Linear/linear rational spline interpolation*

Abstract For a strictly monotone function y on [a, b] we describe the construction of an interpolating linear/linear rational spline S of smoothness class C 1. We show that for the linear/linear rational splines we obtain ¦S(xi ) − y(xi )¦8 = O(h 4) on uniform mesh xi = a + ih, i = 0,…, n. We prove also the superconvergence of order h3 for the first derivative and of order h2 for the second derivative of S in certain points. Numerical examples support the obtained theoretical results.


Introduction
For a smooth function y and interpolating linear/linear rational spline S it is known that S − y ∞ = O(h 3 ), for the proof see, e.g., [8]. The linear/linear rational splines of class C 1 have the same accuracy as the classical quadratic splines. In some cases, the error is less for the quadratic splines and in some cases, the error is less for the linear/linear rational splines. For the quadratic splines, the expansions on subintervals via the derivatives of the smooth function to interpolate could be found, e.g., in [4,5]. They give the superconvergence of the spline values and its derivatives in certain points. We will study such a problem in the case of linear/linear rational spline interpolant.
Note that, linear/linear rational splines, being strictly monotone or constant everywhere, cannot interpolate nonmonotone data. For consistent data, the linear/linear rational spline interpolant of class C 1 always exists and is unique [9]. Let us mention that O(h 2 ) convergence rate of quadratic spline collocation for boundary value problems is based on superconvergence property of interpolating splines. This was discovered in [3] and developed extensively in [7,10]. Polynomial interpolants are used to establish convergence results of quasi-interpolants, see e.g., [6] and the references therein. For linear/linear rational spline histopolation, convergence rates could be found in [2].
Let us point out that, while the interpolation problem is a linear one, the linear/linear rational spline interpolation is, in nature, a nonlinear method because it leads to a nonlinear system with respect to the spline parameters. It was shown in [1] that any strict convexity preserving interpolation method having certain regularity properties cannot be linear. Hopefully, similar result should also hold for strict monotonicity preservation.

Interpolation by Linear/Linear Rational Splines
with knots x i = a + ih, i = 0, . . . , n, h = (b − a)/n, n ∈ N. We also need the points ξ i = x i−1 + h/2, i = 1, . . . , n. Linear/linear rational spline on each which allows to express uniquely a i , c i and d i via the spline values and to have the representation

(2.2)
This also gives for According to the representation (2.2) we have 3n parameters to determine for constructing the spline. We require for C 1 continuity of S on [a, b] which involves 2(n − 1) conditions, namely that S and S ′ must be continuous at all interior knots x 1 , . . . , x n−1 . When the continuity of S is guaranteed by the representation (2.2), the continuity of S ′ at interior knots leads to the equations Given the dataȳ i , i = 1, . . . , n, let us require that the interpolation conditions S(ξ i ) =ȳ i , i = 1, . . . , n, (2.6) are satisfied. In addition, we impose some boundary conditions, e.g.: and a combination with one condition from (2.7) and another from (2.8) at different endpoints is also allowed. We will specify the choice of numbers α 1 , α 2 , α 3 and α 4 in the following section. Replacing the valuesS i , i = 1, . . . , n, from (2.6) in the internal equations (2.5) and considering them with two boundary conditions we obtain a nonlinear system with respect to the unknowns S 0 , . . . , S n .

Expansions of the Interpolant
We derive our superconvergence rate results basing on the expansions of the interpolant which will be established in this section. First, we analyze the nonlinear system with respect to the unknowns S 0 , . . . , S n .
Let us write equations (2.5) with replaced valuesS i from (2.6) in the form introducing at the same time functions ϕ i . Then the system consisting of the boundary conditions (2.7) and the internal equations (3.1) can be written as Suppose now that we have a function y : [a, b] → R to interpolate andȳ i = y(ξ i ), i = 1, . . . , n. Denote y i = y(x i ), i = 0, . . . , n, similar notation will be used in the case of derivatives. At (3.1) the Taylor expansion gives Suppose in the following that y ∈ C 4 [a, b]. Let us expand y i−1 ,ȳ i ,ȳ i+1 and y i+1 at the point x i by Taylor formula up to the forth derivative as with the error terms O(h 4+α ) in the case y IV ∈ Lip α, 0 < α ≤ 1. Then direct calculations yield The entries in the matrix ϕ ′′ i consisting of the second order partial derivatives of ϕ i are of order O(h). This with the help of . Taking now into account (3.4), (3.5) and the order of the error term in (3.3), system (3.2) reduces to It is known, see, e.g., [8,9] that a linear/linear rational spline interpolant exist only if y is strictly monotone or constant everywhere. Thus, we assume that y ′ (x) > 0 for all x ∈ [a, b] or y ′ (x) < 0 for all x ∈ [a, b] which means that y is strictly monotone. Consider (3.6) as a linear system with respect to the unknowns S i − y i , i = 0, . . . , n. Then its matrix has the diagonal dominance in rows for sufficiently small h. We look for the solution such that where the continuous function ψ(y) and numbers β i will be specified later. The Replacing now (3.7) in the internal equations of (3.6) we get Determine the function ψ(y) so that the coefficient at h 6 is equal to 0, i.e., Now we may write (3.6) as follows This system has the matrix form Aβ = g, whereβ = (β 0 , . . . , β n ) and where A −1 is the matrix norm corresponding to the uniform vector norm. In total, we have which could be also transformed into the form Our next aim is to establish the expansions of interpolant S and its first and second derivatives on the whole particular interval. First, we write the representation (2.2) with obvious notations A and B in the form Then usingS i =ȳ i and Taylor expansions at ξ i in the fractional term, we arrive at the expansion for Clearly, the expansion (3.11) at x = x i or t = 1/2 coincides with (3.10). From (2.3), proceeding similarly, we get (3.12) and, finally, from (2.4) Similar reasoning allows us to establish the expansions (3.10)-(3.13) in the case of boundary conditions (2.8) provided we pose them in the form 14) We have proved the following theorem. They are given, e.g., in [4] in a slightly different form We can check directly that here the superconvergence takes place at the same points as for linear/linear rational spline interpolants.

Numerical Examples
We   Table 3. Numerical results for y(x) = x −2 , ε ′′ n = S ′′ (z i ) − y ′′ (z i ), i = 1, 2. The "tridiagonal" nonlinear system to determine the values of S i was solved by Newton's method and the iterations were stopped at S k − S k−1 ∞ ≤ 10 −10 , S k being the sequence of approximations to the vector S = (S 0 , . . . , S n ). The errors ε n = S(z i ) − y(z i ) and ε ′′ n = S ′′ (z i ) − y ′′ (z i ) were calculated in certain superconvergence points z i . Results of numerical tests are presented in Tables 1-3.