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Natural and clamped cubic splines

What is the difference between natural and clamped Cubic Splines?

Solve the following problems with a clear explanation.

[1] A natural cubic spline S on [0,2] is defined by

S(x) = { S0(x) = 1 + 2*x - x^3 , if 0 <= x <= 1
S(x) = { S1(x) = 2 + b*(x-1) + c*(x-1)^2 + d*(x-1)^3 , if 1 <= x <= 2

Find b, c and d.

[2] A clamped cubic spline S for a function f is defined by

S(x) = { S0(x) = 1 + B*x + 2*x^2 - 2*x^3 , if 0 <= x <= 1
S(x) = { S1(x) = 2 + b*(x-1) - 4*(x-1)^2 + 7*(x-1)^3 , if 1 <= x <= 2

Find f'(0) and f'(2).

Solution Preview

Natural and Clamped Cubic Spline are same in all the conditions except two -

Natural Cubic Spline requires that
S0''(x0) = SN_1''(xn) = 0 , N_1 represents the subscript value (n-1)

and Clamped Cubic Spline requires that
S0'(x0) = f'(x0)
SN_1'(xn) = f'(xn) , N_1 represents the subscript value (n-1)

[1] S0'(x) = 2 - 3x^2
S0''(x) = - 6x

S1'(x) = b + 2c(x-1) + 3d(x-1)^2
S1''(x) = 2c + 6d(x-1)

...

Solution Summary

Solution shows every step methodically while determining the value of unknown constants and boundary values.

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