Spline Based Series for Sine and Arbitrarily Accurate Bounds for Sine, Cosine and Sine Integral
©© Roy Howard 2020
Print Date: 28/10/20
Spline Based Bounds for Sin/Cos Spline Based Series for Sine and Arbitrarily AccurateBounds for Sine, Cosine and Sine Integral
Roy M. Howard
School of Electrical Engineering, Computing and Mathematical Sciences, Curtin University, GPO BoxU1987, Perth, 6845, Australia.email: [email protected]
17 June 2020Abstract
Based on two point spline approximations of arbitrary order, a series of functions that define lower bounds for and , over the interval , with increasingly low relative errors and smaller relative errorsthan published results, are defined. Second, fourth and eighth order approximations have, respectively, maximumrelative errors over the interval of , and . New series for the sinefunction, which have significantly better convergence that a Taylor series over the interval , are pro-posed. Applications include functions that are upper bounds for the sine function, upper and lower bounds for thecosine function and lower bounds for the sine integral function. These bounded functions can be made arbitrarilyaccurate.
Keywords:
Jordan’s inequality, trigonometric inequalities, spline approximation, sine integral function.
MSC (2020): x sin x sin x Spline Based Bounds for Sin/Cos
Print Date: 28/10/20 © Roy Howard 2020
1 Introduction
Research to establish upper and lower bounds for over the interval has a long history (e.g.Sándor & Bencze 2005, Mortici 2011) and Qi et al. 2009 provides a detailed review. Fundamental results includeJordan’s inequality (1) the Cusa-Huygens inequality (2) and the Redheffer inequality (Redheffer 1969): (3)
Similar, and related inequalities, which are indicative of published bounds, are detailed in Table 1.1. The graphsof the relative errors in the bounds defined by the first, second, fourth, fifth, eighth and tenth inequalities stated inthis table are shown in Figure 1. The relative error of the other bounds specified in Table 1.1 lie within the limitsof the bounds shown in Figure 1. The relative errors associated with these bounds typical have maximum values,over the interval , in the range of to . Some of the bound are ‘sharp’, i.e. exact, at the pointszero and . The majority of the bounds have relative errors that are monotonically increasing and have maxi-mum values at .Applications of such bounds include defining bounds to the Riemann zeta function , e.g. Luo et al., 2003,and bounds on some trigonometric ratios, e.g. Wu, 2004.
Table 1.1
Published upper and lower bounds for , . ć , 1965Upper: Cusa-Huygens3 Bhayo, 2017, eqn. 1.64 Bercu, 2016, Lemma 2.15 Hua, 2016, eqn. 3.16 Yang, 2016, eqn. 2.147 Yang, 2014, eqn. 548 Chen, 2015, eqn 4.79 Chen, 2015, eqn 4.910 Chen, 2015, eqn 5.8 x sin x --- x sin x --------------- 1 x x sin x --------------- 2 x cos+ 3------------------------- x x – x +----------------- x sin x --------------- x x sin x x x cos+ 2------------------------- 2 x cos+ 3------------------------- x cos x cos+ 3------------------------- x cos ----------------------------------- x cos ----------------------------------- x –1 x +--------------------------- 1 x – 11 x +1 x +-------------------------------------------------------2 23 x x sin720--------------------------- x tan x -------------------------–+ 2 k x x sin -------------------------- x tan x -------------------------–+ k
128 16 – 16 +=2 --- x x –6--------exp p o x cos p o p o x p x cos+14 x cos+----------------------------- p ------= x cos+14 x cos+-----------------------------9 6 x cos+14 x cos+----------------------------- r r ln 14 9 ln-----------------------= x cos+14 x cos+-----------------------------2 x cos k x –+ 3 k x –------------------------------------------ k –------------------= x cos k x –+ 3 k x –------------------------------------------ k Roy Howard 2020
Print Date: 28/10/20
Spline Based Bounds for Sin/Cos Naturally, it is of interest to determine bounds with lower relative errors. Zhu (2008a, Theorem 5) proposed thefollowing general form for bounds of , : (4) and, as indicated in Figure 2, the relative error in these bounds can be made arbitrarily small by utilizing increas-ingly high orders. This result was generalized in Zhu 2008b. These bounds build on the results of Li, 2006 (Theo-rem 2.1) and an alternative, but equivalent form, was published by Nui et al. (2008). Nui et al. 2010 provides ageneralization and showed that the equation forms specified by Nui and Zhu are equivalent (Proposition 3). Thesebounds, for orders zero to two, are: (5)(6) (7) (cid:1)(cid:1)(cid:2)(cid:2)(cid:1) (cid:1)(cid:2) (cid:2)(cid:1) (cid:1) (cid:1)(cid:2) (cid:2)(cid:1) (cid:1) (cid:1) (cid:1) (cid:1)(cid:2) (cid:2) (cid:2) (cid:2) (cid:2)(cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1)(cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2)(cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1)(cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) - - - Figure 1
