Quintic Spline

In algebra, a quintic function is a function of the form

:$g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\,$

where , , , , and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero. In other words, a quintic function is defined by a polynomial of degree five.

Because they have an odd degree, normal quintic functions appear similar to normal cubic functions when graphed, except they may possess one additional local maximum and one additional local minimum. The derivative of a quintic function is a quartic function.

Setting and assuming produces a quintic equation of the form:
:$ax^5+bx^4+cx^3+dx^2+ex+f=0.\,$
Solving quintic equations in terms of radicals (''n''th roots) was a major problem in algebra from the 16th century, when cubic and quartic equations were solved, until the first half of the 19th century, when the impossibility of such a general solution was proved with the Abel–Ruffini theorem.

Finding roots of a quintic equation

Finding the roots (zeros) of a given polynomial has been a prominent mathematical problem.

Solving linear, quadratic, cubic and quartic equations by factorization into radicals can always be done, no matter whether the roots are rational or irrational, real or complex; there are formulae that yield the required solutions. However, there is no algebraic expression (that is, in terms of radicals) for the solutions of general quintic equations over the rationals; this statement is known as the Abel–Ruffini theorem, first asserted in 1799 and completely proved in 1824. This result also holds for equations of higher degrees. An example of a quintic whose roots cannot be expressed in terms of radicals is .

Some quintics may be solved in terms of radicals. However, the solution is generally too complicated to be used in practice. Instead, numerical approximations are calculated using a root-finding algorithm for polynomials.

Solvable quintics

Some quintic equations can be solved in terms of radicals. These include the quintic equations defined by a polynomial that is reducible, such as . For example, it has been shown that

:$x^5-x-r=0$

has solutions in radicals if and only if it has an integer solution or ''r'' is one of ±15, ±22440, or ±2759640, in which cases the polynomial is reducible.

As solving reducible quintic equations reduces immediately to solving polynomials of lower degree, only irreducible quintic equations are considered in the remainder of this section, and the term "quintic" will refer only to irreducible quintics. A solvable quintic is thus an irreducible quintic polynomial whose roots may be expressed in terms of radicals.

To characterize solvable quintics, and more generally solvable polynomials of higher degree, Évariste Galois developed techniques which gave rise to group theory and Galois theory. Applying these techniques, Arthur Cayley found a general criterion for determining whether any given quintic is solvable. This criterion is the following.

Given the equation
:$ax^5+bx^4+cx^3+dx^2+ex+f=0,$
the Tschirnhaus transformation , which depresses the quintic (that is, removes the term of degree four), gives the equation

:$y^5+p y^3+q y^2+r y+s=0$,

where

:$\beginp &= \frac\\ q &= \frac\\ r &= \frac\\ s &= \frac\end$

Both quintics are solvable by radicals if and only if either they are factorisable in equations of lower degrees with rational coefficients or the polynomial , named , has a rational root in , where

:$\begin P = &z^3-z^2(20r+3p^2)- z(8p^2r - 16pq^2- 240r^2 + 400sq - 3p^4)\\ &- p^6 + 28p^4r- 16p^3q^2- 176p^2r^2- 80p^2sq + 224prq^2- 64q^4 \\ &+ 4000ps^2 + 320r^3- 1600rsq \end$

and

:$\begin \Delta = &-128p^2r^4+3125s^4-72p^4qrs+560p^2qr^2s+16p^4r^3+256r^5+108p^5s^2 \\ &-1600qr^3s+144pq^2r^3-900p^3rs^2+2000pr^2s^2-3750pqs^3+825p^2q^2s^2 \\ &+2250q^2rs^2+108q^5s-27q^4r^2-630pq^3rs+16p^3q^3s-4p^3q^2r^2. \end$
Cayley's result allows us to test if a quintic is solvable. If it is the case, finding its roots is a more difficult problem, which consists of expressing the roots in terms of radicals involving the coefficients of the quintic and the rational root of Cayley's resolvent.

In 1888, George Paxton Young described how to solve a solvable quintic equation, without providing an explicit formula; in 2004, Daniel Lazard wrote out a three-page formula.

Quintics in Bring–Jerrard form

There are several parametric representations of solvable quintics of the form , called the Bring–Jerrard form.

During the second half of the 19th century, John Stuart Glashan, George Paxton Young, and Carl Runge gave such a parameterization: an irreducible quintic with rational coefficients in Bring–Jerrard form
is solvable if and only if either or it may be written
:$x^5 + \fracx + \frac = 0$
where and are rational.

In 1994, Blair Spearman and Kenneth S. Williams gave an alternative,
:$x^5 + \fracx + \frac = 0.$

The relationship between the 1885 and 1994 parameterizations can be seen by defining the expression
:$b = \frac \left(a+20 \pm 2\sqrt\right)$
where . Using the negative case of the square root yields, after scaling variables, the first parametrization while the positive case gives the second.

The substitution , in the Spearman-Williams parameterization allows one to not exclude the special case , giving the following result:

If and are rational numbers, the equation is solvable by radicals if either its left-hand side is a product of polynomials of degree less than 5 with rational coefficients or there exist two rational numbers and such that
:$a=\frac\qquad b=\frac.$

Roots of a solvable quintic

A polynomial equation is solvable by radicals if its Galois group is a solvable group. In the case of irreducible quintics, the Galois group is a subgroup of the symmetric group of all permutations of a five element set, which is solvable if and only if it is a subgroup of the group , of order , generated by the cyclic permutations and .

If the quintic is solvable, one of the solutions may be represented by an algebraic expression involving a fifth root and at most two square roots, generally nested. The other solutions may then be obtained either by changing the fifth root or by multiplying all the occurrences of the fifth root by the same power of a primitive 5th root of unity, such as
:$\frac.$

In fact, all four primitive fifth roots of unity may be obtained by changing the signs of the square roots appropriately; namely, the expression
:$\frac,$
where $\alpha, \beta \in \$, yields the four distinct primitive fifth roots of unity.

It follows that one may need four different square roots for writing all the roots of a solvable quintic. Even for the first root that involves at most two square roots, the expression of the solutions in terms of radicals is usually highly complicated. However, when no square root is needed, the form of the first solution may be rather simple, as for the equation , for which the only real solution is

: x=1+\sqrt \left(\sqrt right)^2+\left(\sqrt right)^3-\left(\sqrt right)^4.

An example of a more complicated (although small enough to be written here) solution is the unique real root of . Let , , and , where is the golden ratio. Then the only real solution is given by

: -cx = \sqrt + \sqrt + \sqrt - \sqrt \,,

or, equivalently, by

:x = \sqrt \sqrt \sqrt \sqrt ,,

where the are the four roots of the quartic equation

:$y^4+4y^3+\fracy^2-\fracy-\frac=0\,.$

More generally, if an equation of prime degree with rational coefficients is solvable in radicals, then one can define an auxiliary equation of degree , also with rational coefficients, such that each root of is the sum of -th roots of the roots of . These -th roots were introduced by Joseph-Louis Lagrange

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