On an Improvement of the Numerical Application for Cardano's Formula in Mathematica Software

Abstract

-The aim of this paper is to develop a program for determining, by symbolic description, the roots of each real cubic polynomial on the basis of the well known Mathematica software. We have obtained a program completely satisfying our expectations and even more. For example, for many tested cases of the cubic polynomials, on the way of comparing the description of the roots of these polynomials received by using our program with their trigonometric form obtained as a result of geometric discussion or respective trigonometric transformations, we have got some new attractive relations of algebraic and trigonometric nature. By applying our elaborated program we can also decide, symbolically, whether the given cubic polynomial is a Ramanujan cubic polynomial (one of two kinds). Moreover, in case of these polynomials we have got many new additional pieces of information essentially completing the facts discovered by now. Additionally, we have solved some, posed ad hoc, theoretical problem. Index Terms-cubic polynomials, Ramanujan cubic polynomials, Cardano's formula.

I. INTRODUCTION

While preparing papers [ 12 ], [ 14 ] a part of computations has been done with the aid of Mathematica software. To our surprise, the program did not manage to derive, in the way as we expected, the symbolic transformations needed for determining the zeros of polynomials, especially the real cubic and quartic polynomials. We had to make by hand a part of the final transformations. For example, for the equation x3 12x + 13 = 0 (1) Mathematica program gave us the following set of solutions q3 1

2 q3 1

2 q3 1 2

13 + ip87 2 1 + ip3

13 + ip87 2 1 ip3 13 + ip87 + r3 1

2

13 + ip87 ;

ip3 1 + ip3 r3 1 r3 1 2 2 13 + ip87 ; 13 + ip87 : We decided to change this situation which resulted in elaborating the appropriate computer procedure calculating the roots of all real cubic polynomials starting from the Cardano’s formula.

Copyright c 2016 held by the authors.

The program, developed by us, gave the following form of the solutions of equation (1): 4 cos arctan ; 4 sin arctan p87 ! 13 6 The possibly excessive optimism about the effectiveness of symbolic computations realized by our program has been suppressed by other examples. For instance, for the equation x3

12x + 11 = 0; Mathematica answer are the following solutions 1; 1 1 3p5 :

2 To the contrast our program produces the following trigonometric form of the same solutions

1 3p15 ! 1 3p15 ! 4 cos arctan ; 4 sin arctan ; 3 11 6 3 11 but only numerically Mathematica verified the equality 1 = 4 sin

arctan = 4 cos +

arctan 6 3

In the third section of this paper we present a number of such completely unexpected trigonometric relations obtained by comparing the ”model” decompositions of some cubic polynomials with the decompositions obtained by us by applying the introduced Cardano’s type formulae (implemented for our computer procedure).

It should be also emphasized that the mentioned model polynomials and their decompositions were usually obtained in result of some trigonometric transformations of the known trigonometric relations (see [ 5 ], [ 17 ], [ 18 ], [ 19 ], [ 28 ], [ 33 ], [ 34 ]). In consequence, these two different ways of decomposing the cubic polynomials resulted in many interesting equalities of trigonometric nature shedding a new light on many quantities, mysterious till now. For example, we discovered that the values of cos 2k7 , k 2 N, are strictly connected with the values of arctan(3p3), whereas values of cos 2k7 = cos 2k+72 , k 2 N, are strictly connected with the values of arctan p313 – see Section 3.

Finally, in the fourth section we describe the recently popular subject matter concerning the Ramanujan cubic polynomials. Discussion executed for the goal of preparing this paper resulted in creating a new paper devoted to these polynomials [ 2 ] – the obtained there original result is announced at the end of Section 4.

As the Authors, we want to emphasize that our aim now is to initiate the investigations on the continuation of this work where we intend to concentrate on developing the algorithm for determining the roots of polynomials belonging to the selected families of polynomials of higher orders, for which such description of the roots is known (the examples are quartic polynomials, the modified Chebyshev polynomials [ 35 ] and the selected quintics [ 8 ], [ 26 ]).

II. ZEROS OF CUBIC POLYNOMIALS – FINAL FORMULAE From the Cardano procedure for finding the (complex) zeros of real cubic polynomial (see for example [ 11 ], [ 16 ], [ 20 ], [ 28 ], [ 35 ] – two last papers consider also because of their historical connotations):

p(z) := z3 + az + b it follows that p(z) = 0 if and only if z = u + v where u 2 q3 2b + p , v 2 q3 2b p and = 2b 2 + a3 3, and where the respective complex roots (of the second and third order) are kept in mind. Let us discuss three cases with respect to the sign of discriminant .