Graph of the magnitude of the relative errors in the upper andlower bounds to as defined by the first, second, fourth, fifth,eighth and tenth order functions defined in Table 1.1. x sin x x re x lower 1 upper 1,2lower 2lower 4upper 10 upper 4 lower 10lower 8 upper 8upper 5 lower 5 upper 5 x sin x x k x – kk n n x – n + x sin x --------------- k x – kk n a k kk n – x – n n ----------------------------------- + k k k --------------- k k k -------------------------------- k – = ---= -----= n --- x – --------------------+ x sin x --------------- 2 --- 1 2 – ------------------- x – + --- x – -------------------- 12 –16 ----------------- x – + + x sin x --------------- 2 --- x – -------------------- 1 3 – ------------------- x – + + --- x – -------------------- 12 –16 ----------------- x – –16 ----------------- x – + + + x sin x --------------- --- x – -------------------- 12 –16 ----------------- x – ----- 1 3 ---– 12 –16 -----------------– x – + + + (cid:1)(cid:1)(cid:2)(cid:2)(cid:1) (cid:1)(cid:2) (cid:2) - - - - Figure 2
Relative error in upper and lower bounds to ,of orders zero to two, as defined by Equation 5 to Equation 7. x sin x x re x lower 0upper 0lower 1upper 1 lower 2upper 2 Spline Based Bounds for Sin/Cos
Print Date: 28/10/20 © Roy Howard 2020
This paper utilizes the two point spline based function approximation, detailed in Howard 2019, to specify a seriesof upper and lower bounds for and , over the interval , with increasingly low relativeerrors. Analysis associated with the bounds leads to new series for the sine function which have significantly bet-ter convergence, over the interval , than a Taylor series and the published series proposed by Zhu 2008aand Nui 2008. Applications include functions that define upper and lower bounds for the cosine function andlower bounds for the sine integral function. These are can be made arbitrarily accurate.Section 2 details miscellaneous results that underpin subsequent analysis. Section 3 details a spline based seriesfor the sine function and in Section 4 it is shown that these series are lower bounds, of increasing accuracy. Sec-tion 5 details applications including new series for the sine function, upper bounds for the sine function, boundsfor the cosine function and a lower bound for the sine integral function. Conclusions are detailed in section 6.
For a function defined over the interval , an approximating function has a relative error, at a point ,defined according to (8)
The relative error bound for the approximating function , over the interval , is defined according to (9)
The notation is used.Mathematica has been used to facilitate analysis and to obtain numerical results. In general, relative error boundresults for approximations have been obtained by sampling the appropriate interval at equally spacedpoints.
The following results underpin later analysis and discussion:
The default reference for bounds for the sine function, and hence , is the Taylor series defined accordingto (10)
A Taylor series yields alternating lower and upper bounds over the interval as the order is increased(e.g. Qi, 2009, Section 1.6). The bounds are increasingly accurate as can be seen in the results shown, along withother results, in Figure 3. The relative error bounds, over the interval , for various orders of Taylor seriesapproximations, are detailed in Table 2.1. The rate of convergence of the Taylor series serves as a reference.
Table 2.1
Relative error bounds forTaylor series approximations to over the interval .
Order of approx. Relative error bound
1: upper bound3: lower bound5: upper bound x sin x sin x f f A x re x f A x f x ---------------–= f A re B max re x : x = f k x t kk dd f x = 1000 x sin x x sin x x x x k x k k !----------------------------- + + += k x sin0 Roy Howard 2020
Print Date: 28/10/20
Spline Based Bounds for Sin/Cos The following lemma details when upper bounds to a set function can be generated from a sequence of converg-ing lower bounds :
Lemma 1 Upper Bound from a Lower Bound
Given a sequence of convergent lower bounds to a function over an interval , a sequence ofupper bounds can be defined over the interval according to (11) provided , where (12)
Proof
Consider and the error (13)
When it follows that as required.
Lemma 2 Relationship Between Sine and Cosine Bounds
The bounds , for the sine function imply the following bounds for the cosinefunction: (14)
Proof
As and , for , it follows that
7: lower bound9: upper bound13: lower bound17: lower bound33: upper bound
Table 2.1
Relative error bounds forTaylor series approximations to over the interval .
