A New Quintic Spline Method for Integro Interpolation and Its Error Analysis

In this paper, to overcome the innate drawbacks of some old methods, we present a new quintic spline method for integro interpolation. The method is free of any exact end conditions, and it can reconstruct a function and its first order to fifth order derivatives with high accuracy by only using the given integral values of the original function. The approximation properties of the obtained integro quintic spline are well studied and examined. The theoretical analysis and the numerical tests show that the new method is very effective for integro interpolation.

The method in [2] was based on the quintic Hermite-Birkhoff polynomials.The method was very complicated because it mainly required solving two linear systems.Furthermore, besides the integral values (2), the method must use seven additional exact end conditions in terms of y(x 0 ), y (x 0 ), y (x 1 ), y (x n−1 ), y (x n ), y (x 0 ) and y (x n ).Later, a new algorithm was given in [18] to simplify the construction of integro quintic spline.It mainly required solving two linear three-diagonal systems.It was kind of simpler than that of [2].However, the algorithm needed five special and proper exact end conditions in terms of y(x 0 ), y (x 1 ), y (x n−1 ), y (x 1 ) and y (x n−1 ).The method in [14] was based on quintic B-splines.It was also very simple because it took advantage of the good properties of quintic B-splines.However, five additional exact end conditions in terms of y(x 0 ), y(x 1 ), y(x 2 ), y(x n−1 ) and y(x n ) must be provided.In other words, these methods all need exact end conditions.This is an obvious drawback of them.New simple methods that are not dependent on exact end conditions are desired.
In [17], we have studied an effective method that was not dependent on any exact end conditions.We first obtained n + 1 approximate function values at the knots and four approximate boundary derivative values from the integral values (2) and then used them to study a modified quintic spline interpolation problem.However, the method also had its own drawbacks.On the one hand, it needed n + 5 artificial values, which brought higher computational cost; on the other hand, the obtained quintic spline did not agree with the given integral values (2) over the subintervals.In [19], a local integro quintic spline method was given.It was also not dependent on exact end conditions and was able to produce good approximations.However, the obtained local integro quintic spline also did not agree with the given integral values (2) over the subintervals.Hence, these methods also need improvements.New attempts on this problem are still necessary.
In this paper, we aim to develop a new effective method to overcome the above-mentioned drawbacks.We will first construct six artificial end conditions by using a similar technique to [17] and use them together with the integral values (2) to get a new kind of integro quintic spline; then, we will theoretically analyze and numerically examine the approximation properties of the new integro quintic spline.The new method is very effective, and it has the following advantages.
(I) The method is free of any exact end conditions, and it only requires five artificial end conditions, which can be easily obtained by simple computations from several integral values.(II) The computational procedure of the method is concise and easy to implement.(III) The obtained quintic spline agrees with the given integral values (2) over the subintervals.(IV) The obtained quintic spline can provide satisfactory approximations to y (k) (x), k = 0, 1, 2, 3, 4, 5.
Hence, this method is very applicable for the integro interpolation problem.The remainder of this paper is organized as follows.In Section 2, we compute some artificial end conditions by using several integral values; in Section 3, we construct our new integro quintic spline with five artificial end conditions; the approximation abilities of the integro quintic spline are theoretically studied in Section 4 and numerically tested in Section 5; finally, we conclude our paper in Section 6.

Integro Quintic Spline Interpolation with Five Artificial End Conditions
In this section, we will use the given integral values (2) and the artificial end conditions in Section 2 to construct an integro quintic spline.Five additional independent conditions are needed.To use the results of (10) and ( 12) sufficiently, we will directly use the hybrid result of (15).
We look for the quintic spline s, which satisfies the following conditions: x j+1 x j s(x)dx = I j , j = 0, 1, . . ., n − 1, ( 17) and: It belongs to the spline space of C 4 quintic piecewise polynomial functions on the uniform partition ∆ (1), so s can be expressed as a linear combination of the quintic B-splines associated with the extended partition of ∆ (1) with knots a + ih, −5 ≤ i ≤ n + 5, i.e.,: where (see, e.g., [6,14,20,21]): For the sake of completeness, we give in Table 1 the values of B i at the knots in (x i−3 , x i+3 ).Furthermore, we have the following integro properties: x j+1 x j i , k = 0, 1, 2, 3, 4, at the knots lying in the interior of the support of B i .
Proof.We will prove that the determinant of matrix A is nonzero.We will perform some proper elementary transformations to A in order to verify |A| = 0. Let C(i) denote the i-th column and R(i) denote the i-th row of a matrix obtained by an elementary row or column transformation.We first perform n + 4 elementary column transformations to A.
Step 1: Then, we get: . We continue to perform the following elementary row transformations to A 1 .
Evidently, the new method is free of exact end conditions and is easy to implement.Furthermore, the obtained quintic spline s satisfies the conditions given in (2).

