Construction of local integro quintic splines

In this paper, we show that the integro quintic splines can locally be constructed without solving any systems of equations. The new construction does not require any additional end conditions. By virtue of these advantages the proposed algorithm is easy to implement and effective. At the same time, the local integro quintic splines possess as good approximation properties as the integro quintic splines. In this paper, we have proved that our local integro quintic spline has superconvergence properties at the knots for the first and third derivatives. The orders of convergence at the knots are six (not five) for the first derivative and four (not three) for the third derivative.


Introduction
Recently, some authors have devoted much attention to constructing integro splines [1,2,4,6,7,9,10,16,17,18,19].In particular, integro quintic splines approach over a uniform grid was discussed in [2, 7, 10 and 18].Interpolation by quintic integro spline method was given in [2] based on the quintic "Hermite-Birkhoff interpolation polynomial".But his method requires solving three systems of linear equations by using seven additional end conditions.Also, Behforooz has not investigated the derivatives approximation for y(x).In [10] Xu and Lang discussed the reconstruction of the quintic spline by using integral values, which does not need any additional endconditions.The methods in [2,7,10,18] require a system of linear equations to be solved.At the same time, the approximation by local spline is interesting not only from the theoretical but also practical point of views.Our local splines possess as good approximation properties as the interpolating ones, and are easy to construct; see [3,15].It should be pointed out that the terms of integro spline and histospline indeed are identical.The histosplines were studied mainly from the shape-preserving point of view.There are many papers on constructing histosplines preserving convexity, monotonicity, positivity.Algorithm for constructing them either leads to the problem of programming with minimal curvature [8,11] or to the system of equalities and inequalities which solved by the staircase algorithm; see [12,13] and references therein.But the approximation properties of these histosplines are not considered, whereas in last years are intensively studied the approximation properties of integro-splines.The aim of this paper is to construct the local integro quintic spline which does not require additional end conditions, and we do not need to solve a system of linear equations.Our new method has some advantages over the above mentioned methods.We highlight them as follows: I. Our method works without any additional end conditions, as in [10].II.It does not require solving any system of linear equations, in contrast to the methods in [2,10,18].III.It gives analytic formulae for derivatives of the quintic spline at the knots obtained from the given data.
IV.The construction of quintic spline is given in the piecewise polynomial representation and in its B-spline representation, as well.
The rest of this paper is organized as follows.We present some preliminary results of the integro quintic spline in Section 2. In Section 3, we give the construction of a new local integro quintic spline and analyze its error estimate.Section 4 is devoted to the construction of the local integro quintic spline in term of B-spline.Numerical results and discussion are presented in Section 5.

Integro quintic spline
Suppose that the interval [a, b] is partitioned by the following k + 1 equally spaced points We assume that the integral values I i of the function y(x) on the subintervals [x i−1 , x i ] are known and they are equal to Our integro quintic spline S(x) ∈ C 4 [a, b] satisfies the conditions in [2] x i For simplicity, we will use the following notations: In each subinterval where [2], The following properties are some relations between m's, n's and I's values; see [2,18], ) and ) where (2.10) The proof of the following theorem is given in [18].
Then we have ) Using the Taylor expansion of N i and y i , we obtain the following relations and y By subtracting (2.13) and (2.14), using (2.12), we get ) (2.17) Similarly, using (2.1) and the Taylor expansion of quintic spline S(x) at the point x = x i , we find i−0 ), (2.18) ) Proof.Using the well known formulae [5] y in (2.17), we obtain From (2.5) and (3.23) we obtain, Substituting m i from (2.4) into (3.24),we obtain Thus, we have three-diagonal systems (3.23) and (3.25) for m i and n i instead of (2.8) and (2.9).By using (3.25) in the expression ) 2) two and four times, we find that ) we get ) ) Using this equality we find that Substituting the last expression into (3.34) and (3.35), we obtain ) (see e.g.[18], for the proof of Theorem 1 and 2).That means that the Lemma 3.

Error analysis for local integro quintic spline
The replacement of S i , m i , and n i used in the quintic spline S(x) by their approximations given in the previous subsection, (we neglect the small terms) is denoted here by Ŝi , mi , and ni respectively.So these values can be computed without loss of accuracy and solving the linear systems in [18].Moreover, the new construction we propose does not require any additional end conditions compared to the quintic spline considered in [2,18].Now, we study the interpolant defined by where, (3.45) Note that the combinations of I j to approximate m i and n i found in the last paragraph of the previous subsection coincide with (3.44)-(3.45).The remainder Ŝ0 and Ŝk are determined from the properties (2.7), in which S i is replaced by Ŝi , For the values of I j and y ∈ C 6 [a, b], the following property holds

Then we have mi
) Proof.Using the Taylor expansion of y(x) ∈ C 7 [a, b] in (2.1) we obtain where It is easy to show that (3.61) Using the Taylor expansion of y ′ i−1 and y ′ i+1 , we have  Theorem 3.3 shows that our local integro quintic spline possesses superconvergence orders in the approximating function's first and third-order derivative values at the knots.The proof of the following lemma can be found in Zavyalov (see [14], page 334).