Order of approx. Relative error bound x sin0 ff L f L f L f L f f kU x f kL x f k L x –= x k k L x kL x x kL x f x f kL x –= x f kU x f kL x f k L x –= kU x f kU x f x – 2 f kL x f k L x – f x –= =2 f kL x f x – f x f k L x – + 2 kL x – k L x += = k L x kL x f kU x f L x x sin f U x x f L y – y cos f U y – y x – sin x cos= x sin x – cos= x Spline Based Bounds for Sin/Cos
Print Date: 28/10/20 © Roy Howard 2020 (15)
When considering analysis of the sine function the basic choices are between considering over the interval or over the interval . When establishing approximations to the sine function, the for-mer is preferred whilst the latter facilitates establishing convergence of approximations. The appropriate relation-ships between the two forms are detailed below:
Lemma 3 Scaling Relationships
With approximating functions and , and error functions and , defined according to (16) the linear transformations and , for the case of ,, are (17)
The relative errors in the approximations and , respectively, are: (18) and the relationships between the two relative errors are (19)
Proof
Consider, for example: (20)
It is clear that establishing the relative error for an approximation to for is equivalent toestablishing the relative error for the scaled approximation to for .
A order, two point, spline approximation to a function , which is at least a order differentiable over theinterval , has been detailed in Howard 2019 and is f L x x sin f U x f L x x –cos f U x x f L y – y cos f U y – y x –= x y –= y x sin0 x sin 0 1 f f y x x sin f x x += = x y x x sin f x x += = x x x = x x = f x f x = f x f x = y x y --- x = y x y x = x --- x = x x = x sin f x x sin f x re x f x x sin---------------–= re x f x x sin----------------------------–=re x re x = re x re x =re x f x x sin----------------------------– 1 f x x sin----------------------------– re x = = = x sin x x sin x n th f n th Roy Howard 2020
Print Date: 28/10/20
Spline Based Bounds for Sin/Cos (21) This general result can be utilized to define a sequence of functions that approximate, with increasing accuracy,the sine function:
Theorem 3.1 Spline Series for Sine
A order spline series approximation to , for the interval , and based on the points and, is (22)
The approximations, of order zero to fourth, are: (23)(24)(25)(26)(27)
Higher order approximations can readily be defined.
Proof
These results arise from direct application of Equation 21 for the case of , the interval ,, and the result (28) f n x x – n – n ---------------------------- x – k k !------------------- f k n i + ! i ! n ! ----------------- i n k – x – i – i ------------------- k n += x – n – n ---------------------------- 1– k x – k k !-------------------------------- f k n i + ! i ! n ! ----------------- i n k – x – i – i ------------------- k n x n th x sin 0 f n x x ------– n k !---- k x k n i + ! i ! n ! ----------------- i n k – x ------ i k n += 2 x ------ n k k !------------- k k x ------– k n i + ! i ! n ! ----------------- i n k – x ------– i k n n f x x ------= f x x ------ 1 x ------ 1 x –+= f x x ------ 1 3 x --------- 1 4 x – 192 --------- 1 x + += f x x x --------- 1 2 x ------------ 1 15 x –+ += 4480 ------------ 1 9 x ------------ 1 x – f x x x ------------ 1 5 x –+= 26880 --------------- 1 4 x --------------- 1 7 x –+80640 --------------- 1 16 x --------------- 1 x + f x x sin= 0 = f k x x k k Spline Based Bounds for Sin/Cos
Print Date: 28/10/20 © Roy Howard 2020
The graphs of the relative errors in the first to fourth order spline based approximations, defined in Theorem 3.1,are shown in Figure 3 and the associated relative error bounds are detailed in Table 3.1. The error functions for thefirst to fourth order approximations are shown in Figure 4. These results, when compared with Taylor seriesapproximations and the approximations defined by Zhu 2008a (specified in Equation 4 and whose relative errorsare shown in Figure 2), demonstrate better convergence. The approximations are exact at the points .The errors in the approximations, for orders one to four, are shown in Figure 4 and are positive which suggests theapproximations can serve as lower bounds for the sine function. The proof of this is detailed below.
To show that the spline approximations for , as detailed in Theorem 3.1, are lower bounds, the first andsecond order approximations are considered.
Table 3.1
Relative error boundsfor spline based approximations to over the interval .