First we assume that < 0 (which implies a < 0). Then u 2 r a 3 2 cobs ' . Hence, if we set a = 3 2, = . Finally we obtain the following

0 and 2 cos ' > 0, then 3 2z + (2 cos ') 3 = 0 , , z = 2p p For example, we obtain (below we assume that > 0, moreover, the respective values of sine and cosine functions can be found in [ 20 ], [ 21 ], [ 28 ]): z3 3 2z p2 3 = 0 8<> 2 cos 12 = pp3+2 1 ;

p , z = = > : (

p2 ; 1 p

p2 3 ; z3

3 2z , z = =

We only need to discuss two more (rather trouble-free) cases.

If > 0, then we have precisely one real zero and two conjugate complex roots z1 = u

v; z2 = ei 23 u z3 = z2; e i 23 v = (u v) + i

(u + v);

By using the cubic polynomials (and their decompositions) from papers [ 17 ], [ 18 ], [ 28 ], [ 30 ], [ 33 ], [ 34 ], [ 35 ] and by applying the discussed here Cardano’s formulae for the roots of cubic polynomials, we obtained the completely unexpected trigonometric identities (of course we present here just few selected examples denoted by A – H). Let us explain only that all the presented equalities are the consequence of the fact that the decompositions of cubic polynomials, given in the cited above papers, result, in the first place, from the appropriate trigonometric transformations (also by geometric discussion) and not from the application of the Cardano’s formulae.

We obtain 2 cos 3z2

4z

As an example we propose to examine the following polynomial 2 Y x k=0 2 cos 2k = x3 + 1 + 2 cos 3 ssiinn((9 ==22)) x2 + 2 cos 3 2 cos 4 1 + sisnin(1(3 ==2)2) x sin 8 ; sin which gives for

= 2 =7; 2 =9 the polynomial of type A and polynomial x3

3 3x + 1 = Y k=1 x 2 cos 2k =9 which can be simply generated on the basis of discussion presented item E, respectively. cos 27 = 1 cos 87

2 cos To the contrast, we note that 6 2 p7 sin 7 6 4 p7 sin 7

F)

To the contrast, we note that z3 + p3z2 3z p33 = z

cos sin

26 + 2p13 Moreover, surprisingly the following relation holds p13 4 8

2 cos 0.577727 3 13 The above polynomial is called the Perrin polynomial, also called as the Siegel’s polynomial (see [ 7 ], [ 34 ]). Constant 0 1 plays an important role in estimating the Mahler measure M (f ) of polynomials f over C, which are not reciprocal. This estimation is of the form M (f ) 0 1 and is optimal (see [ 23 ], [ 25 ]). To the contrast let us note that 0 is the only positive root of polynomial 3 + 2 1 (the same fact holds for the polynomial 5 + 1 = ( 3 + 2 1)( 2 + 1)) and := arcsin We note that the convergence of the above nested radical from Theorem 3.2 in [ 15 ] holds. Moreover, we observe that the similar equality is fulfilled also for the only positive root zk of the polynomial zk z 1, k 3, q having the form zk = k 1 + pk1 + pk1 + : : :.

In order to emphasize the importance of this section let us introduce two more decompositions of the cubic polynomials, which we have not found in literature till now and which essentially complete the decompositions presented in items F) and G).

So from (10) and (11) we can obtain the decompositions z 2 sin z 2 sin Remark III.1. In this section we have dealt with expressions of the form cos 13 arctan and sin 13 arctan . Let us notice that one can find in literature some very interesting decompositions of such expressions on the nested square roots. For example, in [ 3 ], [ 4 ] we can find the following Ramanujan formula and = z3

z = z3 and where the sequence of signs ; +; +; : : :, appearing in this nested radicals, has period 3. In case a = 44 we obtain (see equalities in examples A, C, F for possible connections): lim an = 2 + 2p21 sin n!1

V. Shevelev and R. Wituła in papers [ 1 ], [ 24 ], [ 29 ], [ 31 ] have distinguished and discussed the so called Ramanujan’s cubic polynomials and Ramanujan’s cubic polynomials of the second kind, denoted for shortness by RCP and RCP2, respectively. And all this could happen thanks to the great Indian mathematician Srinivasa Ramanujan who proposed the proof of the following equalities (see [ 22 ]): It is easy to connect the above equations with the following problem: for which cubic polynomials Q(x) (with all real roots) of the form

Q(x) = (x 1)(x 2)(x 3) = x3 + px2 + qx + r there exists a function f (p; q; r) that possesses possibly ”simple” algebraic form and for which the following equality holds p3 1 + p3 2 + p3 3 = p3f (p; q; r): Although some attempts to solve this interesting problem have been undertaken (see for example the respective Ramanujan Theorem in [ 4 ] or in the second Notebook of Ramanujan [ 22 ]), only the mentioned above V. Shevelev and R. Wituła succeeded in distinguishing the appropriate families of cubic polynomials and in describing properties of these polynomials (see also the paper [ 1 ]). Let us only recall that if the following conditions are satisifed 8 r 6= 0; < p p3r + 3 p3r2 + q = 0; (12) : 2b 2 + a3 3 < 0; where a := q p2 ; b := 2 p3 1 pq + r;

