Isolation Intervals of the Real Roots of the Parametric Cubic Equation and Improved Complete Root Classification

The isolation intervals of the real roots of the real symbolic monic cubic polynomial $p(x) = x^3 + a x^2 + b x + c\,\,$ are found in terms of simple functions of the coefficients of the polynomial (such as: $-a$, $-a/3$, $-c/b$, $\pm \sqrt{-b}$, when $b$ is negative), and the roots of some auxiliary quadratic equations whose coefficients are also simple functions of the coefficients of the cubic. All possible cases are presented with clear and very detailed diagrams. It is very easy to identify which of these diagrams is the relevant one for any given cubic equation and to read from it the isolation intervals of the real roots of the equation. A much-improved complete root classification, addressing the signs (together with giving the isolation intervals) of the individual roots, is also presented. No numerical approximations or root finding techniques are used. Instead of considering the discriminant of the cubic, criterion for the existence of a single real root or three real roots is found as conditions on the coefficients of the cubic, resulting from the roots of the auxiliary quadratic equations. It is also shown that, if a cubic equation has three real roots, then these lie in an interval $I$ such that $\sqrt{3}\sqrt{a^2/3 - b} \le I \le 2 \sqrt{a^2/3 - b}$, independent of $c$. A detailed algorithm for applying the method for isolation of the roots of the cubic is also given and it is illustrated through examples, including the full mathematical analysis of the cubic equation associated with the Rayleigh elastic waves and finding the isolation intervals of its real roots.


Introduction
The explicit formulae for the roots of the cubic equation were discovered during the Renaissance [1] and since then, there has been significant research on the theory of equations -see, for example, the reading list [2] from 1933. For more recent and comprehensive reference, see [3]- [7]. The general cubic equation Ax 3 + Bx 2 + Cx + D = 0 can be depressed by the coordinate translation x → x − B/3A and dividing by A afterwards to obtain the depressed cubic equation x 3 + px + q = 0, for which p = (3AC − B 2 )/(3A 2 ) and q = (2B 3 − 9ABC + 27A 2 D)/(27A 3 ). The Cardano cubic formulae apply to the depressed cubic equation and yield its roots. The roots of the original equation can then be recovered by making the inverse coordinate transformation. Despite of the existence of the explicit Cardano cubic formulae, it is not always easy to apply them. For example, when the roots of a real cubic equation are all real and distinct, the Cardano formulae give the roots in a form involving cube roots of complex numbers and the cube root of a general complex number cannot be expressed in the form a + bi, where a and b involve only real radicals [6]. This is the so called casus irreducibilis. The cube root of a complex number appears in result of the necessity to introduce an imaginary number in the Cardano formulae by taking the square root of a negative number. Secondly, when the coefficients of the equation are parameters or functions of some parameters, the above renders the Cardano formulae practically inapplicable. The first remedy in such situations would be to study the root locations, that is, to find intervals which contain the roots of the equation. If a finite interval is found such that only one root of the equation lies in it, then this interval is called isolation interval. The Descartes Rule of Signs [8] from 1637 was the first step in the direction of locating the roots of an equation. This rule yields the isolation of more than one root over x > 0 (or over x < 0, when x is replaced by −x in the equation) with some indeterminacy: the number of positive real roots of an equation with real coefficients cannot be greater than the number of the variations of sign in the sequence of its coefficients (Descartes' original formulation). In 1876, Gauss [9] found a more precise formulation: the number of positive real roots (counted with their multiplicities) is equal to the number of the variations of sign or is less than that number by a positive even integer. Newton's rule [10], giving an upper limit on the number of positive and negative roots, could also be hard to apply due to need to analyse the signs of the associated parametric quadratic elements. There has also been an extensive amount of research in another direction: on the determination of the number of roots of an equation over some prescribed interval. For example, this can be done using Sturm's [11] or Budan's [12] theorems, see also [6]. The Sturm sequence for the monic cubic polynomial p(x) = x 3 + ax 2 + bx + c is where rem[p i (x)/p i+1 (x)] denotes the remainder of the division of the polynomial p i (x) by the polynomial p i+1 (x) and is the discriminant of p(x). [The product of the squares of the differences of the roots x i of a monic polynomial is called discriminant of the polynomial. For the cubic, it is Sturm's theorem [6] allows the determination of the number of real roots of p(x) = 0 between a and b. This number is equal to the excess of the number of variations of sign of p(x), p 1 (x), p 2 (x), and p 3 for x = a over the number of variations of sign of these for x = b (the vanishing terms are ignored). Cheng and Lin [13] consider a nonlinear function with n parameters and determine if none, some or all of the roots of the function lie in a specified subregion of the domain of the function for a partition of the parameter region.