Order of approx. Relative error bound: Spline based. - - - - - Figure 3
Graphs of the relative error in the spline approximations to, of orders one to four, as well as Taylor series approximationsof orders one, three, five, seven and nine. x sin x order 1order 2order 3order 41 f k x x sin---------------– Taylor 1 Taylor 3 Taylor 5Taylor 7Taylor 9 x sin 0 x sin Roy Howard 2020
Print Date: 28/10/20
Spline Based Bounds for Sin/Cos The proof, that the first order spline approximation, as defined by Equation 24, is a lower bound for overthe interval , is based on showing that the associated error function is positive over this interval. Con-sider the scaled error function (29) which has the property of being positive on the interval and being such that :
Theorem 4.1 Error in First Order Spline Approximation to Sine
Using the notation , the error function defined by Equation 29, can be written as a summation of positiveterms according to (30)
The series converges for and the coefficients can be defined iteratively according to (31)(32) and (33)
Figure 4
Graph of the errors, , in the firstto fourth order spline approximations to . k x x sin f k x –= x sin x x x x x x sin0 x x sin f x –= x
0= = t x = t c t t – c t t – c t t – c t t – c t t – c t t – + + + + + += c k t k t – pk = p k =3 k = = c k t k t – k k = t t c c c c
3– 2 c c –= = c
4– 3 c c c – c
9– 7 c c –= = c k c k c k –= k c k c k c k – k k k !----------+= k c k c k c k – c k += k c k c k c k – c k – c k k k k !----------–+= k Spline Based Bounds for Sin/Cos
Print Date: 28/10/20 © Roy Howard 2020
By construction the coefficients are positive and decrease according to , , and it is the casethat for .
Proof
The proof is detailed in Appendix 1.
The first few coefficients are: , , , ,, and .
As , , it follows that is a lower bound for , . The scalingrelationships specified in Lemma 3 then imply that is a lower bound for , .
The second order spline approximation, as defined by Equation 25, is also a lower bound for the sine function.The proof of this is based on considering the scaled error function (34) which has the property of being positive on the interval and being such that :
Theorem 4.2 Error in Second Order Spline Approximation to Sine
Using the notation , the error function defined by Equation 34, can be written as a summation of positiveterms according to (35)
The coefficients can be defined iteratively according to (36)(37)(38) c k c k k x x c c c c c = c = x x f x x sin x f x x sin x x x sin f x –= x
0= = t x = t c t t – c t t – c t t – c t t – c t t – c t t – + + + + + += c k t k t – pk = p k =4 k = = c k t k t – k k = c c c c c c – c
10– 3 c c c – c
15– 5 c c – 4 c – c –= = c
36– 25 c c – 3 c – Roy Howard 2020
Print Date: 28/10/20
Spline Based Bounds for Sin/Cos (39) By construction the coefficients are positive and decrease according to , .
Proof
The proof is detailed in Appendix 2.
The first few coefficients are: , , , and.
As , , it follows that is a lower bound for , . The scalingrelationships specified in Lemma 3 then imply that is a lower bound for , .
The proof that a given higher order spline approximation defined in Theorem 3.1 is also a lower order bound for, can be proved in an analogous manner to that detailed in the proofs for Theorem 4.1 and Theorem 4.2.
For the case of a set relative error bound of for an approximation to , over the interval , asecond order approximation, as specified by Equation 25, is required. The achieved relative error bound is. The following second order approximation with the approximate coefficients (40) yields a relative error bound of . In comparison, a fifth order Taylor series approximation (Table 2.1)has a relative error bound of . To achieve a relative error bound of , a fourth order approximation, as specified by Equation 27, is requiredand the achieved relative error bound is . The following fourth order approximation, with the resolu-tion defined by the given coefficient accuracy, yields a relative error bound of : (41)
A ninth order Taylor series approximation (Table 2.1) yields a relative error bound of . c k c k c k – c k += k c k c k c k – 2 c k – c k k k k !----------–+= k c k c k c k – 4 c k c k –+= k c k c k c k – c k c k – k k k !----------+ += k c k c k k c c c c = c = x x f x x sin x f x x sin x x sin 0.001 x sin 0 f x x x x x f x x x x x x –+ + + x x + 3.54 10 Spline Based Bounds for Sin/Cos
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For relative error bounds of and , spline approximations of orders six and eight are required. Therequired approximations are: (42)(43)
The achieved relative error bounds, respectively, are and . Higher precision in thecoefficients leads to relative error bounds, respectively, of and . For comparison, athirteenth and a seventeenth order Taylor series approximation have, respectively, relative error bounds of and .
Theorem 5.1 Lower Bounds for Sine
The spline based series for , defined in Theorem 3.1, are lower bounds for on the interval. Thus, for example, the first and second order spline approximations detailed in Theorem 3.1, yield thebounds: (44)(45) and similar results hold for higher order spline approximations. There is equality at the points . The maxi-mum relative errors associated with these lower bounds are detailed in Table 3.1.
Proof
This result follows from Theorem 4.1 and Theorem 4.2 and, similarly, for higher order spline approximations.