Lemma 3.3. If the matrix of the system Ax
then the first and third order derivatives superconvergence property also holds for S(x) in Theorem 2.1.
Proof.Since it holds true for m i , n i in place of mi , ni in (3.63), using the end conditions (2.11), we get ).The end conditions (2.11) and the equations (3.65), (3.66) form the following linear system, Adding row one multiplied by -1 to row two and adding row k − 1 multiplied by -1 to row k − 2, we get a system with diagonally dominant matrix.By Lemma 3.3, the S(x)'s superconvergence order in the approximating function's first-order derivative values at the knots is proved.The proof for the third-order derivative is similar and omitted here.

B-spline representation of the local integro-quintic spline
Besides (2.2), the quintic integro spline was given in [10] was based on the B-spline representation.Now, we consider the local construction of B-spline representation of spline which satisfies (2.1).To this end, we extend the uniform grid by ten additional equidistant knots where B j (x) is quintic B-spline, which is locally supported on [x j−3 , x j+3 ].By virtue of the properties of quintic B-spline we have (i = 0, 1, ) ) In [17] we gave an analytical formula for α j in the case of a local integro cubic spline.As for local integro quintic spline it is also possible to find such formula.To drive the formula for α j we will use the following one (see e.g.[8] The B-spline representation of the local integro-quintic spline is, Using (3.50) and (4.75), the remainder coefficients in (4.76) are determined from (4.69) and (4.70).

Numerical examples and discussion
In this section we present some numerical results to test the accuracy and efficiency of our method.The tested functions are y 1 (x) = exp(x), y 2 (x) = sin(πx), and y 3 (x) = 1 x+2 on the interval [a, b] = [0, 1].Let Ŝ(x) be the local quintic spline given by (3.40) for y 1 (x), y 2 (x), and y 3 (x).Numerical experiments of the proposed method are performed in 64 decimal digits of precision.It should be noted that (3.40) is preferably used to make the error estimation rather than B-spline representation (4.76).The respective maximum absolute errors are given in Tables 1, 2 and 3 In order to verify the convergence rate we calculate the Runge coefficient β r : for each r from 0 to 5. In these tables, the last two entries on each column are used to compute β r .Theoretically, the rate of convergence of our method should be β r = 6 − r (r = 0, 2, 4), β 1 = 6, and β 3 = 4. From the numerical experiments (Tables 1-3) we can see that the values of β r correspond to theoretical ones.To make the comparison we use the results in [10].By referring to this paper, we remark that our new results are better than those shown in [10].The differences can clearly be seen from Tables 1 and 4. Finally, we estimate the  [10]).The maximum absolute errors of s1 (x) for y 1 (x) = e x .k MAE (0) (k) MAE (1) (k) MAE (2) (k) MAE (3) (k) MAE (4) (k) 10 2.994 × 10

Conclusion
In this paper, we construct the local integro quintic splines without solving any systems of equations.This construction does not require any additional end conditions and it is easy to implement.They possess as good approximation properties as the integro quintic spline.The superconvergence orders of the function's first and third-order derivatives at knots are confirmed by numerical experiments.From Tables 1-3 we can see that the results are more reliable and better capture the asymptotic behavior of the error.

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5 )of local integro quintic spline 3 . 1
Some properties of integro-quintic spline Lemma 3.1.If the assumptions of Theorem 2.1 are fulfilled then for i = 3, 4, • • • , k − 3, we have 1 and Lemma 3.2 are valid under the conditions (3.39).The remainder approximation values of m i and n i for i = 1, 2 and i = k − 2, k − 1 are determined from (3.23) and (3.25) respectively.Then the values m i and n i for i = 0 and i = k are determined from (2.4) and (2.5) for i = 1 and i = k − 1 respectively.Thus, we find all m i and n i for i = 0, 1, • • • , k without solving any system of equations and without loss of accuracy.From (2.7) for i = 1 and i = k we can determine S 0 and S k .For the values of M i and N i at the end points for i = 0 and i = k we can use (3.30) and (3.31) for i = k and (3.32) and (3.33) for i = 0.