Generating C1 class function with hermit interpolation?

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Flatron 2011-11-2

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https://ww2.mathworks.cn/matlabcentral/answers/20161-generating-c1-class-function-with-hermit-interpolation

Firstly thanks for all reply.

Problem: I want to generate a c1,c2,c3 class polinom(function,curve) with hermite interpolation. Given: a, b begining and end of my function. y(a),y(b), y'(a),y'(b),y''(a),y''(b),y'''(a),y'''(b). Example: http://dl.dropbox.com/u/27560194/picture.jpg

How i can do this in matlab?

Thank you, very much.

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Answer 1

Walter Roberson 2011-11-2

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You have 8 constraints, so you need a polynomial with 8 coefficients, which is a 7th degree polynomial. So construct

syms c1 c2 c3 c4 c5 c6 c7 c8
syms x
Y = sym('x->c1*x^7+c2*x^6+c3*x^5+c4*x^4+c5*x^3+c6*x^2+c7*x+c8');

Now let the constraint values be C01, C02, C11, C12, C21, C22, C31, C32 -- which I numbered with the continuity order as the first digit and the second digit is 1 for "a" and 2 for "b".

syms C01 C02 C11 C12 C21 C22 C31 C32

Now we make use of the symbolic operator D to express the constraints:

ics = '{Y(a)=C01,Y(b)=C02,D(Y)(a)=C11,D(Y)(b)=C12,D([1,1],Y)(a)=C21,D([1,1],Y)(b)=C22,D([1,1,1],Y)(a)=C31,D([1,1,1],Y)(b)}';

and expand the polynomial function and allow the D expressions to be evaluated:

eqnsys = simplify(subs(ics,'Y',Y));

And what we have left is a simple system of equations that we can solve() for:

S = solve(eqnsys,C01,C02,C11,C12,C21,C22,C31,C32);
SS = structfun(@simplify, S, 'Uniform', 0);

And now you will have SS.c1, SS.c2 and so on, in symbolic form, expressed in terms of C01, C02, etc.

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Walter Roberson 2011-11-3

So rename variables if they bother you.

The above code already accounts for the continuity. Polynomials are continuously differentiable, and your question specifically asked for a "polynomial" to be built, so this code builds the minimum polynomial that satisfies the derivative conditions at the given locations. Whether you call the derived coefficients c1, c2, c3, c4, c5, c6, c7, c8, or Marcia, Carol, Greg, Jan, Peter, Cindy, Mike, and Bobby, really does not matter.

Here, allow me to make the only necessary change to the code shown:

Y = sym('x->Marcia*x^7+Carol*x^6+Greg*x^5+Jan*x^4+Peter*x^3+Cindy*x^2+Mike*x+Bobby');

Now you will have SS.Marcia, SS.Carol, and so on, in symbolic form, expressed in terms of C01, C02, etc.

Flatron 2011-11-3