Consider the result stated in Theorem 4.1 which specifies a series for the error function . Convergence of thisseries implies a new series for the sine function:
Theorem 5.2 Series Approximation for Sin Based on First Order Spline Approximation
The error function associated with a first order spline approximation yields the following series for the sine func-tion that is valid for the interval : (46) where and the coefficients are defined in Theorem 4.1.The first few terms are 10 f x x x x x – 7.78 10 x x + + + + x x – 3.06 10 x x + + f x x x x x x + + + x x – 1.26122 10 x x + + +7.21759 10 x x – 7.0722 10 x x + +4.39 10 x sin x sin0 ------ 1 x ------ 1 x –+ x sin x --------------- x ------ 1 3 x --------- 1 4 x – 192 --------- 1 x + + x sin x --------------- x x sin 2 x ------ 2 x ------ 1 2 x ------– c k x ------ k x ------– pk + + p k =3 k = == c + = c c Roy Howard 2020
Print Date: 28/10/20
Spline Based Bounds for Sin/Cos (47) Proof
The proof is detailed in Appendix 3.
The sine function can be approximated by taking the first terms in the series defined in Theorem 5.2, i.e. (48)
The graphs of the relative errors associated with the approximations to , which are defined by the first toninth terms in the series approximations, are shown in Figure 5. The associated relative error bounds are detailedin Table 5.1.
Table 5.1
Relative error bounds for the first and secondorder series approximations to . x sin 2 x ------ 2 x ------ 1 2 x ------– 2 1– x ------ 1 2 x ------– x ------ x ------– + += 4– 3 x ------ x ------–
9– 7 x ------ x ------– + + nx sin 2 x ------ 2 x ------ 1 2 x ------– c k x ------ k x ------– pk n + + p k =3 k = = x sin - - - Figure 5
Graphs of the relative errors in the approximation to, as defined by Equation 48, for the cases of the first toninth terms. x sin x x x sin n Spline Based Bounds for Sin/Cos
Print Date: 28/10/20 © Roy Howard 2020
The series for the error function defined in Theorem 4.2 converges and this underpins the definition of the fol-lowing series for the sine function:
Theorem 5.3 Series Approximation for Sin Based on Second Order Spline Approximation
The error function associated with a second order spline approximation yields the following series for the interval: (49) where (50) and with the coefficients being defined in Theorem 4.2. The first few terms are (51)
Proof
The proof is detailed in Appendix 4.
The sine function can be approximated by taking the first terms in the series defined in Theorem 5.3, i.e. (52)
The graphs of the relative errors in the approximations, for the case of two to nine terms, are shown in Figure 6.The associated relative error bounds are detailed in Table 5.1.
Higher order series for can be generated in a similar manner and, as the results in Table 5.1 indicate, withlower relative error bounds. x sin 1 x ------– – c k x ------ k x ------– pk + p k =4 k = == c c c
10– 2 c c x sin 1 x ------– – 1– x ------– x ------ 1 2 x ------– + += 10– 2 x ------ x ------–
10– 3 x ------ x ------– + + nx sin 1 x ------– – c k x ------ k x ------– pk n + p k =4 k = = - - - - - - Figure 6
Graphs of the relative errors in the approximations to as defined by Equation 52, for orders two to nine. x sin x x x sin Roy Howard 2020
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Spline Based Bounds for Sin/Cos The approximations for , specified by Theorem 3.1, Theorem 5.2 and Theorem 5.3, of order two, are: (53)(54)(55) with respective relative error bounds over the interval of , and . Ingeneral, the series specified by Theorem 3.1 have better convergence that the other two series and a simpler form.
Consistent with Lemma 1, the requirement for an upper bound for the sine function, based on the first and secondorder spline approximations and , as defined by Equation 24 and Equation 25, is for (56) where and are defined by Equation 30 and Equation 35. When this is the case the following upper boundcan be defined:
Theorem 5.4 Upper Bound for Sine
The lower bounds and , as defined by Equation 24 and Equation 25, define an upper bound for according to (57) i.e. (58)
The error in this bound, as defined by (59) is shown in Figure 7. Note that the error is close to the error in the lower bound, , associated with, which is shown in Figure 4.
Proof
The proof is detailed in Appendix 5.
Higher order approximations for the upper bound of the sine function follow in an analogous manner. For exam-ple x sin f x x ------ 1 3 x --------- 1 4 x – 192 --------- 1 x + += f x x ------ 2 x ------ 1 2 x ------– 2 1– x ------ 1 2 x ------– x ------ x ------– += f x x ------– – 1– x ------– x ------ 1 2 x ------– + += 10– 2 x ------ x ------– f f x ------ 2 x ------ x f f f U x sin f U x f x f x –= x ------ 1 x – 176 --------- 1 13 x --------- 1 4 x – 384 --------- 1 x + + x sin x --------------- x U x f U x x sin–= x U x x f x Spline Based Bounds for Sin/Cos
Print Date: 28/10/20 © Roy Howard 2020 (60) are valid upper bounds with and being defined by Equation 25 to Equation 27. The graphs of the errorsin these bounds, denoted, respectively, and , are shown in Figure 7.
