# ERROR BOUNDS FOR KRONROD EXTENSION OF GENERALIZATIONS OF MICCHELLI-RIVLIN QUADRATURE FORMULA FOR ANALYTIC FUNCTIONS.

1. Introduction. The Gaussian quadrature formula with multiple nodes

(1.1) [mathematical expression not reproducible]

with respe

(1.2) [mathematical expression not reproducible]

which has algebraic degree of precision 2sn + 2n + 1. The nodes [[tau].sub.V] are the same as in (1.1), and the [[tau].sub.j] are the zeros of the monic polynomial

[mathematical expression not reproducible]

where [U.sub.n-1] is the Chebyshev polynomial of the second kind of degree n - 1. A nice and detailed survey of Kronrod rules in the last fifty years is provided by Notaris .

Error bounds for the Micchelli-Rivlin quadrature formula, and then for (1.1), for functions being analytic on confocal ellipses that contain the interval [-1,1] in the interior, have been considered in  and , respectively. In this paper, our aim is to do the same for the quadrature formula (1.2).

2. The remainder term of the Kronrod extension of the generalization of Micchelli-Rivlin quadrature formula for analytic functions. Let f be an analytic function on a domain D which contains the interval [-1,1] in its interior, and let [GAMMA] be a simple closed curve in D surrounding [-1,1]. Assume that we know the values of the function f and its derivatives [f.sup.(i)], i = 1,2,..., 2s - 1, at the nodes [x.sub.1], [x.sub.2],..., [x.sub.n] in the interval [-1,1] and that we also know the values of the function f at the nodes [y.sub.1], [y.sub.2], ..., [y.sub.n]+1 in the interval [-1,1] satisfying

y1 < x1 < y2 < x2 < * * * < [y.sub.n] < [x.sub.n] < [y.sub.n]+1.

Let

t2v = [x.sub.v], v = 1, 2,..., n, [t.sub.2v-]1 = [y.sub.v], v = 1, 2,. .., n + 1.

Using a result by Goncarov , the error in the Hermite interpolation of the function f can be written in the form

(2.1) [mathematical expression not reproducible]

where the [pounds sterling][i.sub.,v] are the basis functions for Hermite interpolation, [s.sub.v] = s if [t.sub.v] G {[x.sub.1],..., [x.sub.n]}, [s.sub.v] = 1/2 if [t.sub.v] [euro] {[y.sub.1],..., [y.sub.n+1]}, and

[mathematical expression not reproducible]

Let [x.sub.v] be the zeros of the Chebyshev polynomial [T.sub.n], i.e., [x.sub.v] = [[tau].sub.v] for v = 1,..., n. For v = 1,..., n - 1, let [[eta].sub.v] be the zeros of the Chebyshev polynomial [U.sub.n-1]. Set y1 = [[tau].sub.1] = - 1, [y.sub.v[+.sub.1] = [[tau].sub.v+1] = [[eta].sub.v], for v = 1,2,..., n - 1, and [y.sub.n]+1 = [tau] n+1 = 1. Thus, [y.sub.v] = [[tau].sub.v] are the zeros of the polynomial ([t.sup.2] - 1)[U.sub.n-1], which are the corresponding nodes in (1.2); see . In this case, by multiplying (2.1) by [omega](t)[T.sub.n](t), where [omega](t) = 1/A/1 - [t.sup.2], and integrating with respect to t over ( - 1,1), we get a contour integral representation of the remainder term in (1.2):

(2.2) [mathematical expression not reproducible]

where the kernel is given by

(2.3) [mathematical expression not reproducible]

and

(2.4) [mathematical expression not reproducible]

The modulus of the kernel is symmetric with respect to the real axis, i.e., it holds that |[K.sub.n,s](z)| = |Kn,s(z)|. Also, note that, due to the symmetry of the Jacobi Polynomials [T.sub.n](z) and [U.sub.n](z), we have |[K.sub.ns](-z)| = |[K.sub.n.sub.s](z)|. Hence, the modulus of the kernel is symmetric with respect to both axes.