Another vein in the research on the theory of equations are the root classification and the complete root classification of parametric polynomials. These refer to the provision of all possible cases of the polynomial roots and consist of the list of the multiplicities of all roots, in the case of root classification, and the root classification, together with the conditions which the equation coefficients should satisfy for each case of the root classification, in the case of complete root classification. The root classification and the complete root classification do not determine the location of the roots. For the very simple case of a depressed cubic polynomial x 3 + px + q, the complete root classification is [6], [14]: where δ (3) = −4p 3 − 27q 2 is the discriminant of x 3 + px + q and the lists in the figure brackets give the real root multiplicities. This paper proposes a method with which the isolation intervals of the roots of the symbolic monic cubic polynomial p(x) = x 3 + ax 2 + bx + c are found. The criteria for existence of a single real root or three real roots are found as conditions on the coefficients of the cubic equation resulting from the roots of some auxiliary quadratic equations. The endpoints of these isolation intervals are found as roots of the auxiliary quadratic equations or simple functions of the coefficients of the cubic: −a, −a/3, −c/b, ± √ −b (when b is negative), etc. Also presented in this paper is a much-improved complete root classification, giving the conditions (as well as the isolation intervals) for three positive roots, for two positive and one negative root, for one positive and two negative roots, for three negative roots, for a single positive root, and for a single negative root.
All results are precise -no numerical approximations or root finding algorithms have been used. One of the examples given in this work provides a thorough analysis of the parametric cubic equation associated with the Rayleigh elastic waves. The isolation intervals of its real roots are found for different values of the parameter of the equation. This illustrates how the proposed method allows one to analyze parametric cubic equations which are a very common occurrence in all branches of mathematics, sciences and engineering. The isolation intervals of the real roots of the general parametric quartic equation and its complete root classification have been studied in [15] and those of the quintic equation -in [16], and, with diminishing degree of determinacy, recursively for the general polynomial, -in [17].

The Auxiliary Quadratic Equations
Consider the general monic cubic polynomial In turn, the discriminant of this quadratic is Given that the leading coefficient of ∆ (3) is negative, if b > a 2 /3, then ∆ (2) < 0 for all a. Therefore, ∆ (3) < 0 for all a, all b > a 2 /3, and all c. Hence, the polynomial x 3 + ax 2 + bx + c with b > a 2 /3 for any a and for any c will have only one real root (and two complex conjugate roots). The monic cubic polynomial x 3 + ax 2 + bx + c with b < a 2 /3 may have either one real root or three real roots (they are distinct if ∆ (3) > 0 and there is a double root if ∆ (3) = 0). To determine which of these occurs, one needs to solve the inequality ∆ (3) ≥ 0. Firstly, consider the quadratic equation ∆ (3) = 0, that is This is the first auxiliary quadratic equation. Its roots are: where Clearly, for any a and b < a 2 /3, when c 2 < c < c 1 , the cubic polynomial x 3 + ax 2 + bx + c will have three distinct real roots (as its discriminant ∆ (3) will be positive).