The results detailed in Lemma 2 allow the bounds for the cosine function to be defined by utilizing the boundsspecified for the sine function:
Theorem 5.5 Lower Bounds for Cosine
First to fourth order spline based approximations for over the interval , as specified inTheorem 3.1, lead to the following lower bounds for the cosine function: (61)(62)(63)(64)
Figure 7
Graphs of the errors in the functions and ,which defined second, third and fourth order upper bounds to. f U f U f U x sin x U x U x U x f U x f x f x –= x f U x f x f x –= x f f f U U x sin 0 g y ------ 1 y – 16 ------ 1 y += g y y ------ 1 y – 240 --------- 1 7 y --------- 1 y –+= g y y --------- 1 3 y – 2688 ------------ 1 13 y –+= 4480 ------------ 1 17 y ------------ 1 y + g y y y ------------ 1 2 y –+ += 26880 --------------- 1 7 y --------------- 1 13 y +–80640 --------------- 1 31 y --------------- 1 y – Roy Howard 2020
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Spline Based Bounds for Sin/Cos
These results follow from the result stated in Lemma 2 of , and the splineapproximations for the sine function stated in Theorem 3.1. Together these imply where, . Alternatively, they can be directly derived from Equation 21 for the caseof .
The relative error bounds in the approximations for the cosine function defined in Theorem 5.5 are identical to therelative error bounds associated with the lower bounds for the sine function which are detailed in Table 3.1.
Upper bounds for the cosine function can be generated consistent with the results stated in Theorem 5.4. Forexample, the result , implies (65)
The error in this bound is (66) and, thus, has the same bound as whose graph is shown in Figure 7. Higher order upper bounds can be definedin a similar manner.
The list of inequalities for over the interval is similar in scope to that for with the Koberinequality (Kober 1944) serving as a reference inequality. Qi et. al. (2009) provides a useful overview. In general,the bounds proposed in this paper are closer bounds than published results and can be made arbitrary accurate.
Bounds for readily lead to bounds for the sine integral function (e.g. Lv et al. 2017, Proposition 5; Zeng& Wu 2013, Theorem 9) which is defined according to (67)
The lower bound published by Lv 2017 is (68)
Integration of the spline series for defined by Theorem 3.1 (after dividing by ) leads to the followingapproximations:
Theorem 5.6 Spline Based Approximations for Sine Integral
First to fourth order approximations for the sine integral function, which are also lower bound functions, are: (69) f L y – y cos y g k y y cos g k y f k y – = k f x x cos= g U y g y g y –= y ------ 1 y --------- 1 9 y – 480 --------- 1 7 y --------- 1 y –+ + y cos y U y g U y y cos– 2 g y g y – y cos–= =2 f y – f y –– y –sin– U y –= = U y cos 0 x sin x sin x Si x sin ---------------- d x =2 x x sin+ 3--------------------------- x x x cos 3 x sin–+ 9 -----------------------------------------------------------– Si x x sin xh x x ----- 1 x --------- 1 x –+= Spline Based Bounds for Sin/Cos
Print Date: 28/10/20 © Roy Howard 2020 (70)(71)(72)
Higher order approximations can similarly be generated.
Proof
First to fourth order spline based approximations for the sine integral, and for the interval , can be gener-ated from the approximations detailed in Theorem 3.1 for the sine function according to (73)
The stated results then follow from the definitions for given in Theorem 3.1. As , are lowerbounds for the sine function, it then follows that the approximations for the sine integral function are also lowerbounds.
Graphs of the relative error in the approximations defined in Theorem 5.6 are shown in Figure 8 and show theimprovement over the lower bound proposed by Lv 2017 for orders three and higher. Bounds on the relative errorare tabulated in Table 5.2 and clearly can be made arbitrarily small by increasing the order of approximation.
Table 5.2
Relative error boundsfor the spline based approximationsto the sine integral function overthe interval .
Order of approx. Relative error bound h x x --------- 1 3 x ------ 1 4 x – 1925 --------- 1 x + += h x x x --------- 1 2 x ------------ 1 15 x –+ += 22403 ------------ 1 9 x ------------ 1 x – h x x x ------------ 1 5 x –+= 4480 ------------ 1 4 x --------------- 1 7 x –+10080 --------------- 1 16 x --------------- 1 x + 0 h k x f k ------------ d x = k f k f k k Roy Howard 2020
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Spline Based Bounds for Sin/Cos
Upper bound functions for the sine integral can be generated from the lower bounds and consistent withLemma 1.
