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    • The quartic equation for time-passengers ...



    Johan van Waveren Hudde (1628 - 1704); Dutch Mathematician - born in Amsterdam.

    Hudde studied the art of mathematics at the University of Leiden (oldest university of the Netherlands, founded 1575 by Willen van Oranje. Willem was a leader of the resistance against the Spanish influence in the Netherlands. He was killed by a catholic fanatic in 1584). Hudde made very important contributions in many areas like polynomial roots, minima and maxima etc. His manuscripts had a crucial influence on Leibniz's introduction to calculus. Hudde was the first who introduced an algebraic solution for the cubic equation (this was 1657).

  • Euler's modification of Hudde's method for the quartic equation

    • x4 + px2 + qx + r = 0. |p| > 0

    Any solution x we can write as x = 0.5*(k + h + m). k,h,m some complex numbers

    In other words:
    4x2 = (k2 + h2 + m2) + 2(kh + km + hm)
    16x4 = (k2 + h2 + m2)2 + 4(k2 + h2 + m2)(kh + km + hm) + 4(k2h2 + k2m2 + h2m2) + 8khm(k + h + m).
    Using the free dimensions of x we set: (i) k2 + h2 + m2 = -2p and (ii) khm = - q. From the quartic equation follows: (iii) k2 + h2 + m2 = p2 - 4r. So we got to the cubic resolvent of our quartic equation: t3 + 2pt2 + (p2 - 4r)t - q2 = 0. (solutions t1,t2 and t3). ⇒ k = t10.5, h = t20.5, m = t30.5. But because of (ii) not all signs can be chosen arbitrarily when generating x.

    2x1 = t10.5 + t20.5 + t30.5
    2x2 = t10.5 - t20.5 - t30.5
    2x3 = - t10.5 + t20.5 - t30.5
    2x4 = - t10.5 - t20.5 + t30.5
    /P normed polynomial x4 + ax3 + ...
    `a`b`c`d set' 0.25 1 1 1f*1_P
    `p`q`r set'(b-6f*l),(c-2*a*b-4*l),d+a*(neg c)+a*b-3*l:a*a
    /solving cubic resolvent
    .cub.s:1f,(2*p),((p*p)-4*r),neg q*q
    /all solutions of the resolvent in the complex plane
    t:.cub.r
    /find a square root of t
    tx:.complex.arootz (flip t;2)
    /condition (ii)
    if[(signum first .complex.prdz tx)=signum q;tx:(1 1 -1)*/:tx]
    /final shift and solution
    sol:((neg a),0f)+0.5*flip sum flip M:(flip tx)*/:(1 1 1;1 -1 -1;-1 1 -1;-1 -1 1)

    Example: x4 - 10x3 + 34x2 + 54x - 495 = 0

    So:
    P:P%first P:1 -10 34 54 -495
    ⇒ p = -3.5 ; q = 99 ; r = -264.6875
    ⇒ resolvent: t3 - 7t2 + 1071t - 9801 = 0
    ⇒ solutions: x1 = - 3 , x2 = 5 , x3,4 = 4 ± 4.123106i = 4 ± i*170.5

    Lodovico Ferrari (1522 - 1565); Italian Mathematician - born in Bologna.

    Lodovico was an orphan since early childhood and his uncle Vincenzo took care of Lodovico; but facts are that Vincenzo looked at Lodovico as kind of equity only, and in order to get rid of him - he sent Lodovico (14 years old) to Cardano as a servant. Cardano very soon recognized Lodovico's very keen mind, but in addition Lodovico could read and write. Cardano soon freed him from all servant duties and introduced Lodovico in Latin, Greek and mathematics. Lodovico Ferrari was the first one solving the quartic equation(under Cardano's supervision). Lodovico supported Cardano in his famous dispute with Nicolas Tartaglia. Lodovico Ferrari became later a wealthy man as tax assessor to the governor of Milan. Lodovico retired from his professional life and moved back to Bologna in the year 1564, his natal city, where his widowed sister Maddalena lived. Shortly after his return Lodovico dies in the year 1565 surprisingly. His mentor Cardano (who suspects arsenic poisoning by greedy Maddalena, she is also the only beneficiery) noticed that Maddalena also did not show any sign of grief during Lodovicos funeral. Cardano seems confused about the fact that no book is left from Lodovicos possessions, too. Maddalena marries only 2 weeks later after Lodovico's death, but her husband leaves her shortly after the wedding and disappears with all her Lodovico's heritage. Maddalena will die in poverty.

