Appendix 1. Cardano s Method

Transcription

1 Appendix 1 Cardano s Method A1.1. Introduction This appendix gives the mathematical method to solve the roots of a polynomial of degree three, called a cubic equation. Some results in this section can be found, for instance, in [ART 04]. As a useful extension, we also give the methodology to determine the roots of a polynomial of degree four, called a quartic equation. The roots of a cubic equation, like those of a quartic equation, can be found algebraically. It can be shown that this property is no more valid, in general, for a quintic equation (equation of fifth-order) or equations of higher degrees. This is known as the Abel Ruffini theorem, which was first published in 184. Referring to the French dictionary Le Robert, the complete method for the general resolution of a cubic is probably due to Tartaglia (Niccolo Fontana, , also called Tartaglia) from his works concluded in 157, based on the first approach of Gerolamo Cardano ( ). In 159, Tartaglia revealed his method to Cardano on the condition that Cardano would never reveal it. Some years later, Cardano learned about Ferro s prior work and published Ferro s method in his book Ars magnasive de regulis algebraicis, liner unus in These works, which are produced by the Italian mathematics school, are also based on: Rafael Bombelli ( ) was the one who finally managed to address the resolution of polynomial equations with imaginary numbers.

2 80 Reinforced Concrete Beams, Columns and Frames Lodovico Ferrari ( ), as Cardano s student, gave the solution of a quartic equation, which was published in one chapter of Ars magnasive de regulis algebraicis, liner unus written by Cardano in Scipione del Ferro ( ) first discovered the method to solve the canonical form of a cubic equation (x + px + q = 0), the first step toward the more general method of a cubic equation. In the following, we use the mathematical function sgn(x) for the sign function of a real x, and we also use: x = x 1/.sgn(x) with x =sgn(x).x [A1.1] A1.. Roots of a cubic function method of resolution A1..1. Canonical form We consider the cubic equation with real coefficients: gt () = at + bt + ct+ d= 0witha 0 [A1.] Each term can be divided by the first coefficient, leading to: t + b a t + c a t + d a = 0 [A1.] This cubic equation can be factorized as: (t + b a ) + ac - b a which is known as the canonical form: (t + b a ) + a d + b - 9abc a = 0 [A1.4] f(x) = x + px + q = 0 by setting x = t + b a ; p = ac - b a and q = a d + b - 9abc a [A1.5] This canonical equation is solved from the introduction of two numbers y and z such that x= y + z are roots of the cubic equation f(x) = 0, by imposing some constraint equalities:

3 Appendix 1 81 y + z = -q yz = - p f(y + z) = 0 y + z = -q yz = - p (y + z ) + (p + yz)(y + z) + q = 0 y + z = -q yz = - p [A1.6] The following change of variable can be chosen as: Y = y Z = z and then Y + Z = - q (YZ) 1/ = - p [A1.7] Knowing the sum and the product of Y and Z, these numbers are necessarily the roots U 1 and U of the quadratic equation: U + qu - p = 0. If U 1 and U are known, then y and z are calculated from (e kip U 1 ) 1/ and (e kip U ) 1/, which should be associated by a pair such that the product yz is a real number. We can distinguish several possible cases using the discriminant concept, depending on the sign of D = q + 4p or equivalently, depending on the sign of 4p + q. A1... Resolution one real and two complex roots Case 4p + q > 0 (one real and two complex conjugate roots for f(x) = 0). This case includes the case p = 0. In this case, both U 1 and U are real numbers: U 1 = - q + q + 4p = - q + q 4 + p and U = - q - q 4 + p [A1.8]

4 8 Reinforced Concrete Beams, Columns and Frames To have the product yz as a real number, the possible couples (y, z) (or equivalently (z,y)) are then: ( U 1 ; U ), ( j U 1 ; j U ), ( j U 1 ; j U ) [A1.9] where j denotes the complex number that is the cubic root of unity. The solutions in x are then: x 1 = - q + q 4 + p + - q - q 4 + p x = j - q + q 4 + p + j - q - q 4 + p with j=e iπ/ [A1.10] x = j - q + q 4 + p + j - q - q 4 + p In reinforced concrete design, we are only concerned with real solutions, and then only x 1 will be of interest, which finally leads to the root of the initial cubic equation [A1.], as: t = 1 a 1 a - a d+b -9abc - a d+b -9abc - + a d+b -9abc + (ac-b ) + a d+b -9abc + (ac-b ) b - a t = 1 a [A1.11] In the specific case p = 0, this real root is simply reduced to - a d-b +9abc - b a

5 Appendix 1 8 A1... Resolution two real roots Case 4p + q = 0 (one real and one double real roots for f(x) = 0). In this case, U 1 and U are real numbers with U 1 = U = - q = q p = U. The product yz being real, the possible couples (y, z) (or equivalently (z,y)) are given by: ( U ; U ), (j U ;j U ), (j U ;j U ) [A1.1] where j denotes the complex number that is the cubic root of unity. Using the fundamental property 1 + j + j = 0, the solutions in x are given by: simple root: x 1 = - q = q p ; double root: x = x = - q p [A1.1] The real roots of the initial cubic equation g = 0 in t (equation [A1.]) are then: t 1 = q p t = t = - q p - b a = 9a d + b - 4abc a(ac - b ) - b a - 9ad + bc = (ac - b ) and [A1.14] A1..4. Resolution three real roots Case 4p + q < 0 (three real roots for f(x) = 0). In this case, U 1 and U are conjugate imaginary numbers. It can be checked that if y is a cubic root of the complex number U 1, then z = - p y is the imaginary conjugate number of y; and x = y+z is a real number. Practically, we use the fact that a necessary and sufficient condition for two polynomial equations to have the same roots is that the coefficients of these polynomial equations are proportional. We use the equality 4 cos a - cos

