The natural algorithmic approach of mixed trigonometric-polynomial problems
- Tatjana Lutovac^{1},
- Branko Malešević^{1}Email author and
- Cristinel Mortici^{2, 3, 4}
DOI: 10.1186/s13660-017-1392-1
© The Author(s) 2017
Received: 28 February 2017
Accepted: 2 May 2017
Published: 18 May 2017
Abstract
Keywords
mixed trigonometric-polynomial functions Taylor series approximations inequalities algorithms automated theorem provingMSC
41A10 26D05 68T15 12L05 41A581 Introduction and motivation
In this way, complicated trigonometric expressions can be reduced to polynomial or rational expressions, which can be, at least theoretically, easier studied (this can be done using some software for symbolic computation, such as Maple).
MTP functions currently appear in the monographs on the theory of analytical inequalities [15, 16] and [5], while concrete MTP inequalities are employed in numerous engineering problems (see, e.g., [17, 18]). A large class of inequalities arising from different branches of science can be reduced to MTP inequalities.
It is notable that many of the above-mentioned analyses and treatments of MTP inequalities are all rather sophisticated and involve complex transformations and estimations. Almost all approaches are designed for ’pen and paper analysis’ and many of them are ripe for automation, being formally defined in precise detail, and yet somewhat overwhelming for humans.
Notwithstanding, the development of formal methods and procedures for automated generation of proofs of analytical inequalities remains a challenging and important task of artificial intelligence and automated reasoning [19, 20].
The aim of this paper is to develop a new algorithm, based on the natural approach method, for proving MTP inequalities by reducing to polynomial inequalities.
Although transformation based on the natural approach method has been made by several researchers in their isolated studies, a unified approach has not been given yet. Moreover, it is interesting to note that just trigonometric expressions involving odd powers of cosx were studied, as the natural approach method cannot be directly applicable for the function cos^{2} x over the entire interval \((0, \pi/2)\). Our aim is to extend and formalize the ideas of the natural approach method for a wider class of trigonometric inequalities, including also those containing even powers of cosx, with no further restrictions.
Notice the logical-hardness general problem under consideration. According to Wang [21], for every function G defined by arithmetic operations and a composition over polynomials and sine functions of the form \(\sin \pi x\), there is a real number r such that the problem \(G(r)=0\) is undecidable (see [22]). In 2003, Laczkovich [23] proved that this result can be derived if the function G is defined in terms of the functions \(x,\sin x\) and \(\sin (x\sin x^{n})\), \(n=1,2,\ldots \) (without involving π). On the other hand, several algorithms [24, 25] and [26] have been developed to determine the sign and the real zeroes of a given polynomial, so that such problems can be considered decidable (see also [22, 27]).
2 The natural approach method and the associated algorithm
The following two lemmas [8] related to the Taylor polynomials associated with sine and cosine functions will be of great help in our study.
Lemma 1
- (i)If \(n=4s+1\), with \(s\in \mathbb {N}_{0}\), thenand$$ T_{n}(x)\geq T_{n+4}(x)\geq \sin x \quad \textit{for every } 0 \leq x\leq \sqrt{(n+3) (n+4)}; $$(3)$$ T_{n}(x)\leq T_{n+4}(x)\leq \sin x \quad \textit{for every } {-}\sqrt{(n+3) (n+4)}\leq x\leq 0. $$(4)
- (ii)If \(n=4s+3\), with \(s\in \mathbb {N}_{0}\), thenand$$ T_{n}(x)\leq T_{n+4}(x)\leq \sin x \quad \textit{for every } 0 \leq x\leq \sqrt{(n+3) (n+4)}; $$(5)$$ T_{n}(x)\geq T_{n+4}(x)\geq \sin x \quad \textit{for every }{-}\sqrt{(n+3) (n+4)}\leq x\leq 0. $$(6)
Lemma 2
- (i)If \(n=4k\), with \(k\in \mathbb{N}_{0}\), then$$\begin{aligned}& T_{n}(x)\geq T_{n+4}(x)\geq \cos x \\ & \quad\textit{for every } {-}\sqrt{(n + 3) (n + 4)} \leq x \leq \sqrt{(n + 3) (n + 4)}. \end{aligned}$$(7)
- (ii)If \(n=4k+2\), with \(k\in \mathbb{N}_{0}\), then$$\begin{aligned}& T_{n}(x)\leq T_{n+4}(x)\leq \cos x \\& \quad \textit{for every } {-}\sqrt{(n + 3) (n + 4)} \leq x \leq \sqrt{(n + 3) (n + 4)}. \end{aligned}$$(8)
According to Lemmas 1-2, the upper bounds of the approximation intervals of the functions sinx and cosx are \(\varepsilon _{1}= \sqrt{(n_{1}+3)(n_{1}+4)}\) and \(\varepsilon _{2}=\sqrt{(n_{2}+3)(n_{2}+4)}\), respectively. As \(\varepsilon_{1}> \frac{\pi}{2}\) and \(\varepsilon _{2}> \frac{\pi}{2}\), the results of these lemmas are valid, in particular, in the entire interval \(( 0, \frac{\pi}{2} )\).