Applying Holder's inequality to (2.2) yields

(2.5) [mathematical expression not reproducible]

where 1 < r < +00, 1/r + 1/r' = 1, and

In the case r = +00 and r' = 1, the estimate (2.5) reduces to

[mathematical expression not reproducible]

which leads to the error bound

(2.6) [mathematical expression not reproducible]

where [pounds sterling]([GAMMA]) is the length of the contour [GAMMA] . We refer to it as the L[infinity]-error bound. On the other hand, for r = 1 and r' = +00, the estimate (2.5) reduces to

(2.7) [mathematical expression not reproducible]

which is stronger than the estimate (2.6). We refer to (2.7) as the [L.sup.1] -error bound.

In this paper, we take the contour [GAMMA] to be an ellipse [[epsilon].sub.[rho]] with foci at the points +1 and the sum of its semi-axes [rho] > 1, i.e.,

The choice of the family of ellipses [[epsilon].sub.[rho]] as basic contours of integration is natural when dealing with analytic functions in a neighborhood of [-1,1] since they are the level curves of the Green's function for the region C \ [-1,1] with a pole at infinity. When [rho] [right arrow] 1+, the ellipse [[epsilon].sub.[rho]] shrinks to the interval [-1,1], and when [rho] [right arrow] 00, the interior of [[epsilon].sub.[rho]] approaches the whole complex plane (which is useful when we deal with entire integrands such as those in Section 6).

In the sequel we present three types of bounds:

* In Section 3, we derive L[infinity] -error bounds by means of contour integration techniques, applying essentially a method introduced by Gautschi and Varga in . Here, one has to calculate

[mathematical expression not reproducible]

* In Section 4, [L.sup.1]-error bounds are derived, which are stronger than the L[infinity]-error bounds and require an estimate for

[mathematical expression not reproducible]

* In Section 5, we derive the bounds resulting from an expansion of the remainder term [Rn.sub.s] (f) in the form

[mathematical expression not reproducible]

following the method introduced by Hunter in  and then estimating [alpha]([2.sub.s]+3)n[+.sub.k] using the result of Elliott in .

Finally, numerical examples illustrating these three estimates are given in Section 6.

In the case of standard Gaussian quadrature formulas (with simple or multiple nodes), L[infinity]-error bounds are considered by Gautschi and Varga , Schira , Milovanovic and Spalevic , Pejcev and Spalevic , and others. For the mentioned quadrature formulas, error bounds analogous to those in Sections 4 and 5, are considered by Hunter , Milovanovic and Spalevic , and others. Here we also mention the general method of estimating the error in Gauss-Turan quadrature formulas for functions analytic inside ellipses proposed by Spalevic .

3. L[infinity] -error bounds. In order to find an explicit formula for the kernel (2.3), we determine the integral (2.4). Substituting t = cos [theta] into (2.4) yields

[mathematical expression not reproducible]

We use formula 1.320.7 in  for [(cosn[theta]).sup.2s+1] to obtain

[mathematical expression not reproducible]

which is transformed into

[mathematical expression not reproducible]

where [mathematical expression not reproducible] := 0 for k < 0. Using formula 3.613.1 in  (see also [3, p. 1176]), we obtain

[mathematical expression not reproducible]

[mathematical expression not reproducible]

where v = z - \J [z.sup.2] - 1.

Substituting z = 1/2 u + u into the above expression (where u = 1/v) yields

(3.1) [mathematical expression not reproducible]

Finally, since[T.sub.n](z) = ([u.sup.n] + [u.sup.n])/2 and [U.sub.n-1](z) = ([u.sup.n] - [u.sup.n])/(u - [u.sup.1]) (see, e.g., ), from (2.3) and (3.1), we get

(3.2) [mathematical expression not reproducible]

Let u = [rho]e and [mathematical expression not reproducible]. In order to find the modulus of the kernel, set

[mathematical expression not reproducible]

Then

[mathematical expression not reproducible]

The graphs of the functions [theta] [right arrow]- |[K.sub.ns](z)| for certain values of n, s, and [rho] are displayed in Figure 3.1. Since the modulus of the kernel is symmetric with respect to both axes, it suffices to consider the interval [0,[pi]/2]. We can now state the following result.