  • Ferrari's method for the quartic equation

    • Issue point is the normed quartic equation: x4 + a1x3 + a2x2 + a3x + a4 = 0
    By adding to both sides 0.25a12x2 + 0.25ξ2 + (x2+0.5a1x)ξ using a "free" variable ξ we will receive on the left side a full square and (0.25a12 - a2 + ξ)x2 + (0.5a1ξ - a3)x + 0.25ξ2 - a4 on the right side. This leaves us with the cubic equation: ξ3 - a2ξ2 + (a1a3 - 4a4)ξ - (a12a4 - 4a2a4 + a32) = 0. It is not necessary to know all roots of the cubic equation, we need only one root ξ0, and the right side will always turn into: (Ux + V)2. After re-ordering we end up with the factorization: (x2 + 0.5a1x + 0.5ξ0 + Ux + V)(x2 + 0.5a1x + 0.5ξ0 - Ux - V)

    Example: 15x4 - 82x3 + 4858x2 - 26650x - 5525 = 0

    /eliminate numerical noise
    flatn:{@[x;where 1e-12>abs x;:;0f]}
    /Polynomial P:P%first P:15 -82 4858 -26650 -5525
    `a`b`c`d set' 1_P
    .cub.s:1f,(neg b),((a*c)-4*d),(4*b*d)-(c*c)+d*a*a
    /back to Renaissance
    \l cardanoTartagliaFerro.q
    /solutions t
    t:.cub.r
    /choosing a real solution ..
    zeta:first first t
    /Left side of the square
    L:1f,0.5*a,zeta
    /addressing the context of quadratic equations
    w:`.quadr.m`.quadr.n`.quadr.q
    w set' flatn ((neg b)+zeta+0.25*a*a;(neg c)+0.5*a*zeta;(neg d)+0.25*zeta*zeta)
    /quadr.equation result complex-valued r=(RE,IM)
    \l quadr.q
    /full square handling, i.e. if a = c = 0
    cofactor:1f;factor:.quadr.m;if[1e-13>factor;factor:1f;cofactor:0f]
    /"sqrt factor" undoing the normalization
    l:(sqrt factor)*0f,cofactor,neg first first .quadr.r (U,V values)

    /solutions 1 and 2
    w set' L+l
    \l quadr.q
    l12:.quadr.r

    /solutions 3 and 4
    w set' L-l
    \l quadr.q
    l34:.quadr.r


    l12 ... ± 18.02776i (= ± 5i*130.5)
    l34 ... 0.2 and 5.66666...

    Other Examples ...

    Sample 1: x4 - x - 1 = 0 : x1 = 1.220744, x2 = -0.7244919, x3,4 = -0.2481261 ± 1.033982i
    Sample 2: x4 + 13x2 + 36 = 0 : x1,2 = ± 2i , x3,4 = ± 3i
    Sample 3: x4 - 73x3 + 537x2 - 1115x + 650 = 0 : x1 = 1, x2 = 2, x3 = 5, x4 = 65
    Sample 4: x4 + 7x3 - 38x2 + 42x - 264 = 0 : x1 = 4, x2 = -11, x3,4 = ± i*60.5
    Sample 5: x4 + 2x3 + 2x2 + 10x + 25 = 0 :x1,2 = -2 ± i , x3,4 = 1 ± 2i


    Joseph Louis Lagrange (1736 - 1813); Italian/French Mathematician - born in Turin, died in Paris.

