after I understood it I failed to get the idea of fnder(pp,2) and how it was necessary when 'variational' was already used to set the second derivative to zero. And now after I have inserted 'variational' I get an error that 'Matrix dimensions must agree'.

[Walter Roberson]

pp2 = fnder(pp,2);

fnplt(pp2)

is not necessary for the computation. The reason it was used was to illustrate that the second derivative was truly 0 at the end points.

I see you are using R2020b, which is after R2018* that John first illustrated the problem with, so we can tell that you do not happen to be using a version of MATLAB too old for 'variational' to be an option.

Is the error in the csape() call? If so then what size() are your x, y, and (if used) conditions ?

[Mali Yaser]

The error is in the following. If I remove that line which is intended to plot the error function, it would work without a problem.

plot(xx, values - xx,'r')

[Walter Roberson]

@Mali Yaser

There a couple of possibilities for how "Matrix dimensions must agree" could occur in that statement.

The simplest possibility is that xx and values are vectors but they do not have the same orientation, and that you are using R2015a or earlier. (If you are using R2015b or later then that situation would not lead to that message, but you would get a plot you did not expect.)

Another possibility is that xx is a vector but values is not a vector, and the non-singular sizes do not match. For example, a (1 x 3) minus a (3 x 2) would not be permitted, but a (3 x 1) minus a (3 x 2) would be permitted because the mis-match in sizes, 1 vs 2, only involves singular dimensions. (3 x 1) minus (3 x 2) would be equivalent to repmat(the_3x1, [1 2]) - the_3x2

Beyond that... both could be matrices with mismatched sizes.

If values and xx are both vectors, make sure they have the same orientation -- either both row vectors or both column vectors.

Answer 1

[Torsten]

One google search and so many hits, e.g.

http://de.mathworks.com/matlabcentral/answers/387177-plot-natural-cubic-spline#answer_309106

The page

http://de.mathworks.com/help/curvefit/csape.html

also answers the question about the spline coefficients.