THEOREM 3.1. For each n [euro] N, n > 1, and each seN, there exists [[rho].sub.0] = [[rho].sub.0](n, s) such that

[mathematical expression not reproducible]

for each [rho] > [[rho].sub.0].

Proof. We have to show that the inequality [mathematical expression not reproducible] holds, i.e., that for each [rho] greater than some [[rho].sub.0] = [[rho].sub.0](n, s), it holds that I = [aBCD.sup.2s] - [Abcd2.sup.s] < 0, where A, B, C, D denote the values of a, b, c, d for [theta] = 0.

The expression I = I([rho]) is a polynomial of degree 12sn + 4n + 2 with leading coefficient 2C[s.sub.0](cos 2[theta]-1), which is negative for [theta] [member of] (0,[pi]/2]. Therefore I([rho]) is negative for sufficiently large [rho].

For n = 1 the corresponding result is slightly different.

THEOREM 3.2. Letn = 1.

a) For s = 1, there exists [[rho].sub.0] such that

[mathematical expression not reproducible]

for each [rho] > [[rho].sub.0].

b) For s > 1, there exists [[rho].sub.0] = [[rho].sub.0](s) such that

[mathematical expression not reproducible]

for each [rho] > [[rho].sub.0].

Proof. For n = 1 we have

[mathematical expression not reproducible]

Let A, B, C, D be the values of a, b, c, d for [theta] = 0, i.e.,

[mathematical expression not reproducible]

Then we have

[mathematical expression not reproducible]

and

[mathematical expression not reproducible]

where [mathematical expression not reproducible], and [C.sub.s,1]/[C.sub.s,0] = [2.sub.s]/s+3. The coefficient in the expression [aBCD.sup.2s] - [Abcd.sup.2s] at [[rho].sup.12s][+.sup.6] equals

[mathematical expression not reproducible]

which is negative for each [theta] [member of] (0,[pi]/2] and s = 1.

Now we consider the values A, B, C, D of a, b, c, d for [theta] = [pi]/2, i.e.,

[mathematical expression not reproducible]

The leading coefficient of the expression I = [aBCD.sup.2s] - [Abcd.sup.2s], similarly to the previous case, is

[mathematical expression not reproducible]

which is negative for each [theta] [member of] [0,[pi]/2) and s > 1.

4. [L.sup.1] -error bounds. In order to derive an [L.sup.1] -error bound, we use (2.7) and study the expression

(4.1) [mathematical expression not reproducible]

where the kernel [K.sub.n,s] (z) is given by (3.2).

Let z = 1/2(u + [u.sup.-1]), where u = [[rho][e.sup.i[theta]]. Denoting [a.sub.j] = 1/2([rho]([P.sup.j] + [[rho].sup.-1]), for j [member of] N, we obtain

[mathematical expression not reproducible]

Substituting (3.2), the previous equations, and [mathematical expression not reproducible] (see ) into (4.1), we get

(4.2) [mathematical expression not reproducible]

Let [mathematical expression not reproducible]. The squared modulus of the sum in (4.2) equals (cf. )

[mathematical expression not reproducible]

where

[mathematical expression not reproducible]

i.e.,

(4.3) [mathematical expression not reproducible]

Hence, from (4.2) we get

[mathematical expression not reproducible]

Since the integrand is periodic, this expression reduces to

(4.4) [mathematical expression not reproducible]

Using [5, eq. 3.616.7], one finds

(4.5) [mathematical expression not reproducible]

where

We are now ready to prove the main result.

THEOREM 4.1. For the expression [Ln.sub.js] ([[epsilon].sub.[rho]]) given by (4.1), it holds that

(4.6) [mathematical expression not reproducible]

where

(4.7) [mathematical expression not reproducible]

Proof. Upon applying Cauchy's inequality to (4.4), we obtain

(4.8) [mathematical expression not reproducible]

where the coefficients [A.sub.k], for k = 0,1,..., s, are defined by (4.3). One finds

[mathematical expression not reproducible]

and, by (4.3) and (4.5),

[mathematical expression not reproducible]

where [Q.sub.s]([[rho].sup.4n]) is given by (4.7). Thus, (4.8) reduces to (4.6).

REMARK 4.2. Let x = [[rho].sup.4n]. The first few polynomials [Q.sub.s] (x) in (4.7) are

[mathematical expression not reproducible]

Note that deg [Q.sub.s] = 3s - 1.

