Summation of piecewise functions in matlab - matlab

Question

I have function y, whish is the sum of piecewise functions f0_basis and f1_basis (defined on [r_i-1; r_i+1]), multipled by number coefficients alpha and beta:
y=#(r) 0;
for j=1:1:N
y1= #(r) f0_basis(r ,j).*alpha(j)+f1_basis(r ,j).*beta(j);
y=#(r) y1(r)+y(r);
end
fplot(y,[a,b])
The fplot gives me what I want (a=1 and b=1.1):
However when I try, to find the value of y(r_i) - it always gives me the same wrong result for (a<r_i<b)
y(r_i)=-0.0016
which doesnt reperesents the plot above. Though if I try y(a) and y(b) - it gives me the right result.
What is wrong? I want to simply create an array y_array(i) = y(r_i).
N=11
a=1; b=1.1
delta_r=(b-a)/(N-1)
r_grid(i)=a+delta_r.*(i-1);
function y=f0_basis(r ,i)%i - номер середины узла
ri=r_grid(i);
r1=ri-delta_r;
r2=ri+delta_r;
y1=(2*((r-ri)/delta_r).^3-3*((r-ri)/delta_r).^2+1).*(r>=ri & r<=r2 & r2<=b);%правый конец
y2=(-2*((r-r1)/delta_r).^3+3*((r-r1)/delta_r).^2).*(r>=r1 & r<=ri & r1>=a);%левый конец
y3=0;%в сотальных случаях
y=y1+y2+y3;
end
function y=f1_basis(r ,i)%ri - середина узла
ri=r_grid(i);
r1=ri-delta_r;
r2=ri+delta_r;
r_div_1=(r-ri)/delta_r;
r_div_2=(r-r1)/delta_r;
y1=delta_r*(r_div_1.^3-2*(r_div_1.^2)+r_div_1).*(r>=ri & r<=r2 & r2<=b);%правый конец
y2=delta_r*(r_div_2.^3-r_div_2.^2).*(r>=r1 & r<=ri & r1>=a);%левый конец
y3=0;%в сотальных случаях
y=y1+y2+y3;
end
Alpha:
-0,000828144527478242
-0,000823822186226137
-0,000819532164240625
-0,000815314580859248
-0,000811187867518122
-0,000807162471718638
-0,000803245019928998
-0,000799439804012876
-0,000795750103056819
-0,000792175886650016
-0,000788726122705251
Beta:
-1,14781261976785e-05
-0,00257727396861173
-0,000627562853139588
0,000211869596648955
0,000683549308474220
0,000987694942326156
0,00119967671089174
0,00136043282028271
0,00145954455875083
0,00163805699746171
-0,000147024199470849
UPDATE:
it seems that y(r_i)=-0.0016 for only r_grid(i) array. If I try some r not from r_grid(i) array - it works fine.

Answer 1

I think your problem is in the functions f0_basis and f1_basis. You compute 2 terms inside each function (the 3rd one is null), each term is subject to some constraints. But the constraints are overlapping for one point, if r==ri. In this case, you are adding two equal terms, but nowhere else. That's why you have strange results at r_grid(i).
So change the tests, for example r>=r1 & r<ri & r1>=a for y2.