Approximation Properties
In this section, we study the approximation properties of the integro quintic spline s obtained in Section 3.
The coefficient matrix is strictly diagonally dominant.The infinity norm of its inverse is bounded.Hence, (43) is proven.

Numerical Tests
In this section, we test the approximation properties of the new integro quintic spline.Our tests are performed by MATLAB.
We take: and: as two illustrative examples.Furthermore, y 1 will be used in the comparison of our method with some other methods.
The absolute errors at the knots are defined as follows: and: The numerical convergence orders of the absolute errors at the knots are defined by: , k = 0, 1, . . ., 5.
The numerical convergence orders in these tables accord with the theoretical expectation.By Theorem 2, E 0 (x i , n) and E 1 (x i , n) are of sixth order convergent (see Tables 2, 3, 8 and 9 for the numerical convergence orders), E 2 (x i , n) and E 3 (x i , n) are of fourth order convergent (see Tables 4, 5, 10 and 11 for the numerical convergence orders), E 4 (x i , n) and E 5 (x i , n) are of second order convergent (see Tables 6, 7, 12 and 13 for the numerical convergence orders).Table 5.The absolute errors of the third order derivatives of y 1 at the knots and the numerical convergence orders.Moreover, all of the absolute errors in these tables are very satisfactory and well accepted.Making a further observation on these tables, we find that the errors at the inner knots are much better than the errors at the left endpoint and the right endpoint.The numerical phenomenon is natural and reasonable, because we only make use of n integral values (2) and do not make use of any exact end conditions.It shows that the influence of the artificial end conditions on the inner errors is limited.In fact, the inner approximation errors are mainly determined by the given n integral values in (2), while the boundary errors are mainly effected by the artificial end conditions.It is checked that our inner errors of y 1 in Tables 2-6 are similar to the ones in [2,14,18], which are obtained by using five or seven additional exact end conditions.It shows that our new method can obtain satisfactory approximation results by using fewer data than the methods in [2,14,18].The performance is very encouraging.
Finally, we give some discussion on fifth order derivative approximation.We remark that we use: to approximate y (5) (x i ) in this paper, i = 1, 2, . . ., n − 1. See Tables 7 and 13 for our numerical results of the fifth order derivatives.Take y 1 as a comparison example.See Table 14 for the comparison of the maximum absolute errors of the fifth order derivatives y (5) 1 (x i ) obtained by our current method and the methods in [18,19].Obviously, our results are very accurate and surprising because they are obtained by only using the integral values (2) with no exact end conditions, while the results of [18] are obtained by using the integral values (2) and five additional exact end conditions (y(x 0 ), y (x 1 ), y (x n−1 ), y (x 1 ) and y (x n−1 )), as well.Hence, our approximation method for the fifth order derivatives at the inner knots is more preferable.

Conclusions
In this paper, an effort that is different from the ones in [1,2,[13][14][15][16][17][18][19] is made to construct a new kind of integro quintic spline without exact end conditions.The demands of exact end conditions in many old methods, such as [1,2,14,15,18], for integro interpolation have been relaxed and deleted in the new method.The good feature makes the current method possess wider applications than many other methods.Moreover, the method is easy to apply, and the obtained integro quintic spline has satisfactory approximation abilities in approximating a function and its first order to fifth order derivatives.Hence, the new method is very effective for integro interpolation.

Table 1 .
The values of B

Table 3 .
The absolute errors of the first order derivatives of y 1 at the knots and the numerical convergence orders.

Table 4 .
The absolute errors of the second order derivatives of y 1 at the knots and the numerical convergence orders.10 −7 1.957 × 10 −9 4.236 × 10 −10

Table 7 .
The absolute errors of the fifth order derivatives of y 1 at the knots and the numerical convergence orders.

Table 9 .
The absolute errors of the first order derivatives of y 2 at the knots and the numerical convergence orders.

Table 10 .
The absolute errors of the second order derivatives of y 2 at the knots and the numerical convergence orders.

Table 12 .
The absolute errors of the fourth order derivatives of y 2 at the knots and the numerical convergence orders.