5. Error bounds based on an expansion of the remainder term. If f is an analytic function in the interior of [[epsilon].sub.[rho]], then it can be expanded as

(5.1) [mathematical expression not reproducible]

[mathematical expression not reproducible]

where

[mathematical expression not reproducible]

This series converges for all z in the interior of [[epsilon].sub.[rho]]. The prime symbol on the summation sign means that the first term of the sum is to be halved.

For the expansion of the kernel we rewrite (3.2) as

(5.2) [mathematical expression not reproducible]

Thus, for |[u.sup.2n|] < 1, we have

(5.3) [mathematical expression not reproducible]

[mathematical expression not reproducible]

whereas the other term is expanded as

(5.4) [mathematical expression not reproducible]

[mathematical expression not reproducible]

Here [x] denotes the integer part of a real number x. From (5.2), (5.3), and (5.4), we get

[mathematical expression not reproducible]

where

v

andp = - (2s + 3)n - 2((j + k)n + i) - 1. The coefficient of [u.sup.(-2s+3)]n-2(an+b)-1] with 0 [less than or equal to] a and 0 [less than or equal to] b [less than or equal to] n - 1in the above expression equals

(5.5) [mathematical expression not reproducible]

[mathematical expression not reproducible]

so we obtain

(5.6) [mathematical expression not reproducible]

THEOREM 5.1. The remainder term [R.sub.n,s](f) can be represented in the form

[mathematical expression not reproducible]

where the coefficients [[epsilon].sup.(s).sub.n,k] are independent of f. Furthermore, [mathematical expression not reproducible] for j = 0,1,...

Proof. Substituting (5.1) and (5.6) in (2.2) gives

[mathematical expression not reproducible]

According to [6, Lemma 5], this reduces to

[mathematical expression not reproducible]

with

(5.7) [mathematical expression not reproducible]

Obviously, [mathematical expression not reproducible] Finally, from (5.5) and (5.7) we find

[mathematical expression not reproducible]

Since

(5.9) [mathematical expression not reproducible]

in order to obtain explicit error estimates for general s, we need to know the sign of [[epsilon].sup.(s).sub.n,k] This requires some computation. We first note that [C.sub.s,l] = 0 for l > s and

[mathematical expression not reproducible]

Thus, the sum in (5.8) can be rewritten as

(5.10) [mathematical expression not reproducible]

where

[mathematical expression not reproducible]

and

[mathematical expression not reproducible]

The sum [S.sub.2] can be calculated using [15, Lemma 3], yielding

(5.11) [mathematical expression not reproducible]

However, we could not determine the sum [S.sub.1] explicitly. Since we only need the sign of [S.sub.1] - [S.sub.2], it will be enough to find sufficiently accurate bounds for [S.sub.1]. An upper bound for [S.sub.1] is provided by the following statement.

LEMMA 5.2. For each j [member of] [N.sub.0] and s [member of] N, s > 1, we have

(5.12) [mathematical expression not reproducible]

Proof. We use induction on j. For j = 0 and j = 1 the inequality (5.12) trivially holds, as it is equivalent to

[mathematical expression not reproducible]

respectively.

Suppose that (5.12) holds for some j [member of] [N.sub.0]. We shall prove that it also holds for j + 2, i.e.,

[mathematical expression not reproducible]

Since

[mathematical expression not reproducible]

it suffices to verify that

[mathematical expression not reproducible]

but this can be simplified to

[mathematical expression not reproducible]

which is evident for s > 1.

We shall use this result to derive a lower bound for [S.sub.1].

LEMMA 5.3. For each j [member of] [N.sub.0] and s [member of] N, s > 1, we have

(5.13) [mathematical expression not reproducible]

Proof. Lemma 5.2 for j - 1 instead of j gives

[mathematical expression not reproducible]

which implies that

[mathematical expression not reproducible]

This is equivalent to the desired inequality.