For any a and b < a 2 /3, when c = c 1 or c = c 2 , the discriminant ∆ (3) of the cubic polynomial x 3 + ax 2 + bx + c will be zero and, hence, the polynomial will have three real roots, two of which equal. The roots in this case can be easily determined using the Viète formulae for the cubic equation x 3 + ax 2 + bx + c = 0 in the case of a double root µ i and a simple root ξ i (with i = 1 when c = c 1 and i = 2 when c = c 2 ), that is, using 2µ i + ξ i = −a, µ 2 i + 2µ i ξ i = b, and µ 2 i ξ i = −c i (for i = 1, 2). The first formula gives ξ i = −a − 2µ i . Substituting into the second, leads to This is the second auxiliary quadratic equation.
In the regime b ≤ a 2 /3, the real roots of equation (11), namely, the double root µ 1,2 of the cubic with c = c 1,2 , are The corresponding simple real root of the cubic with c = c 1,2 is Finally, for any a and b = a 2 /3, one has c 1 = c 2 = a 3 /27. The cubic polynomial with c = a 3 /27 will be exactly (x + a/3) 3 and it will have a triple real root −a/3. For any other value of c, when b = a 2 /3, one has ∆ (3) = −(a 3 − 27c) 2 /27, which is negative. The cubic in this case will have only one real root −a/3 + 3 a 3 /27 − c (found by completing the cube). Note that equation (11) is nothing else but the equation for the critical points of the cubic polynomial x 3 +ax 2 +bx+c. The two "extreme" cubics, namely x 3 +ax 2 +bx+c 1 and x 3 + ax 2 + bx + c 2 , are such that the graph of each of them is tangent to the abscissa at the double root, that is, for the "extreme" cubics the local extrema (the maximum of the cubic with c = c 2 and the minimum of the cubic c = c 1 , both for b < a 2 /3) and, also, the saddle of the cubic with b = a 2 /3 and c = a 3 /27 lie on the abscissa. Note also that the free terms of the two "extreme" cubics are c i = c − p(µ i ).
The cubic with c = c 0 = −2a 3 /27 + ab/3 has three real roots. These are ρ 0 = −a/3 (which is the first coordinate projection of the inflection point of this particular cubic as well as of the general one) and ρ 1,2 which are equidistant from ρ 0 and are given by Note that the critical points µ 1,2 = −a/3±( √ 3/3) a 2 /3 − b of the cubic polynomial are equidistant from its inflection point. In the case of a cubic equation with three real roots, the length of the interval where all roots are, varies between its minimum value, achieved when c = c i (i = 1, 2), namely between |µ i − ξ i | = √ 3 a 2 /3 − b and its maximum value, 2 a 2 /3 − b achieved when c = c 0 , see also [18]. That is, if a cubic equation has three real roots, then the length of the interval I which contains all three roots satisfies -independent of the equation parameter c. This "root harness" is another constraint on the roots of the cubic and works in conjunction with the isolation intervals of the roots.

The Method
The essence of the method, based on the ideas of [17], is to re-write the cubic equation x 3 + ax 2 + bx + c = 0 as and seek the intersection points of the "sub-cubic" is that it has a simple root at −a and a double root at zero. The analysis of the intersection points, that is, the real roots of the cubic equation, is done by studying x 2 (x + a) in specific ranges of a and the straight line −bx − c in specific ranges of b, while allowing the variation of the free term c, namely, allowing the straight line to "slide" vertically and, in this process, reveal the different root scenarios. The isolation intervals of the real roots of the cubic can be read graphically by analysing the position of the roots relative to a number of fixed points from a set of five suitable parallel straight lines. These straight lines are determined as follows.
Firstly, all these straight lines have the same slope −b as the straight line −bx − c on the right-hand side of (16). Two of these straight lines are the ones which are tangent to the graph of the lefthand side x 2 (x + a) of (16). They are given by −bx − c 1,2 and they intersect the ordinate at −c 1,2 .