3 27 3 then Q(x) is a RCP and the following identities hold (the last expressions in all three formulae below are prepared for the algorithmic applications): p3x1 + p3x2 + p3x3 = p 6r1=3 + 3(9r pq)1=3 1=3 = sgn p

6r1=3 + 3 sgn(9r p 6r1=3 + 3 sgn(9r pq)j9r pq)j9r

pqj1=3 pqj1=3 1=3 ; or where r 2 R n f g

0 . On the other hand we know that the polynomials belonging to families RCP and RCP2 share many common analytical-algebraic properties (see [ 14 ], [ 24 ]).

Let us present now the examples of cubic polynomials with indicating the Ramanujan classes of polynomials they belong: 1 p(x) = x3 +3x2 3 p32x+1 is the RCP2 polynomial which is not the RCP one. It has the following zeros (we apply here our formulae (5)): x1 = 1 2 q and next, by applying the Cardano’s formulae (5) we get p3x1 = p323 q61 + p32 cos 31 arctan 13 q4 p32 (this identity results directly from formula (15)). It means that the following ”unexpected” polynomial identity holds Moreover, by ”numerical experiment” we find that 2 q(x) = x3 +x2 2x 1 = (x 2 cos 27 )(x 2 cos 47 )(x 2 cos 87 ) – an example of the RCP polynomial which is not the RCP2 one. 3 The polynomials from examples B) – H) presented in Section 3 are neither RCP polynomials nor RCP2 ones. Announcement

While preparing this section we have solved unexpectedly one more problem. We have proven that the polynomials (17) (18) R(x; p) := x3 + 9px2 + 23 1

V. CONCLUSION

In this paper we have presented the algorithms for determining the roots of cubic complex polynomials. The proposed algorithms generate the descriptions of these roots with the aid of radicals of trigonometric functions. The examples of testing polynomials, presented in this paper, have been originally generated by using the direct methods different than the given here algorithms. In consequence, by applying the algorithms presented here we received, the most often, the new and different symbolic descriptions of the sought roots of the given cubic polynomials. It gave us the possibility, by comparing the obtained and the testing descriptions, to reveal many new identities and relations. Additionally some conjecture arose about the possible existence of description of the values of functions cos 13 arctan , sin 13 arctan in the form of radicals of variable , where is also a radical defined on the set of rational numbers. We intend to discuss this problem in a separate paper. Moreover, let us notice that our algorithm verifies whether the given cubic polynomial belongs to the RCP or RCP2 class.

Also in a separate paper (see [ 36 ]) we intend, as we declared at the end of Introduction, to extend the discussion of the symbolic description of the roots of cubic polynomials, undertaken in this paper, for the roots of quartic polynomials and the selected polynomials of higher order. Next, we plan to adapt them numerically since we count on some better results then the ones obtained with the aid of Mathematica software.

References

[1] S. Barbero , U. Cerruti , N. Murru , M. Abrate: Identities involving zeros of Ramanujan and Shanks cubic polynomials , J. Integer Seq . 16 ( 2013 ), article 13.8 .1.

[2] B. Bajorska-Harapin ´ska, M. Pleszczyn´ski, R. Wituła, When RCP generates RCP2, that is one more property of the Ramanujan polynomials , in preparation.

[3] B.C. Berndt: Ramanujan's Notebooks , Part IV , Springer, New York 1994 .

[4] B.C. Berndt , S. Bhargava: Ramanujan - for Lowbrows , Amer. Math. Monthly 100 No 7 ( 1993 ), 644 - 656 .

[5] W.A. Beyer , J.D. Louck , D. Zeilberger: Math Bite: A generalization of a curiosity that Feynmann remembered all his life , Math. Magazine 69 No 1 ( 1993 ), 43 - 44 .

[6] B.N. Delone: Algebra (Theory of algebraic equations), in Mathematics, its subject, methods and significance , Part I , Academy of Science Press, Moscow 1956 .

[7] A. Dubickas , K.G. Hare , J. Jankauskas: There are no two non-real conjugates of a Pisot number with the same imaginary part , arXiv:1410.1600v1 [math.NT].

[8] D.S. Dummit: Solving solvable quintics , Math. Comp. 57 ( 1991 ), 387 - 401 .

[9] H.B. Dwight : Tables of Integrals and Other Mathematical Data , The MacMillan Company, New York 1961 .