(Now we can compare S1 with S2 . Since it holds that [mathematical expression not reproducible] and [mathematical expression not reproducible], equation (5.11) is equivalent to

(5.14) [mathematical expression not reproducible]

From (5.13) and (5.14) we get

[mathematical expression not reproducible]

where

[mathematical expression not reproducible]

The previous expression is positive if and only if

[mathematical expression not reproducible]

For s > 1, all coefficients of I as a polynomial in j are positive, and hence, I > 0 whenever j [member of] N0, so [S.sub.1] - [S.sub.2] > 0, which together with (5.10) implies

(5.15) [mathematical expression not reproducible]

for [member of] N, s > 1, and j G N0. Moreover, if s = 1, it follows from (5.8) that

[mathematical expression not reproducible]

and [[epsilon].sup.(1).sub.n,2] =0 otherwise, so (5.15) remains valid in this case. Thus we have proved the following result.

LEMMA 5.4. For [mathematical expression not reproducible] defined by (5.8) we have

(5.16) [mathematical expression not reproducible]

[mathematical expression not reproducible]

for s G N and j G N0.

Now, from (5.9) we shall derive an explicit bound for |[R.sub.n,s] (f) |.

In general, the Chebyshev coefficients [[alpha].sub.k], in (5.1) are unknown. However, Elliott 

described a number of ways to estimate or bound them. In particular, under our assumptions,

(5.17) [mathematical expression not reproducible]

Substituting (5.8) and (5.17) into (5.9) gives

(5.18) [mathematical expression not reproducible]

where

(5.19) [mathematical expression not reproducible]

and [mathematical expression not reproducible] are defined by (5.8). Although the [mathematical expression not reproducible] are sums themselves, it turns out that F([rho]) can be simplified to a single finite sum.

LEMMA 5.5. For F([rho]) given by (5.19) with [rho] > 1, it holds that

(5.20) [mathematical expression not reproducible]

Proof. From (5.3) we find

[mathematical expression not reproducible]

(5.21) [mathematical expression not reproducible]

[mathematical expression not reproducible]

Let [mathematical expression not reproducible]. From (5.21) and (5.20) we obtain

[mathematical expression not reproducible]

which reduces to (5.19), using (5.8) and (5.16).

Now we can formulate the main result.

THEOREM 5.6. For se [member of], the estimate (5.18) can be expressed in the form

(5.22) [mathematical expression not reproducible]

6. Numerical examples. Let us now compute the integral

[mathematical expression not reproducible]

by using the quadrature formula (1.2) for two entire functions.

Denoting the bounds in Sections 3, 4, and 5 by [mathematical expression not reproducible], respectively, we find

(6.1) [mathematical expression not reproducible]

(6.2) [mathematical expression not reproducible]

where [[rho].sub.0] is defined in Theorem 3.1 and the Bj, for i = 1,2, 3, are defined below. Numerical experiments show that for all n and all s, the corresponding values of [[rho].sub.0] (n, s) are very close to 1 (in most cases they are less than 1.1).

The length of the ellipse in (2.6) can be estimated by (cf. [17, Eq. (2.2)])

v

where [a.sub.1] = 1/2([rho] + [[rho].sup.-1]). Therefore from (2.6), Theorem 3.1 and (6.1) for n > 1, we get

[mathematical expression not reproducible]

From (2.7), (4.6), and (6.2) we get

[mathematical expression not reproducible]

where [Q.sub.s] are defined by (4.7). Finally, from (5.22) and (6.2) we obtain

[mathematical expression not reproducible]

We have calculated these bounds for some values of n, s, and [omega]. "Error" is the actual (sharp) error and [I.sub.u] is the exact value of the integral. All computations reported in this paper were carried out in MATLAB with high-precision arithmetic. The computations were carried out with 150 significant decimal digits.

EXAMPLE 1. Let

[mathematical expression not reproducible]

Since the function [f.sub.0] is entire, the obtained estimates hold for [E.sub.[rho]], [rho] > 1. One finds

[mathematical expression not reproducible]

The results are reported in Table 6.1.

EXAMPLE 2. Let

[mathematical expression not reproducible]

Similarly to the previous example, the obtained estimates hold for [E.sub.[rho]], [rho] > 1. We have

[mathematical expression not reproducible]

The results are reported in Table 6.2.