    Lagrange Lagrange was born as the oldest of 11 children (of which only 2 survived to adulthood). Lagrange (he was an autodidact, and did not have the opportunity to study with world-class mathematicians) is without any doubt one of the most gifted mathematicians of the 18th century. With the age of 16 he became professor of mathematics at the Artillery School in Turin. Mathematics appears not to be Lagrange's first choice: If I had been rich, I probably would not have devoted myself to mathematics. (Lagrange's father lost a lot of money in financial speculations) His name is written in marble regarding modern mechanics, algebra, number theory and the calculus of variations (in 1756, probably the most valuable of Lagrange's achievements). This list is by far incomplete. He is an icon(that's still an understatement) of what we call today "mathematical physics". The famous Swiss mathematician Leonhard Euler made an attempt to bring Lagrange to the Berlin Academy of Sciences. But Lagrange refused that offer (in one of his letters from 1765 he states:"It seems to me that Berlin would not be at all suitable for me while M Euler is there"). Lagrange's different route to solutions, however, shows that he was looking for different methods than those of Euler, for whom Lagrange had the greatest respect. But in 1766 Euler leaves Berlin for Saint Petersburg and Lagrange becomes director of mathematics at the Berlin Academy from 1766 to 1787. (based on the invitation of the Prussian King Frederick the Great, 1712 - 1786). In the years 1770 and 1771 (while still in Prussia) he analyzed methods solving equations of higher degrees (3,4 and higher). He showed that the roots of an algebraic equation can be expressed quite easily by the roots of the so-called resolvent, an expression introduced by Lagrange. However, Lagrange's beautiful results meant an unexpected disappointment for the traditional mathematical world: The resolvent of a cubic equation is of grade 2, the resolvent for the quartic equation is of grade 3, but the resolvent for a quintic equation is of grade 6. These results were the issue point for serious doubts regarding the general solvability of equations higher than grade 4.(at least there was no such traditional solution pattern the old mathematicians had in mind) Lagrange's results were the pre-requisite for the following statement from the ingenious mathematician Evariste Galois: There is an algebraic equation of grade 5 with rational coefficient, which is not solvable over the field R of the real numbers. In 1786 (after the death of the Prussian King Frederick) Lagrange accepted the offer of Louis XVI to move to Paris. Lagrange became member of the French Academy of Sciences. In Paris he suffered from heavy Melancholy (caused by his restlessness which damaged his nervous system) and it was ironically the French Revolution which freed him from his lethargy (at least temporarily) Many of Lagrange's colleagues were beheaded during the French Revolution , like the famous chemist and aristocrat Antoine Laurent de Lavoisier who died by the guillotine in 1794. Lagrange is filled with bitterness about Lavoisiers execution: It took them only an instant to cut off that head, but France may not produce another like it in a century. Lagrange survived the wildest time of the French Revolution, but he was not entirely noncritical, he openly qualified the writings of the revolution leader Jean-Paul Marat as "empty talk". Marat was furious about that and he tried to take revenge during the revolutionary years. Napoleon Bonaparte protected the scientific development in France (for rather military reasons). Together with Pierre-Simon Laplace, Adrien-Marie Legendre and Sylvestre Francois Lacroix, he (Lagrange) converted from the politics of the Directorate to those of Napoleon. Napoleon named Lagrange to his Legion of Honor and made him a count of the Empire in 1808. Napoleon often consulted Lagrange in technical and philosophical matters. On 3 April 1813 Lagrange was awarded the Grand Croix of the Ordre Impérial de la Réunion, but only one week later he dies. Still in 1813 he was interred in the Pantheon as a father of France.

  • Lagrange's approach for the quartic equation

    • Let be x1,....,xn complex numbers... and In the index set {1,..,n}

    Let be:(+++)

    a1 = - Σ xi and i ∈ In
    a2 = Σ xixj and i,j ∈ In and i < j
    a3 = - Σ xixjxk and i,j,k ∈ In and i < j < k
    etc..
    an = (-1)n ∏ xi and i ∈ In

    For any symmetric polynomial Θ(x1,x2,....,xn) there is such a polynomial γ(u1,u2,....,un) that Θ(x1,x2,....,xn) = γ[a1(x1,x2,....,xn),a2(x1,x2,....,xn),..., an(x1,x2,....,xn)]

    So let be now σ1,..,σ4 the roots of x4 + a1x3 + a2x2 + a3x + a4 = 0, we are looking for a cubic polynomial with roots ξ1,ξ2 and ξ3 such that

    ξ1 = σ1σ2 + σ3σ4
    ξ2 = σ1σ3 + σ2σ4
    ξ3 = σ1σ4 + σ2σ3

    Important to know at this point: If we permute the σi in σ1σ2 + σ3σ4 we always get only 3 possibly different values: these are ξ1,ξ2 and ξ3 ! (that's the key idea behind that choice)
    The expressions (ξ1 + ξ2 + ξ3) , (ξ1ξ2 + ξ1ξ3 + ξ2ξ3) and ξ1ξ2ξ3 are very basic symmetric polynomials and therefore there are such polynomials Ξ1(a1,a2,a3,a4),Ξ2(a1,a2,a3,a4) and Ξ3(a1,a2,a3,a4) and where the ai fulfill (+++) in respect to the σj.

    ξ1 + ξ2 + ξ3 = a2 = Ξ1(a1,a2,a3,a4)

    ξ1ξ2 + ξ1ξ3 + ξ2ξ3 generates expressions of the form: σi2σjσk , i,j,k ∈ I4 and i,j,k all differ. We find:
    ξ1ξ2 + ξ1ξ3 + ξ2ξ3 = Ξ2(a1,a2,a3,a4) = a1a3 - 4a4

    ξ1ξ2ξ3 generates expressions of the form σi3σjσkσh , i,j,k,h ∈ I4 and i,j,k,h all differ. We find:
    ξ1ξ2ξ3 = Ξ3(a1,a2,a3,a4) = a12a4 - 4a2a4 + a32

    This means ξ1,ξ2 and ξ3 fullfil the cubic equation: ξ3 - a2ξ2 + (a1a3 - 4a4)ξ - (a12a4 - 4a2a4 + a32) = 0

    and consequently we end up with:

    σ1σ2 + σ3σ4 = ξ1
    σ1σ2 · σ3σ4 = a4

    We then continue with ξ2 and ξ3 which leads to the solution of the quartic equation ...

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