Spline based approximations for over the interval have been to detailed and exhibit better con-vergence that comparative Taylor series approximations. It was proved that these approximations are also lowerbounds for and , over the interval , with increasingly high accuracy and with lower rel-ative error bounds than published lower bounded functions. For example, approximations of order two, four andeight have relative error bounds, respectively, of , and . The analysis underpinning the proof that the approximations are lower bounds for the sine function led to newseries for . These have better convergence that Taylor series for over the interval .Applications were detailed and include, first, a sequence of upper bounded functions, with increasing accuracy,for over the interval . Second, a sequence of upper and lower bounded functions for withaccuracy consistent with the corresponding sine approximations. Finally, a sequence of convergent lowerbounded approximations for the sine integral function over the interval .
Acknowledgement
The support of Prof. A. Zoubir, SPG, Technische Universität Darmstadt, Darmstadt, Ger-many, who hosted a visit where the writing of this paper was completed, is gratefully acknowledged.
Appendix 1: Proof of Theorem 4.1
By utilizing a Taylor series expansion for , , and the definition for defined byEquation 24, the error in a first order spline function approximation for is (74) where and etc. The goal is to write as a sum-mation of positive terms only. To this end the following form is considered: - - - - - Figure 8
Graph of the relative error in spline basedapproximations, of orders one to four, for the sine integral functionas well as the relative error in the lower bound specified by Lv(2017) and stated in Equation 68. x order 1order 2order 3order 41 h k x si x -------------– Lv 2017 x sin 0 x sin x sin x x sin x sin 0 x sin 0 x cos0 t sin t f t sin t t sin f t –= t t t t – 19!----- t + + + += e t e t e t e t e t + + + + += e
3– 0.314159= = e – – 0.21676–= = Spline Based Bounds for Sin/Cos
Print Date: 28/10/20 © Roy Howard 2020 (75)
Equating Equation 74 and Equation 75 it immediately follows that . For to be positive, the factor needs to be chosen such that and . This is guaranteed when. Choosing the smallest integer greater than zero such that this is the case yields and for the given values of and the required value is . It then follows that.The ratio of to then is (76) which implies as and . With and with and set, it follows that (77) and the requirement of can be satisfied by choosing such that . With the requirement is for . Choosing the smallest integer greater than zero such thatthis is the case yields and for the defined values of and it is the case that . Itthen follows that .The ratio of to then is (78) which implies . Proceeding in this manner yields the stated form and proof of positive coefficients with decreasing values. Thecoefficients are defined according to (79)(80)(81)(82) t c t t – p c t t – p c t t – p c n t n t – p n + + + + += c t p t – p t – t p + c t p t – t p + c t p t – t p + + + += c t p c – c + t p c p c – c + t + + += c e = 0 c p e p c – c += c e p c += 0 p e c – p e – e e e p c
4– 3 –+ 0.0664249= = c c c c ----- e p c + e ---------------------- e e ----- p + e e ----- e – e -------- 1+ + 1 e – e -------- e – e --------––= = = =0 c c e – e x x – 1 p c c t c t t – t + c t p t – p t – t p + c t p t – t p + + += e t e t c p c – c + t c t p t – t p c t p t – t p + + + + + += e c p c – c + 0= = p c c c – p c += 0 p c c p c c c c p c
9– 7 –+ 0.057682= = c c c c ----- c – p c + c -------------------------- c – c -------- p + 1 c c ----- c c -----–+= = =0 c c c += c
4– 3 c
9– 7 c
15– 6 c
7– 3 c
8– 7 c
17– 15 c
27– 12
185 794 560 ---------------------------------+ + +=
Roy Howard 2020
Print Date: 28/10/20
Spline Based Bounds for Sin/Cos (83)(84)(85) To find a general formula for the coefficients consider the definitions: (86)
It then can be shown that (87)(88)(89)
The iteration formulas (90) then follow.