One can notice that for the results in these examples, all three estimates are within the same range. These results are comparable to those obtained for the quadrature formula with multiple nodes for the Fourier-Chebyshev coefficients; see .

7. Concluding remarks. Three kinds of effective error bounds of the quadrature formulas with multiple nodes that generalize the well-known Micchelli-Rivlin quadrature formula, when the integrand is an analytic function in the regions containing the confocal ellipses, were considered recently in [13, 15]. In this paper, we continue with the analogous analysis for their Kronrod extensions and obtain effective error bounds of them, which is confirmed by the given numerical examples.

Acknowledgments. We are grateful to the referee and Lothar Reichel for careful reading the manuscript and for making suggestions that have improved the paper.

REFERENCES

 B. BOJANOV AND G. PETROVA, Quadrature formulae for Fourier coefficients, J. Comput. Appl. Math., 231 (2009), pp. 378-391.

 D. ELLIOTT, The evaluation and estimation of the coefficients in the Chebyshev series expansion of a functions, Math. Comp., 18 (1964), pp. 82-90.

 W. GAUTSCHI AND R. S. VARGA, Error bounds for Gaussian quadrature of analytic functions, SIAM J. Numer. Anal., 20 (1983), pp. 1170-1186.

 V. L. GONCAROV, Theory of Interpolation and Approximation of Functions, GITTL, Moscow, 1954.

 I. S. GRADSHTEYN AND I. M. RYZHIK, Tables of Integrals, Series and Products, 6th ed., Academic Press, San Diego, 2000.

 D. B. HUNTER, Some error expansions for Gaussian quadrature, BIT, 35 (1995), pp. 64-82.

 C. A. MICCHELLI AND T. J. RIVLIN, Turan formulae and highest precision quadrature rules for Chebyshev coefficients, IBM J. Res. Develop., 16 (1972), pp. 372-379.

 G. V. MILOVANOVIC AND M. M. SPALEVIC, Error bounds for Gauss-Turan quadrature formulas of analytic functions, Math. Comp., 72 (2003), pp. 1855-1872.

 __, An error expansion for some Gauss-Turan quadratures and [L.sup.1] -estimates of the remainder term, BIT, 45 (2005), pp. 117-136.

 __, Kronrod extensions with multiple nodes of quadrature formulas for Fourier coefficients, Math. Comp., 83 (2014), pp. 1207-1231.

 G. V. MILOVANOVIC, R. ORIVE, AND M. M. SPALEVIC, Quadrature with multiple nodes for Fourier-Chebyshev coefficients, IMA J. Numer. Anal., to appear, DOI: 10.1093/imanum/drx067

 S. E. NOTARIS, Gauss-Kronrod quadrature formulae-a survey of fifty years of research, Electron. Trans. Numer. Anal., 45 (2016), pp. 371-404. http://etna.ricam.oeaw.ac.at/vol.45.2016/pp371-404.dir/pp371-404.pdf

 A. V. PEJCEV AND M. M. SPALEVIC, Error bounds of Micchelli-Rivlin quadrature formula for analytic functions, J. Approx. Theory, 169 (2013), pp. 23-34.

 __, The error bounds ofGauss-Radau quadrature formulae with Bernstein-Szego weight functions, Numer. Math., 133 (2016), pp. 177-201.

 __, Error bounds of a quadrature formula with multiple nodes for the Fourier-Chebyshev coefficients for analytic function, Sci. China Math., to appear. doi: https://doi.org/10.1007/s11425-016-9259-5

 M. M. SPALEVIC, Error bounds and estimates for Gauss-Turan quadrature formulae of analytic functions, SIAM J. Numer. Anal., 52 (2014), pp. 443-467.

 R. SCHERER AND T. SCHIRA, Estimating quadrature errors for analytic functions using kernel representations and biorthogonal systems, Numer. Math., 84 (2000), pp. 497-518.

 T. SCHIRA, The remainder term for analytic functions of symmetric Gaussian quadratures, Math. Comp., 66 (1997), pp. 297-310.