[10] S.K. Gupta , W. Szyman´ski: Cubic polynomials with rational roots and critical points , College Math. J. 41 No 5 ( 2010 ), 365 - 369 .

[11] A. Heefer , T. Rothman: On remembering Cardano anew , Math. Intelligencer 36 No 4 ( 2014 ), 53 - 66 .

[12] E. Hetmaniok , P. Lorenc , M. Pleszczyn´ski, R. Wituła: Iterated integrals of polynomials , Appl. Math. Comput. 249 ( 2014 ), 389 - 398 .

[13] J.C. Lagarias: Eulers constant: Eulers work and modern developments , Bull. Amer. Math. Soc . 50 ( 2013 ), 527 - 628 ; arXiv: 1303 . 1856 .

[14] P. Lorenc , M. Pleszczyn´ski, R. Wituła: Representation of the sum of powers of the first n positive integers with multiple integral , [in review]

[15] C.D. Lynd : Using difference equations to generalize results for periodic nested radicals , Amer. Math. Monthly 121 No 1 ( 2014 ), 45 - 59 .

[16] G.P. Matvievskaya : Rene´ Descartes 1596-1650, Nauka, Moscow 1976 (in Russian).

[17] P.S. Modenov : Problems in Geometry, Mir Publishers, Moscow 1981 .

[18] S.G. Moreno , E.M. Garcia-Caballero : A Geometric proof of Morrie's Law , Amer. Math. Monthly 122 ( 2015 ), 168 .

[19] T.J. Osler : Cardan polynomials and the reduction of radicals , Math. Mag. 74 ( 2001 ), 26 - 32 .

[20] F.W.I. Olver , D.W. Lozier , R.F. Boisvert , C.W. Clark: NIST Handbook of Mathematical Functions , Cambridge Univ. Press, 2010 .

[21] A.P. Prudnikov , A.J. Bryczkov , O.I. Mariczev: Integrals and series , Elementary Functions, Nauka, Moscov 1981 (in Russian).

[22] S. Ramanujan: Notebooks (2 volumes) , Tata Institute of Fundamental Research , Bombay 1957 .

[23] A. Schinzel : Solved and unsolved problems on polynomials , 149 - 159 in Panoramas of Mathematics Banach Center Publications , vol. 34 , Warszawa 1995 .

[24] V. Shevelev : On Ramanujan cubic polynomials , South East Asian J. Math. & Math. Sci. 8 ( 2009 ), 113122 .

[25] C.J. Smyth : On the product of the conjugates outside the unit circle of an algebraic integer , Bull. London Math. Soc. 3 ( 1971 ), 169175 .

[26] B.K. Spearman , K.S. Williams : Characterization of solvable quintics x5 + ax + b , Amer. Math. Monthly 101 No. 10 ( 1994 ), 986992 .

[27] J.-P. Tignol : Galois' Theory of Algebraic Equations, World Scientific, New Jersey 2001 .

[28] R. Wituła : Complex Numbers, Polynomials and Partial Fractions Decompositions, Parts 1 , 2 and 3, Silesian University of Technology Press, Gliwice 2010 (in Polish).

[29] R. Wituła : Full description of Ramanujan cubic polynomials , J. Integer Seq . 13 ( 2010 ), article 10.5 .7.

[30] R. Wituła : On Some Applications of Formulae for Sums of Unimodular Complex Numbers , PKJS Publishers, Gliwice 2011 .

[31] R. Wituła : Ramanujan cubic polynomials of the second kind , J. Integer Seq . 13 ( 2010 ), article 10.7 .5.

[32] R. Wituła : Ramanujan type trigonometric formulae , Demonstratio Math. 45 No. 4 ( 2012 ), 779 - 796 .

[33] R. Wituła , E. Hetmaniok , D. Słota , N. Gawron´ska: Sums of the rational powers of roots of the polynomials , International Journal of Pure and Applied Mathematics 85 No 1 ( 2013 ), 179 - 191 .

[34] R. Wituła , P. Lorenc , M. Ro´ z˙an´ski, M. Szweda: Sums of the rational powers of roots of cubic polynomials, Zesz. Naukowe Politechniki S´ la¸skiej, Seria: Matematyka Stosowana z . 4 ( 2010 ), 17 - 34 , http://mat.polsl.pl/zn/zeszyty/z4/znps mat stos 04 2014 str 017- 034 .pdf.

[35] R. Wituła , D. Słota: Cardano's formula, square roots, Chebyshev polynomials and radicals , J. Math. Anal. Appl . 363 ( 2010 ), 639 - 647 .

[36] A. Wro ´bel, E. Hetmaniok, M. Pleszczyn´ski, R. Wituła: Symbolic description of roots of polynomials and their numerical implementation - better than in Mathematica software? , in preparation.