Find the natural cubic spline function St at the points t0

Find the natural cubic spline function S(t) at the points t_0 = 0.t_1 = 1, t_2 = 2, t_3 = 3, with values S(t_0) = 0, S(t_1) = 2, S(t_2) = 1, S(t_3) = 0. Find the expressions of the spline functions in each of the three intervals.

Solution

(2) Since it is a cubic spline function, So let S(t) = a+b(t)+c(t)²+d(t)³ - eq(1)

       Now at t₀=0, we have S(t₀)=0, put this value in eq(1), we have the equation,

    0= a+b(0)+c(0)²+d(0)³ , that is a=o

Now at t₁=1, we have S(t₁) = 2, putting this value in eq(1), we get

0+b+c+d = 2, that is b+c+d =2    ---eq(2),

At t₂ = 2, we have S(t₂) = 1, putting this value in eq(1), we get

0+2b+4c+8d = 1, that is 2b+4c+8d =1   -----eq(3),

Multiplying eq(2) with 2 and subtracting with eq (3), we get

2c+3d =-3   ------eq(4),

Now at t₃ =3, we have S(t₃) = 0, putting this value in eq (1), we get

3b+9c+27d = 0, that is b+3c+9d = 0 -------eq(5),

Now put the value of eq(4) in eq(5), we have

b+2c+6d+c+3d = 0, b-3+c+3d =0, b+c+3d = 3 -------eq(6),

Now subtract eq(2) from eq(6), we get

2d = 1, d = 1/2,

Putting the value of d in eq(4), we have

2c+6d = -3, 2c +6*(1/2) = -3, that is c = -3,

Now the put the value of a, c, d in eq (2),

a+b+c+d = 2, 0+b+(-3)+(1/2) = 2,   that is b = 9/2,

Now put all the values in eq(1), we have the equation,

S(t) = (9/2)t - 3t² +(1/2)t³ this is the required cubic spline functions,