Question

Natural cubic spline: If f is defined at a = x0 < x1 < ? < xn = b, then f has a unique natural spline interpolation S on the nodes x0,x1,…,xn is given by, Sj(x) = aj + bj(x ? xj) + cj(x ? xj)^2 + dj(x ? xj)^3 for j = 0,1,…,n ? 1 ...... (i) To obtain the values of bj,cj, and dj use following formulas. To obtain cj solve the equation Ax = b where, A = [ 1 0 0 ? 0 h0 2(h0 + h1) h1 0 h1 2(h1 + h2) h2 0 ? 0 ? ? 0 1 ], Here hj = xj+1 ? xj b = [ 0 3/h1(a2 ? a1) ? 3/h0(a1 ? a0) ? 3/hn-1(an ? an-1) ? 3/hn-2(an-1 ? an-2) 0 ] and x = [ c0 c1 ? cn ] bj = 1/hj(aj+1 ? aj) ? hj/3(2cj + cj+1), j = 0,1,…,n ? 1 dj = 1/3hj(cj+1 ? cj), j = 0,1,…,n ? 1

          Natural cubic spline:
If f is defined at a = x0 < x1 < ? < xn = b, then f has a unique natural spline interpolation S on the nodes x0,x1,…,xn is given by,
Sj(x) = aj + bj(x ? xj) + cj(x ? xj)^2 + dj(x ? xj)^3 for j = 0,1,…,n ? 1 ...... (i)
To obtain the values of bj,cj, and dj use following formulas.
To obtain cj solve the equation Ax = b where,
A = [
1 0 0 ? 0
h0 2(h0 + h1) h1
0 h1 2(h1 + h2) h2 0
?
0 ? ? 0 1
], Here hj = xj+1 ? xj
b = [
0
3/h1(a2 ? a1) ? 3/h0(a1 ? a0)
?
3/hn-1(an ? an-1) ? 3/hn-2(an-1 ? an-2)
0
] and x = [
c0
c1
?
cn
]
bj = 1/hj(aj+1 ? aj) ? hj/3(2cj + cj+1), j = 0,1,…,n ? 1
dj = 1/3hj(cj+1 ? cj), j = 0,1,…,n ? 1

Natural cubic spline:
If f is defined at a = x0 < x1 < ? < xn = b, then f has a unique natural spline interpolation S on the nodes x0,x1,…,xn is given by,
Sj(x) = aj + bj(x ? xj) + cj(x ? xj)^2 + dj(x ? xj)^3 for j = 0,1,…,n ? 1 ...... (i)
To obtain the values of bj,cj, and dj use following formulas.
To obtain cj solve the equation Ax = b where,
A = [
1 0 0 ? 0
h0 2(h0 + h1) h1
0 h1 2(h1 + h2) h2 0
?
0 ? ? 0 1
], Here hj = xj+1 ? xj
b = [
0
3/h1(a2 ? a1) ? 3/h0(a1 ? a0)
?
3/hn-1(an ? an-1) ? 3/hn-2(an-1 ? an-2)
0
] and x = [
c0
c1
?
cn
]
bj = 1/hj(aj+1 ? aj) ? hj/3(2cj + cj+1), j = 0,1,…,n ? 1
dj = 1/3hj(cj+1 ? cj), j = 0,1,…,n ? 1

Added by Ethan N.

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