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Topic: Cubic Spline Interpolation using MATLAB

Problem statement: Write MATLAB codes to perform the following actions

(a) Read a given number of points from the command prompt and apply cubic spline

interpolation(b) Generate plots of the cubic splines

generated by MATLAB from part (1)(c) Calculate the areas under individual

splines


Solutions

(1) For this code, the points are first read from the user by means of the input function at the prompt. Then, the boundary conditions are applied. • The second derivatives at the starting and

ending points are zero.• The second derivatives and first derivatives at

the shared nodes are equal.• The Spline is continuous at all the nodes.


MATLAB Codeprompt= ' Enter the number of points for which the spline is to be constructed\n';n=input(prompt); fprintf('The number of points for which the spline is to be constructed is %d\n',n); x=[]; y=[]; for i=1:n prompt= 'Enter the x coordinate of point :\n'; x(i)= input(prompt); prompt='Enter the y coordinate of point :\n'; y(i)=input(prompt); end


MATLAB Code (contd.)

for i=1:n-1 h(i)= x(i+1)-x(i); end for i=1:n a(i)=y(i); end for i=2:n-1 A(i)= (((3*(a(i+1)-a(i)))/h(i))-((3*(a(i)-a(i-1)))/h(i-1))); end




I(1)=1; M(1)=0; Z(1)=0; for i=2:n-1 I(i)=2*(x(i+1)-x(i-1))-(h(i-1)*M(i-1)); M(i)=(h(i)/I(i)); Z(i)= ((A(i)-(h(i-1)*Z(i-1)))/I(i)); end I(n)=1; c(n)=0; Z(n)=0;


for j=n-1:-1:1 c(j)=Z(j)-(M(j)*c(j+1)); b(j)=((a(j+1)-a(j))/h(j))-((h(j)*(c(j+1)+(2*c(j))))/3); d(j)=((c(j+1)-c(j))/(3*h(j))); end for j=1:n-1 a(j) b(j) c(j) d(j) end


Solutions(contd.)

(2)The following scripts were used in

the plotting of the three splines. The scripts use the formula


MATLAB Code

CubicSpline2 for i=1:20q(i)=1+(i/10);y1(i)= a(1)+(b(1)*(q(i)-1))+(c(1)*((q(i)-1)^2))+(d(1)*((q(i)-1)^3));endplot(q,y1);hold onclear q;for i=1:10q(i)= 3+(i/10);y2(i)= a(2)+(b(2)*(q(i)-3))+(c(2)*((q(i)-3)^2))+(d(2)*((q(i)-3)^3));end




plot(q,y2);hold onclear q;for i=1:10q(i)= 4+(i/10);y3(i)= a(3)+(b(3)*(q(i)-4))+(c(3)*((q(i)-4)^2))+(d(3)*((q(i)-4)^3));endplot(q,y3);hold onclear q;for i=1:10q(i)= 5+(i/10);y4(i)= a(4)+(b(4)*(q(i)-5))+(c(4)*((q(i)-5)^2))+(d(4)*((q(i)-5)^3));end

MATLAB Code (contd.)plot(q,y4);hold onclear q;plot([1,3,4,5,6],[2,3,2,1,5],'o');hold onAreaSpline(n,x,a,b,c,d)

CubicSpline2for i=1:10q(i)=4+(i/10);z1(i)= a(1)+(b(1)*(q(i)-4))+(c(1)*((q(i)-4)^2))+(d(1)*((q(i)-4)^3));endplot(q,z1);


MATLAB Code (contd.)hold onclear q;for i=1:5q(i)= 5+(i/10);z2(i)= a(2)+(b(2)*(q(i)-5))+(c(2)*((q(i)-5)^2))+(d(2)*((q(i)-5)^3));endplot(q,z2);hold onclear q;for i=1:15q(i)= 5.5+(i/10);z3(i)= a(3)+(b(3)*(q(i)-5.5))+(c(3)*((q(i)-5.5)^2))+(d(3)*((q(i)-5.5)^3));


MATLAB Code (contd.)endplot(q,z3);hold onclear q;for i=1:10q(i)= 7+(i/10);z4(i)= a(4)+(b(4)*(q(i)-7))+(c(4)*((q(i)-7)^2))+(d(4)*((q(i)-7)^3));endplot(q,z4);hold onclear q;for i=1:10q(i)= 8+(i/10);z5(i)= a(5)+(b(5)*(q(i)-8))+(c(5)*((q(i)-8)^2))+(d(5)*((q(i)-8)^3));end



plot(q,z5);hold onclear q;for i=1:20q(i)= 9+(i/10);z6(i)= a(6)+(b(6)*(q(i)-9))+(c(6)*((q(i)-9)^2))+(d(6)*((q(i)-9)^3));endplot(q,z6);hold onclear q;plot([4,5,5.5,7,8,9,11],[6,6,8,3,6,3,9],'x');hold on


Solutions(contd.)Plot of splines with their respective points


Solutions(contd.)The area under each spline was found out by integrating the spline under the

bounds. This is done by the formula:

Where n is the number of splines/. This area was found using MATLAB script AreaSpline.



function D= AreaSpline(n,x,a,b,c,d)Area=0;for j=1:n-1 l(j)= x(j+1)-x(j); Area=Area+ a(j)+(b(j)*(l(j)^2)/2)+(c(j)*(l(j)^3)/3)+(d(j)*(l(j)^4)/4);end D=Area; End