Another straight line of this set is the one that goes through the inflection point of the cubic which coincides with the inflection point of the left-hand side Another straight line is −bx − ab. This line passes through the point at which The final straight line is the separatrix −bx. This corresponds to a cubic with c = 0, that is x 3 + ax 2 + bx. Such cubic has a zero root and two more roots which are given by the roots of the third auxiliary quadratic equation: Note that if b > a 2 /4, the roots λ 1,2 are not real. This means that x 2 (x+a) intersects the straight line −bx only once -at the origin. Alternatively, if b ≤ a 2 /4, then λ 1,2 are both real and hence x 2 (x + a) intersects the straight line −bx, except at the origin, at two more points: one in the first quadrant and one in the third, should b < 0, and one in the second quadrant and one in the fourth, should b > 0 (if b = 0, the straight line −bx is the abscissa itself and it intersects x 2 (x + a) at −a and 0). Hence, if λ 1,2 are not real, that is, if b > a 2 /4, then c 1 and c 2 will have the same sign. This situation is shown on Figures 14 and 15. Note that µ 1,2 ar real only when b ≤ a 2 /3 and the straight lines −bx − c 1,2 exist only in these cases. These are the different regimes for the coefficient b: (1) b < −a 2 /9. Thus, −ab < −c 0 < 0 for a < 0 ( Figure 4) and 0 < −c 0 < −ab for a > 0 ( Figure 5). For both, µ 1,2 are real. The cubic can have either three real roots (for c 2 ≤ c ≤ c 1 ) or only one real root.
(5) 2a 2 /9 < b ≤ a 2 /4. Thus λ 1,2 are both real and c 1 and c 2 have opposite sign: −c 1 < 0 and −c 2 > 0. Also, −c 0 and −ab have the same sign: they are both positive if a is negative ( Figure 12) and both negative if a is positive ( Figure  13). For both µ 1,2 are real. The cubic can have either three real roots (for c 2 ≤ c ≤ c 1 ) or only one real root.
(6) a 2 /4 < b ≤ a 2 /3. Thus λ 1,2 are not real and c 1 and c 2 have the same sign: both are positive if a < 0 ( Figure 14) and both negative if a > 0 ( Figure  15). For both µ 1,2 are real. The cubic can have either three real roots (for c 2 ≤ c ≤ c 1 ) or only one real root.
(7) a 2 /3 < b. Now µ 1,2 are not real. In this case, the cubic can have only one real root for all a and all c - Figure 16 for a < 0 and Figure 17 for a > 0.
Each of these regimes of b is studied for a < 0 (even numbered Figures, starting with Figure 4) and a > 0 (odd numbered Figures, starting with Figure 5). The depressed cubic (with a = 0) is studied separately -it is on Figures 1, 2, and 3. Note also that the 5 parallel straight lines look like the staves and the intersection points between them and left-hand side x 2 (x + a) of (16) look like musical notes. A Pythagorean musical analogy (linking proportion to harmony) can be found and the different cubic equations could be endowed with individual tunes -see [15] for the quartic equation.
4 Algorithm for applying the method (i) For the given cubic x 3 + ax 2 + bx + c, use a and b to determine which of the above ranges (1) to (7) for b applies. This will also put the numbers in the list −c 1 , −c 0 , −c 2 , −ab, 0 in increasing order (note that −c 1 ≤ c 0 ≤ c 2 always).
(iii) Solve the three auxiliary quadratic equations (8), (11), and (18). This will provide c 1,2 , µ 1,2 , and λ 1,2 , respectively. Find also ξ 1,2 from (13).  (viii) In the case of b < 0 and c > 0, the proposed method does not allow one to bind the smallest root x 3 from below. This situation is realized on Figure 1 and Figures  4 to 7. Also, in the case of b < 0 and c < 0, the biggest root x 1 cannot be bound from above. This can be seen on the same Figures. In order to find the isolation intervals of all roots, in these two cases one needs to find a polynomial root bound: an upper bound B U and a lower bound B L . This could be any of the many existing root bounds [3], [4], [6], [7] (which, depending on the coefficients of the polynomial, perform differently). For example, a root bound could be the bigger of 1 and the sum of the absolute values of all coefficients (the Cauchy bound), or one could use the narrower bound [19] which is the bigger of 1 and the sum of the absolute values of all negative coefficients. The bound used in this paper is [6]: 1 + k √ H, where H is the biggest absolute value of all negative coefficients in x 3 + ax 2 + bx + c and k = 1 if a < 0, k = 2 if a > 0 and b < 0, and k = 3 if a > 0 and b > 0, and c < 0 (if a, b, and c are all positive, the upper root bound is zero).
