Math Problem Solver

for the given data

x=1,2,3,4

f(x)=1.5,2,3,3.5

estimate of at x=2.5 and x=3.25 using cubic splines interpolation method

hints :

there will be three cubic cubic splines of the form

s0(x) = a 0 + b0(x-x0) + c0(x-x0)^2 + d0(x-x0)^3

s0(x) = a1 + b1(x-x1) + c1(x-x1)^2 + d1(x-x1)^3

s0(x) = a2 + b2(x-x2) + c2(x-x2)^2 + d2(x-x2)^3

to find 12 constants (unknowns) above we need at least 12 conditions

INTERPOLATING CONDITIONS:

1) s0(x0) = f(x0)

2) s1(x1) = f(x1)

3) s2(x2) = f(x2)

CONTINUITY CONDITIONS:

1) s0(x1) = s1(x1)

2) s1(x2) = s2(x2)

3) s2(x3) = s3(x3)

FIRST DERIVATIVE CONDITIONS:

1) s0'(x1) = s1'(x1)

2) s1'(x2) = s2'(x2)

FIRST DERIVATIVE CONDITIONS:

1) s0''(x1) = s1''(x1)

2) s1''(x2) = s2''(x2)

TWO EXTRA CONDITIONS (BOUNDARY CONDITIONS)

1) s0''(x0) = 0

2) s2''(x2) = 0