6 84 Reinforced Concrete Beams, Columns and Frames a - cos a = 0 which is always true. The unknown y = cos a is a root of 4 y - y - cos a = 0. We are looking for the conditions to have both equations x + px + q = 0 and 4 y - y - cos a = 0 with proportional coefficients. In this case, if x is a root of the first cubic, kx would be the root of the second cubic, with k as a proportional coefficient, leading to the equivalence principle: 1 4 k = - p k = - q cos a for k 0 [A1.15] These two equations are equivalent to the conditions: k = - 4p k cos a = - 4q or equivalently k = k = - - 4p cos a 4q when p < 0 [A1.16] The elimination of k gives: cos a = q p - p which should be comprised between 1 and +1, leading to q p - p q 1-4p 1 with - 4p > 0. We recognize the condition that the discriminant is negative, that is 4p + q 0. In this case, and from equation [A1.16], we have the inverse relationship: k = - 4p and a = 1 Arc cos q p - p + kπ [A1.17] As the roots of the cubic equation 4 y - y - cos a = 0 are y = cos a, the roots of the cubic equation x + px + q = 0 are 1/k proportional to the previous ones (with x 1 < x < x ): x 1 = - p Arc cos cos p - p x = - p Arc cos cos p - p [A1.18]

7 Appendix 1 85 x = - p cos Arc cos q p - p The roots of the initial cubic equation [A1.] g(t) = 0 are then (with t 1 < t < t ): t 1 = a t = a t = a b - ac cos Arc cos b - ac cos Arc cos b - ac sgn(-a) (a d+b -9abc)(b -ac) sgn(-a) (a d+b -9abc)(b -ac) sgn(-a) (a d+b -9abc)(b -ac) π + 4π cos Arc cos - b a - b a - b a [A1.19] A1.. Roots of a cubic function synthesis A1..1. Summary of Cardano s method Considering the cubic equation now expressed in terms of the unknown α that is related to the dimensionless neutral axis position in reinforced concrete design: aα bα cα d = 0 [A1.0] The parameters p and q can be introduced as: 9 p= ac b and q= a d + b abc [A1.1] a a If 4p + q > 0, the unique real solution of the cubic equation is obtained from: q q p q q p b α 1 = [A1.] 4 4 a If p q 4 + < 0, the three real solutions are given by:

8 86 Reinforced Concrete Beams, Columns and Frames q Arccos + π p p p b α1 cos = a q Arccos + 4π p p p b α cos = a q Arccos p p p b α cos = a [A1.] A1... Resolution of a cubic equation example We give here a small example to illustrate our purpose. Let us consider the following cubic equation: + = 0 [A1.4] α α α The coefficients ( abcd,,, ) a =+ 1, b =, c = 1 and d =+ are identified from equation [A1.0] as: [A1.5] We calculate now p and q for determining the nature of the solutions: b 7 bc 0 p c q d b = = and = + = [A1.6] We calculate the discriminant as: 4p q = [A1.] Hence, we have three real solutions for this cubic equation. It can be relevant to compute the following number for the root calculation:

9 Appendix 1 87 q 10 Arccos = Arccos p p 7 7 [A1.8] We then compute the three roots of the cubic from equation [A1.] as: π α1 = cos 1 + = π α = cos 1 + = α = cos + = Of course, it is easy to check that α + 1 α 1 α ( )( )( ) α α α [A1.9] + =. A1.4. Roots of a quartic function principle of resolution We now consider the quartic equation with real coefficients: 4 f( x) = x + ax + bx + cx+ d = 0 witha 0 [A1.0] It can be postulated that the quartic corresponds to the beginning of the square of a second-order polynomial equation like: f(x) = x + a x + y + -y - a 4 + b x + (-ay+c)x+(d-y ) [A1.1] where y is a real number. For the following, we will assume that: - α = -y - a 4 + b and - β = d-y [A1.] y is chosen such that the second trinome of f(x), constituted of the three last terms of f(x), could be considered in a square format. It is then necessary that:

10 88 Reinforced Concrete Beams, Columns and Frames (-ay+c) - 4 -y - a 4 + b (d-y ) = 0 [A1.] We recognize a cubic equation expressed with the unknown y: y - b + ac - b - 4d y - b + abc - b + 8bd - a d - c = 0 [A1.4] which can be solved with the previous Cardano s cubic method. Let y 1, y and y be the three roots of this cubic equation. The parameters α and β will be chosen as: α = y1 + a 4 -b and β = y 1 -d if ay 1 - c 0 and β = - y1 -d if ay 1 - c < 0 [A1.5] Once the cubic root y is calculated y = y 1, the quartic function f(x) has the following form: f(x) = x + a x + y 1 - ( αx+β) = x + a - α x+y 1- β x + a + α x+y 1+ β = 0 [A1.6] Then, the determination of the roots of the quartic function is reduced to the determination of the roots of two quadratic functions.