A1.1 Proof of Convergence
To prove that the new series for , as specified by Equation 75, converges over the interval , con-sider the upper bounds: (91)
The inequalities follow as , for all and the final form, consistent with the convergenceof a geometric series, is valid for . As it follows that the series defining isbounded above on the interval .With positive coefficients, the series defining is bounded below according to , . c
11– 5
92 897 280 ------------------------------+ + += c
12– 11
61 931 520 ------------------------------
81 749 606 400 ------------------------------------------–+ + += c
25– 23
26 542 080 ------------------------------
27 249 868 800 ------------------------------------------–+ + += c c c c c c –= c c c c – c c c –= c c c – c c c – c += c c c – c – c c k c k c k –= k c k c k c k – k k k !----------+= k c k c k c k – c k += k c k c k c k – c k – c k k k k !----------–+= k t t t c t t – c t t – c t t – c t t – c t t – c t t – + + + + + += c t c t c t c t c t c t + + + + + + c t t t t t t + + + + + + c t t –---------- = t c k c k c k kt
0= = t t t t t Spline Based Bounds for Sin/Cos
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Appendix 2: Proof of Theorem 4.2
By utilizing a Taylor series expansion for , , and the definition for defined by Equation 25,the error in a second order spline function approximation for is (92) where and etc. Theapproach is then to proceed in a manner consistent with the first order case detailed in Appendix 1 and the follow-ing form is considered: (93)
Equating Equation 92 and Equation 93, it immediately follows that . For the coefficient to be pos-itive, the factor needs to be chosen such that and . This is guaranteedwhen . Choosing the smallest integer greater than zero such this is the case yields and for the defined values of and the required value is . It then follows that. The ratio of to then is (94) which implies . Proceeding in this manner yields the stated form and proof of positive coefficientswith decreasing values. The coefficients are defined according to (95)(96)(97)(98)(99) t sin t f t sin t t sin f t –= 10– 3 t
15 4 – t
6– 3 t t – 19!----- t + + + += e t e t e t e t e t + + + + += e
10– 3 –+ + 0.0125144= = e
15 4 – – 0.0337717–= = t c t t – p c t t – p c n t n t – p n + + + += c t p t – p t – t p + c t p t – t p + c t p t – t p + + + += c t p c – c + t p c p c – c + t + + += c e = 0 c p e p c – c += c e p c += 0 p e c – p e – e e e p c
15– 5 –+ + 0.00377153= = c c c c ----- e p c + e ---------------------- e e ----- p + e e ----- e – e -------- 1+ + 1 e – e -------- e – e --------––= = = =0 c c c
10– 3 c
15– 5 c
36– 25 c
64– 23 c
36– 14 c
45– 18 c
185 794 560 ---------------------------------+ + + += c
46 448 640 ------------------------------+ + + +=
Roy Howard 2020
Print Date: 28/10/20
Spline Based Bounds for Sin/Cos (100)(101) To find a general formula for the coefficients consider the definitions: (102)
It is then the case that (103) and it can then be readily shown that (104)(105)(106)
The iterative formulas then follow: (107)
The proof that the series defining converges, and is bounded below by zero, follows in an identical manner tothat given in Appendix 1.
Appendix 3: Proof of Theorem 5.2
Equation 29 implies (108) where is defined by Equation 24 and the convergent series for is defined in Theorem 4.1. As can bewritten in the form (109) the required result follows, namely (110) c
78– 33
30 965 760 ------------------------------
81 749 606 400 ------------------------------------------–+ + + += c
91– 39
18 579 456 ------------------------------
27 249 868 800 ------------------------------------------–+ + + += c c c c c c – c
10– 3 c c c – c c c – 4 c – c –= c c c – 3 c – c c c – c += c c c c – c +– c c c – 4 c c –+= c c c – c c – c k c k c k – c k += k c k c k c k – 2 c k – c k k k k !----------–+= k c k c k c k – 4 c k c k –+= k c k c k c k – c k c k – k k k !----------+ += k x sin f x x ------+= x f f f x x ------ 2 x ------ 1 2 x ------– c x ------ 1 2 x ------– + += c x sin 2 x ------ 2 x ------ 1 2 x ------– c k x ------ k x ------– pk + + p k =3 k = == Spline Based Bounds for Sin/Cos
Print Date: 28/10/20 © Roy Howard 2020
Appendix 4: Proof of Theorem 5.3
Equation 34 implies (111) where is defined by Equation 25 and the convergent series for is defined in Theorem 4.2, As can bewritten in the form (112) the required result follows, namely: (113)
Appendix 5: Proof of Theorem 5.4
First, the definitions and , as defined by Equation 24 and Equation 25, imply (114)
Second, utilizing the transformed functions associated with the interval rather than the interval ,the requirement is for , , i.e. for (115) where the positive coefficients and are defined, respectively, in Theorem 4.1 and Theorem 4.2. Consider (116)
Hence, a sufficient condition for , , is for , . This iseasy to confirm, along with the rapid decline toward zero in the sequence . x sin f x x ------+= x f f f x x ------– – 1– x ------– x ------ 1 2 x ------– + + += 10– 2 x ------ x ------– x sin 1 x ------– – c k x ------ k x ------– pk += p k =4 k = = c c c
10– 2 f f f x f x – 2 x ------ 1 3 x --------- 1 4 x – 192 --------- 1 x + + –= x ------ 1 x ------ 1 x –+ x ------ 1 x – 176 --------- 1 13 x --------- 1 4 x --------- 1 x + + += 0 1 t t t c t t – c t t – c t t – c t t – c t t – c t t – + + + + + +2 d t t – d t t – d t t – d t t – d t t – d t t – + + + + + + c k d k c k t k t – p d k t k t – p – p t k c k d k t t – – t k t – p c k d k – t k t – p d k t t – – t k t – p += = c k d k – t k t – p t t t c k d k – 0 k c k d k – k Roy Howard 2020
Print Date: 28/10/20
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