Lemma 3
- (1)Let \(n\in \mathbb {N}\) and \(x\in ( 0,\frac{\pi}{2} ) \). Then$$T_{n}^{ \sin ,0}(x) \geq 0. $$
- (2)Let \(s\in \mathbb {N}_{0}\), \(p\in \mathbb {N}\) and \(x\in ( 0, \frac{\pi}{2} )\). Then$$\bigl( \underline{T}_{4s+3}^{ \sin ,0}(x) \bigr) ^{p}\leq \sin ^{p}{ x}\leq \bigl( \overline{T}_{4s+1}^{ \sin ,0}(x) \bigr) ^{p}. $$
Lemma 4
In contrast to the function sinx and its downward Taylor approximations, in the interval \(( 0, \frac{\pi}{2} ) \) the function cosx and the downward Taylor approximations \(\underline{T}_{ 4k+2}^{ \cos ,0}(x)=\sum_{i=0}^{2k+1}{ \frac{(-1)^{i}x^{2i}}{(2i)!}}, k \in \mathbb {N}_{0}\), require special attention as there is no downward Taylor approximation \(\underline{T}_{ 4k+2}^{ \cos ,0}(x)\) such that \(\cos ^{2}{ x} \geq ( \underline{T}_{ 4k+2}^{ \cos ,0}(x) ) ^{2}\) for every \(x\in ( 0, \frac{\pi}{2} )\).
We present the following results related to the problem with downward Taylor approximations of the cosine function.
Proposition 5
- (1)
For every \(k\in \mathbb{N}_{0} \), the downward Taylor approximation \(\underline{T}_{ 4k+2}^{ \cos ,0}(x)\) is a strictly decreasing function on \(( 0, \frac{\pi}{2} ) \).
- (2)
For every \(k\in \mathbb{N}_{0} \), there exists unique \(c_{k}\in ( 0, \frac{\pi}{2} ) \) such that \(\underline{T}_{ 4k+2}^{ \cos,0}(c_{k})=0\).
- (3)
The sequence \((c_{k} ) _{k\in\mathbb{N}_{0}}\), with \(c_{0}=\sqrt{2}\), is strictly increasing and \(\lim_{k\rightarrow +\infty }{c_{k}}=\frac{\pi }{2}\).
- (4)
For every \(k\in \mathbb {N}_{0}\), there exists \(d_{k}\in ( c_{k}, \frac{\pi}{2} )\) such that \(\cos {d_{k}} = \vert \underline{T}_{ 4k+2}^{ \cos ,0}(d_{k}) \vert \).
- (5)
The sequence \(( d_{k} ) _{k\in \mathbb{N}_{0}}\) is strictly increasing and \(\lim_{k\rightarrow +\infty }{ d_{k}}=\frac{\pi }{2}\).
Proof
(1) The function \(\underline{T}_{ 4k+2}^{\cos ,0}(x)\) is strictly decreasing on \(( 0,\frac{\pi}{2} )\) since, according to Lemma 1, \(( \underline{T}_{ 4k+2}^{ \cos ,0}(x) ) ^{\prime} = -\overline{T}_{4k+1}^{ \sin ,0} (x) \leq 0\).
(2) The existence of \(c_{k}\) follows from the fact that \(\underline{T}_{ 4k+2}^{ \cos ,0}(0) = 1 > 0\) and \(\underline{T}_{ 4k+2}^{ \cos ,0} ( \frac{\pi}{2} ) < \cos { ( \frac{\pi}{2} )} = 0\).