RADA M. MUTAVDZIC ([dagger]), ALEKSANDAR V. PEJCEV ([dagger]), AND MIODRAG M. SPALEVIC ([dagger])

Dedicated to Walter Gautschi on the occasion of his 90th birthday

(*) Received March 11, 2018. Accepted May 27, 2018. Published online on November 13, 2018. Recommended by L. Reichel. Research supported in part by the Serbian Ministry of Education, Science and Technological Development (Research Project: "Methods of numerical and nonlinear analysis with applications" (# 174002)).

([dagger]) Department of Mathematics, University of Belgrade, Faculty of Mechanical Engineering, Kraljice Marije 16, 11120 Belgrade 35, Serbia ({rmutavdzic, apejcev, mspalevic}@mas.bg.ac.rs).

DOI: 10.1553/etna_vol50s20
```Table 6.1 The values of the derived bounds r1([f.sub.0]),
r2([f.sub.0]), rs([f.sub.0]), for some values ofn, s, [omega].

n, s, [omega]  [r.sub.1]    [r.sub.2]    [r.sub.3]    Error
([f.sub.0])  ([f.sub.0])  ([f.sub.0])

8, 1,  1      4.43(-29)    4.37(-29)    4.37(-29)    3.88(-30)
8, 2,  1      1.59(-44)    1.57(-44)    1.57(-44)    1.18(-45)
8, 1,  5      3.54(-14)    3.32(-14)    3.32(-14)    2.94(-15)
8, 2,  5      4.78(-24)    4.56(-24)    4.56(-24)    3.42(-25)
8, 1, 10      6.12(-7)     5.35(-7)     5.35(-7)     4.69(-8)
8, 2, 10      1.94(-14)    1.76(-14)    1.76(-14)    1.32(-15)
8, 1, 15      3.91(-2)     3.17(-2)     3.17(-2)     2.73(-3)
8, 2, 15      2.79(-8)     2.41(-8)     2.41(-8)     1.78(-9)
10, 1,  1      7.55(-39)    7.48(-39)    7.48(-39)    5.95(-40)
10, 2,  1      3.18(-59)    3.16(-59)    3.16(-59)    2.13(-60)
10, 1,  5      1.84(-20)    1.75(-20)    1.75(-20)    1.39(-21)
10, 2,  5      7.36(-34)    7.10(-34)    7.10(-34)    4.77(-35)
10, 1, 10      9.58(-12)    8.61(-12)    8.61(-12)    6.78(-13)
10, 2, 10      3.66(-22)    3.39(-22)    3.39(-22)    2.27(-23)
10, 1, 15      4.25(-6)     3.60(-6)     3.60(-6)     2.81(-7)
10, 2, 15      8.43(-15)    7.51(-15)    7.51(-15)    5.00(-16)
14, 1,  1      1.27(-59)    1.26(-59)    1.26(-59)    8.51(-61)
14, 2,  1      2.38(-90)    2.36(-90)    2.36(-90)    1.35(-91)
14, 1,  5      2.94(-34)    2.84(-34)    2.84(-34)    1.91(-35)
14, 2,  5      3.28(-55)    3.20(-55)    3.20(-55)    1.82(-56)
14, 1, 10      1.46(-22)    1.36(-22)    1.36(-22)    9.09(-24)
14, 2, 10      2.55(-39)    2.41(-39)    2.41(-39)    1.37(-40)
14, 1, 15      3.37(-15)    3.01(-15)    3.01(-15)    2.00(-16)
14, 2, 15      1.59(-29)    1.47(-29)    1.47(-29)    8.29(-31)

n, s, [omega]  [I.sub.[omega]]

8, 1,  1      8.53...(-4)
8, 2,  1      8.53...(-4)
8, 1,  5      5.28...(+0)
8, 2,  5      5.28...(+0)
8, 1, 10      2.38...(+3)
8, 2, 10      2.38...(+3)
8, 1, 15      4.99...(+5)
8, 2, 15      4.99...(+5)
10, 1,  1      4.25...(-5)
10, 2,  1      4.25...(-5)
10, 1,  5      1.25...(+0)
