Discrete spline interpolation - Alchetron, the free social encyclopedia

In the mathematical field of numerical analysis, discrete spline interpolation is a form of interpolation where the interpolant is a special type of piecewise polynomial called a discrete spline. A discrete spline is a piecewise polynomial such that its central differences are continuous at the knots whereas a spline is a piecewise polynomial such that its derivatives are continuous at the knots. Discrete cubic splines are discrete splines where the central differences of orders 0, 1, and 2 are required to be continuous.

Discrete cubic splines

Let x₁, x₂, . . ., x_n-1 be an increasing set of real numbers. Let g(x) be a piecewise polynomial defined by

g ( x ) = { g 1 ( x ) x < x 1 g i ( x ) x i − 1 ≤ x < x i for i = 2 , 3 , … , n − 1 g n ( x ) x ≥ x n − 1

where g₁(x), . . ., g_n(x) are polynomials of degree 3. Let h > 0. If

( g i + 1 − g i ) ( x i + j h ) = 0 for j = − 1 , 0 , 1 and i = 1 , 2 , … , n − 1

then g(x) is called a discrete cubic spline.

Alternative formulation 1

The conditions defining a discrete cubic spline are equivalent to the following:

g i + 1 ( x i − h ) = g i ( x i − h ) g i + 1 ( x i ) = g i ( x i ) g i + 1 ( x i + h ) = g i ( x i + h )

Alternative formulation 2

The central differences of orders 0, 1, and 2 of a function f(x) are defined as follows:

D ( 0 ) f ( x ) = f ( x ) D ( 1 ) f ( x ) = f ( x + h ) − f ( x − h ) 2 h D ( 2 ) f ( x ) = f ( x + h ) − 2 f ( x ) + f ( x − h ) h 2

The conditions defining a discrete cubic spline are also equivalent to

D ( j ) g i + 1 ( x i ) = D ( j ) g i ( x i ) for j = 0 , 1 , 2 and i = 1 , 2 , … , n − 1.

This states that the central differences D ( j ) g ( x ) are continuous at x_i.

Example

Let x₁ = 1 and x₂ = 2 so that n = 3. The following function defines a discrete cubic spline:

g ( x ) = { x 3 x < 1 x 3 − 2 ( x − 1 ) ( ( x − 1 ) 2 − h 2 ) 1 ≤ x < 2 x 3 − 2 ( x − 1 ) ( ( x − 1 ) 2 − h 2 ) + ( x − 2 ) ( ( x − 2 ) 2 − h 2 ) x ≥ 2

Discrete cubic spline interpolant

Let x₀ < x₁ and x_n > x_n-1 and f(x) be a function defined in the closed interval [x₀ - h, x_n + h]. Then there is a unique cubic discrete spline g(x) satisfying the following conditions:

g ( x i ) = f ( x i ) for i = 0 , 1 , … , n . D ( 1 ) g 1 ( x 0 ) = D ( 1 ) f ( x 0 ) . D ( 1 ) g n ( x n ) = D ( 1 ) f ( x n ) .

This unique discrete cubic spline is the discrete spline interpolant to f(x) in the interval [x₀ - h, x_n + h]. This interpolant agrees with the values of f(x) at x₀, x₁, . . ., x_n.