(3) The monotonicity of the sequence \((c_{k})_{k\in \mathbb{N}_{0}}\) is a result of the monotonicity of \(\underline{T}_{ 4k+2}^{ \cos ,0}(x)\) and Lemma 2(ii).
The convergence of the sequence \((T_{n}^{ \cos ,0}(x))_{n\in \mathbb{N}}\) implies the convergence of the sequence \((c_{k})_{k\in \mathbb {N}_{0}}\) to \(\frac{\pi}{2}\).
(4) The function \(\vert \underline{T}^{ \cos ,0}_{4k+2}(x) \vert \) is decreasing on \((0,c_{k})\) and increasing on \((c_{k},\frac{\pi}{2} )\). Based on Lemma 2(ii), it follows that there exists \(d_{k} \in ( c_{k}, \frac{\pi}{2} )\) such that \(\cos{d_{k}} = \vert \underline{T}^{ \cos ,0}_{ 4k+2}(d_{k}) \vert \).
(5) This statement is a consequence of the monotonicity of the sequence \(( c_{k})_{k\in \mathbb{N}_{0}}\) and the increasing monotonicity of the function \(\vert \underline{T}_{4k+2}^{ \cos,0}(x) \vert \) on \((c_{k},\frac{\pi}{2} )\). □
Corollary 6
- (1)
\(\cos ^{2p}{ x}> ( \underline{T}_{4k+2}^{ \cos ,0}(x) )^{2p}\) for every \(x\in (0,d_{k})\);
- (2)
\(\cos ^{2p}{ x}< ( \underline{T}_{4k+2}^{ \cos ,0}(x) ) ^{2p}\) for every \(x\in (d_{k}, \frac{\pi}{2} )\).
Based on the above results, we have the following.
Corollary 7
Let \(k\in \mathbb {N}_{0}\) and \(p\in \mathbb {N}\). Then \(\underline{T}_{ 4k+2}^{ \cos ,0}(x)\) is not a downward approximation of the MTP function \(\cos^{2p}{x}\) on \((d_{k}, \frac{\pi}{2} )\).
In order to ensure the correctness of the algorithm [27, 28] we will develop next in the sequel, the following problem needs to be considered.
Problem
Remark
If cosx appears in odd powers only in the given MTP function \(f(x)\), we take \(\widehat{k}=0\).
One of the methods to solve the problem of downward approximation of the function \(\cos ^{2p}{ x}, p\in\mathbb{N}\) is the method of multiple angles developed in [8]. All degrees of the functions sinx and cosx are eliminated from the given MTP function \(f(x)\) through conversion into multiple-angle expressions. This removes all even degrees of the function cosx, but then sine and cosine functions appear in the form \(\sin \boldsymbol{\kappa} x\) or \(\cos \boldsymbol{\kappa} x\), where \(\boldsymbol{\kappa} x \in ( 0, \boldsymbol{\kappa} \frac{\pi}{2} )\) and \(\boldsymbol{\kappa} \in \mathbb{N}\). In this case, in order to use the results of Lemmas 1-2, we are forced to choose large enough values of \(k \in \mathbb{N}_{0}\) such that \(\sqrt{(k+3)(k+4)} > \boldsymbol{\kappa} \frac{\pi}{2}\). Note that a higher value of k implies a higher degree of the downward Taylor approximations and of the polynomial \(P(x)\) in (2) (for instance, see [10] and [12]).
Several more ideas to solve the above problem are proposed and considered below under the names of Methods A-D. In the following, the numbers \(c_{k}\) and \(d_{k}\) are those defined in Proposition 5.
Method A
If \(\delta < \frac{\pi}{2}\), find the smallest \(k\in \mathbb{N}_{0}\) such that \(d_{k}\in (\delta,\frac{\pi}{2} )\). Then \(\widehat{k}=k\).
Note that Method A assumes solving a transcendental equation of the form \(\cos {x}=\underline{T}_{ 4k+2}^{ \cos ,0}(x)\) that requires numerical methods.
Method B
If \(\delta < \frac{\pi}{2}\), find the smallest \(k\in \mathbb{N}_{0}\) such that \(c_{k}\in (\delta, \frac{\pi}{2} )\). Then \(\widehat{k}=k\).