10, 2,  5      1.25...(+0)
10, 1, 10      1.00...(+3)
10, 2, 10      1.00...(+3)
10, 1, 15      2.74...(+5)
10, 2, 15      2.74...(+5)
14, 1,  1      6.32...(-8)
14, 2,  1      6.32...(-8)
14, 1,  5      4.39...(-2)
14, 2,  5      4.39...(-2)
14, 1, 10      1.19...(+2)
14, 2, 10      1.19...(+2)
14, 1, 15      5.92...(+4)
14, 2, 15      5.92...(+4)

Table 6.2 The values of the derived bounds [r.sub.1]([f.sub.1]),
[r.sub.2]([f.sub.1]), r3(f1), for some values ofn, s, [omega].

n, s, [omega]  [r.sub.1]   [r.sub.2]  [r.sub.3]   Error
([f.sub.1]  ([f.sub.1] ([f.sub.1]

8, 1,  0.5    5.54(-39)   5.50(-39)  5.50(-39)   3.34(-40)
8, 2,  0.5    1.21(-56)   1.21(-56)  1.21(-56)   6.03(-58)
8, 1,  1      3.07(-27)   2.99(-27)  2.99(-27)   1.80(-28)
8, 2,  1      3.81(-40)   3.73(-40)  3.73(-40)   1.85(-41)
8, 1,  5      1.89(-5)    1.16(-5)   1.16(-5)    5.81(-7)
8, 2,  5      8.23(-9)    5.39(-9)   5.39(-9)    2.25(-10)
8, 1, 10      3.14(-1)    9.48(-2)   9.65(-2)    4.06(-3)
8, 2, 10      1.93(-2)    6.80(-3)   7.10(-3)    1.98(-4)
10, 1,  0.5    3.30(-50)   3.29(-50)  3.29(-50)   1.75(-51)
10, 2,  0.5    8.16(-73)   8.12(-73)  8.12(-73)   3.56(-74)
10, 1,  1      1.71(-35)   1.67(-35)  1.67(-35)   8.85(-37)
10, 2,  1      3.73(-52)   3.66(-52)  3.66(-52)   1.60(-53)
10, 1,  5      9.30(-8)    5.96(-8)   5.96(-8)    2.65(-9)
10, 2,  5      2.35(-12)   1.60(-12)  1.60(-12)   5.93(-14)
10, 1, 10      3.36(-2)    1.09(-2)   1.09(-2)    3.95(-4)
10, 2, 10      5.72(-4)    2.09(-4)   2.12(-4)    6.01(-6)
14, 1,  0.5    3.26(-73)   3.25(-73)  3.25(-73)   1.42(-74)
14, 2,  0.5    6.75(-106)  6.73(-106) 6.73(-106)  2.43(-107)
14, 1,  1      1.49(-52)   1.47(-52)  1.47(-52)   6.39(-54)
14, 2,  1      6.70(-77)   6.59(-77)  6.59(-77)   3.68(-78)
14, 1,  5      9.42(-13)   6.38(-13)  6.38(-13)   2.37(-14)
14, 2,  5      5.79(-20)   4.10(-20)  4.10(-20)   2.72(-21)
14, 1, 10      2.32(-4)    8.36(-5)   8.36(-5)    2.45(-6)
14, 2, 10      2.39(-7)    9.50(-8)   9.51(-8)    2.30(-9)

n, s, [omega]  [I.sub.[omega]]

8, 1,  0.5     1.08...(-6)
8, 2,  0.5     1.08...(-6)
8, 1,  1       1.98...(-4)
8, 2,  1       1.98...(-4)
8, 1,  5       3.11...(-1)
8, 2,  5       3.11...(-1)
8, 1, 10       1.07...(+0)
8, 2, 10       1.07...(+0)
10, 1,  0.5    -1.29...(-8)
10, 2,  0.5    -1.29...(-8)
10, 1,  1      -9.13...(-6)
10, 2,  1      -9.13...(-6)
10, 1,  5      -1.72...(-1)
10, 2,  5      -1.72...(-1)
10, 1, 10      -8.80...(-1)
10, 2, 10      -8.80...(-1)
14, 1,  0.5    -1.37...(-12)
14, 2,  0.5    -1.37...(-12)
14, 1,  1      -1.47...(-8)
14, 2,  1      -1.47...(-8)
14, 1,  5      -4.23...(-2)
14, 2,  5      -4.23...(-2)
14, 1, 10       9.03...(-2)
14, 2, 10       9.03...(-2)
```
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