Spline - Interpolation :restart : with(linalg): with(plots): M:=6;

Warning, the protected names norm and trace have been redefined and unprotected

Warning, the name changecoords has been redefined

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t:=[ 0, 2.3, 3.4, 4.45, 6.5, 9.1 ]; s:=[ 0, 10, 20, 30, 50, 75 ] ; v:=[ 0, 29.7, 34.2, 34.8, 35.0, 34]:

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pair := (x,y) -> [x,y] : data_list :=zip(pair , t , s );

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Spline interpolation (nat\374rlicher Rand) :for i from 1 by 1 while i <= M do a[i]:=s[i]; end do:for i from 1 by 1 while i <= M-1 do h[i]:=t[i+1]-t[i]; end do:S:=Matrix(M,M): U:=Vector(M):S[1,1]:=1: S[M,M]:=1: U[1]:=v[1]/3.6: U[M]:=v[M]/3.6:for j from 2 by 1 while j <= M-1 do S[j,j]:= 2*(h[j-1]+h[j]); S[j,j+1]:= h[j] ; S[j,j-1]:= h[j-1] ; U[j]:= 3/h[j]*(a[j+1]-a[j]) - 3/h[j-1]*(a[j]-a[j-1]) ; end do:

b:=evalm(inverse(S) &* U) ;

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for j from 1 by 1 while j <= M-1 do c[j]:=(a[j+1]-a[j])/h[j] - h[j]*(2*c[j]+c[j+1])/3 ; d[j]:=(c[j+1]-c[j])/(3*h[j]) ; end do:eval(b);

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eval(d);

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f:= (x) -> sum( '(a[n] + b[n]*(x-t[n])+ c[n]*(x-t[n])^2+ d[n]*(x-t[n])^3 )*Heaviside(x-t[n])*Heaviside(t[n+1]-x)' , n=1..M-1) ;

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v:=(x) -> eval( diff(f(tau),tau), tau=x);

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plot( [ data_list , f(x) ] , x=0..15 , y=0..100 , style=[point,line] , thickness=2, title="Spline Interpolation" , symbol=circle, symbolsize=16);

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tend:=simplify (fsolve( f(x)=100, x) ); #f(tend); 3.6*v(tend);

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