Method C
If \(\delta < \frac{\pi}{2}\), find the smallest \(k\in \mathbb{N}_{0}\) such that \(\underline{T}_{ 4k+2}^{ \cos,0}(\delta ) \geq 0\). Then \(\widehat{k}=k\).
Method D
2.1 An algorithm based on the natural approach method
- (i)
\(a_{i}(x) =-\beta_{i}x^{p_{i}}(\sin x)^{q_{i}}(\cos x)^{r_{i}}\geq \beta_{i}x^{p_{i}} (\underline{T}^{\sin,0}_{4s_{i}+3}(x) )^{ q_{i}} (-\overline{T}^{\cos,0}_{4k_{i}}(x) )^{ r_{i}}\),
- (ii)
\(a_{i}(x) =-\beta_{i}x^{p_{i}}(\sin x)^{q_{i}}(\cos x)^{r_{i}}\geq\beta_{i}x^{p_{i}} (-\overline{T}^{ \sin,0}_{4s_{i}+1}(x) )^{ q_{i}} (\underline{T}^{ \cos,0}_{4k_{i}+2}(x) )^{ r_{i}}\),
- (iii)
\(a_{i}(x) =-\beta_{i}x^{p_{i}}(\sin x)^{q_{i}}(\cos x)^{r_{i}}\geq - \beta_{i}x^{p_{i}} (\overline{T}^{\sin,0}_{4s_{i}+1}(x) )^{ q_{i}} (\overline{T}^{ \cos,0}_{4k_{i}}(x) )^{ r_{i}}\).
We present the following characteristic [28, 29] for the Natural Approach algorithm.
Theorem 8
The Natural Approach algorithm is correct.
Proof
Every step in the algorithm is based on the results obtained from Lemmas 1-4 and Proposition 5. Hence, for every input instance (i.e., for any MTP function \(f(x)\) over a given interval \({\mathcal {I}}\subseteq (0, \pi/2 )\)), the algorithm halts with the correct output (i.e., the algorithm returns the corresponding polynomial). □
3 Some applications of the algorithm
We present an application of the Natural Approach algorithm in the proof (Application 1 - Theorem 9) of certain new rational (Padé) approximations of the function cos^{2} x, as well as in the improvement of a class of inequalities (20) by Yang (Application 2, Theorem 10).
Application 1
Bercu [7] used the Padé approximations to prove certain inequalities for trigonometric functions. Let us denote by \(( f(x) ) _{[m/n]}\) the Padé approximant \([ m/n ] \) of the function \(f(x)\).
Theorem 9
Proof
Note
Using Padé approximations, Bercu [7, 13] recently refined certain trigonometric inequalities over various intervals \(\mathcal{I}=(0,\delta ) \subseteq (0, \frac{\pi}{2})\). All such inequalities can be proved in a similar way and using the Natural Approach algorithm as in the proof of Theorem 9.
Application 2
In this example we propose the following improvement of (20).
Theorem 10
Proof
Remark on Theorem 10
Corollary 11
Corollary 12
4 Conclusions and future work
The results of our analysis could be implemented by means of an automated proof assistant [31], so our work is a contribution to the library of automatic support tools [32] for proving various analytic inequalities.
Our general algorithm associated with the natural approach method can be successfully applied to prove a wide category of classical MTP inequalities. For example, the Natural Approach algorithm has recently been used to prove several open problems that involve MTP inequalities (see, e.g., [8–12]).
Therefore, in general, for the function \(\cos^{2n}{ x}\), it is possible to determine, depending on the form of the real natural number m, the upward (downward) Taylor approximations \(\overline{T}^{ \cos^{2n}{x}, 0}_{m}(x)\) (\(\underline{T}^{ \cos^{2n} x,0}_{m}(x)\)) that are all above (below) the considered function in a given interval \(\mathcal{I}\). Such estimation of the function \(\cos^{2n}{x}\) and the use of corresponding Taylor approximations will be the object of future research.
Because for every fixed \(a\in ( 1,\frac{3}{2} ) \): \(\alpha _{1}=4(1-a)<0\) and \(\alpha _{2}=-2a<0\).
Declarations
Acknowledgements
The first and the second authors were supported in part by the Serbian Ministry of Education, Science and Technological Development, Projects TR 32023 and ON 174032, III 44006. The third author was supported by a Grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, with the Project Number PN-II-ID-PCE-2011-3-0087.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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