(ix) The "root harness" (15) provides an additional constraint on the roots: the distance from, say, the biggest root to the smallest root is not smaller than √ 3 a 2 /3 − b and not bigger than 2 a 2 /3 − b. This narrows down the isolation intervals of the roots. 8

Isolation Intervals of the Roots of the General Cubic
The isolation intervals of the roots of the parametric cubic polynomial are explicitly given in the Figures in this Section. All possible cases are presented. Plotted on each Figure is the graph of x 3 + ax 2 for a = 0 (depressed cubic - Figures 1 to 3), for a < 0 (even numbered Figures from 4 to 16), and for a > 0 (odd numbered Figures from 5 to 17), a representative example graph of −bx − c, for which b is in its relevant range (depending on a) and for some c (note that the free term c can take any value), together with the set of straight lines with slope −b, as described in Section 3. These are: −bx − c 1,2 (when c 1,2 are real, i.e. for b ≤ a 2 /3, the cubic has three real roots if c 2 ≤ c ≤ c 1 ), −bx − c 0 , −bx − ab, and, when relevant, the separatrix −bx.

Improved Complete Root Classification for the General Cubic Polynomial
Excluding the case of a zero root (which occurs when c = 0 and will not be considered), all possible cases of the Improved Complete Root Classification of the cubic polynomial, with their conditions, relevant Figures, and corresponding root isolation intervals, are presented in the following six tables: (III) Two Positive Roots (0 < x 2 ≤ x 1 ) and One Negative Root ( (IV) One Positive Root (x 1 > 0) and Two Negative Roots ( (V) One Positive Root (x 1 > 0) and a Pair of Complex-Conjugate Roots b a c Figure  Root Isolation (VI) One Negative Root (x 1 < 0) and a Pair of Complex-Conjugate Roots b a c Figure  Root Isolation If one is not interested in the root isolation intervals, recalling that (as only those are needed), the improved complete root classification can be summarized as follows:

(III) Two Positive Roots and
One Negative Root a b c (IV) One Positive Root and Two Negative Roots (V) One Positive Root and a Pair of Complex-Conjugate Roots a b c (VI) One Negative Root and a Pair of Complex-Conjugate Roots a b c Consider the equation One has: a = 3, b = −1/2 and c = −4.
(ii) From (10) and (14) and from using ρ 0 = −a/3, one also finds: Figure 5). These are spread over an interval of length 3.7071 -the upper bound of the "root harness". The cubic equation with c = c 2 = −5.0203 has the double root x 2,3 = µ 2 = −2.0801 and the root x 1 = ξ 2 = 1.1602 ( Figure 5). The distance between these roots is exactly 3.2403 -the lower bound of the "root harness".

Another Example -Rayleigh Waves
The Rayleigh waves are elastic surface waves. These propagate on the surface of solids (decaying exponentially inside the solids). Some seismic waves are Rayleigh waves. A Rayleigh wave is a superposition of two waves propagated independently [20]. These are a combination of compression and dilation, resulting in elliptical motion. The equation of motion is [20] where u = u l + u t is the displacement vector with u l being the displacement in the direction of propagation (the co-called longitudinal wave) and u t is the displacement in a plane, perpendicular to the direction of propagation (transverse wave) [20]. In (21), c is either c l or c t -the velocities of the two waves (often referred to as longitudinal and transverse speeds of sound) [20]. The relationship between these two speeds can be determined as follows. Consider the modulus of hydrostatic compression K (also known as bulk modulus) and the modulus of rigidity µ (also known as shear modulus). Both K and µ are always positive. These are connected [20] through the Lamé coefficient λ, that is: λ = K − 2µ/3. The Young modulus E is given by [20]: E = 9Kµ/(3K + µ).