PENELOPE 2003 - OECD Nuclear Energy Agency

Appendix B
Numerical tools
B.1 Cubic spline interpolation
In this section we follow the presentation of Maron (1982). Suppose that a function
f(x) is given in numerical form, i.e. as a table of values
f i = f(x i ) (i = 1, . . . , N). (B.1)
The points (knots) x i do not need to be equispaced, but we assume that they are in
(strictly) increasing order
x 1 < x 2 < · · · < x N .
(B.2)
A function ϕ(x) is said to be an interpolating cubic spline if
1) It reduces to a cubic polynomial within each interval [x i , x i+1 ], i.e. if x i ≤ x ≤ x i+1
ϕ(x) = a i + b i x + c i x 2 + d i x 3 ≡ p i (x) (i = 1, . . . , N − 1). (B.3)
2) The polynomial p i (x) matches the values of f(x) at the endpoints of the i-th interval,
p i (x i ) = f i , p i (x i+1 ) = f i+1 (i = 1, . . . , N − 1), (B.4)
so that ϕ(x) is continuous in [x 1 , x N ].
3) The first and second derivatives of ϕ(x) are continuous in [x 1 , x N ]
p ′ i (x i+1) = p ′ i+1 (x i+1) (i = 1, . . . , N − 2), (B.5)
p ′′
i (x i+1 ) = p ′′
i+1(x i+1 ) (i = 1, . . . , N − 2). (B.6)
Consequently, the curve y = ϕ(x) interpolates the table (B.1) and has a continuously
turning tangent.