Talk:Cubic function/Archive 3

This is an archive of past discussions about Cubic function. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page.

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Archive 8

Upgrading the page

Through my preceding edits, I have rewritten most of the page in order

to have coherent notation
to improve mathematical correctness
to be as complete as possible without repetitions

There remains three sections which need editing

Section Factorization. This section deals in fact with the case where there is a rational root. I have proposed some improvements which have been refused by DVdm. In any case the notation of this section is not coherent with the remainder of the page.
Section Bipartite cubic. This section has nothing to do with this page. If could be included in cubic curve if this terminology was usual or sourced, which is not the case. Thus I propose to remove this section.
Section Solution in terms of Chebyshev radicals. My opinion is that this section is misplaced: It would be better to include it as an example in the page Chebyshev root. In any case this section needs edition. I propose to remove it or at least to shorten it.D.Lazard (talk) 18:20, 28 October 2010 (UTC)

The rational roots section was two parts, guessing a solution, as it is often used in tutorial problems, and the deflation of the polynomial. Both are general techniques for polynomial equations and not specific for the cubic. And both stand in their own right, deflation is also used in numerical methods, not only for rational solutions. -- No opinion, only expanding the problem definition.--LutzL (talk) 12:20, 29 October 2010 (UTC)

Suggestion: insert these three formulae into the aricle pragraph Trigpnometric (and hyperbolical) method

x_{0}=-{\frac {q}{|q|}}\left|{\sqrt {-{\frac {4p}{3}}}}\cos \left({\frac {1}{3}}\arccos \left|{\sqrt {\left({\frac {1}{2}}q\right)^{2}:\left(-{\frac {1}{3}}p\right)^{3}}}\right|\right)\right|-{\frac {b}{3a}}{\text{ if }}0\leq \left({\frac {1}{2}}q\right)^{2}:\left(-{\frac {1}{3}}p\right)^{3}\leq 1\,.

x_{0}=-{\frac {q}{|q|}}\left|{\sqrt {-{\frac {4p}{3}}}}\cosh \left({\frac {1}{3}}\operatorname {arcosh} \left|{\sqrt {\left({\frac {1}{2}}q\right)^{2}:\left(-{\frac {1}{3}}p\right)^{3}}}\right|\right)\right|-{\frac {b}{3a}}{\text{ if }}\left({\frac {1}{2}}q\right)^{2}:\left(-{\frac {1}{3}}p\right)^{3}>1\,.

x_{0}=-{\frac {q}{|q|}}\left|{\sqrt {\frac {4p}{3}}}\sinh \left({\frac {1}{3}}\operatorname {arcsinh} \left|{\sqrt {\left({\frac {1}{2}}q\right)^{2}:\left({\frac {1}{3}}p\right)^{3}}}\right|\right)\right|-{\frac {b}{3a}}{\text{ if }}p>0\,.

188.127.121.100 (talk) 09:44, 31 October 2010 (UTC)stap (moved to the current discussion by D.Lazard (talk) 17:20, 31 October 2010 (UTC))

The idea is not for us to go and verify these formula's, so do you have a source for this? If not, we obviously can't take it. DVdm (talk) 19:23, 31 October 2010 (UTC)

As mentioned earlier the source is: Weisstein, Eric W. "Cubic Formula." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CubicFormula.html

However, the basic problems are the sign ambiguities and the notation discrepancy.

See formula (18): x^3 + px – q = 0 but in main Wiki-article we have x^3 + px + q = 0.

Therefore sign of q i.e. sign of square root multiplying cos, cosh and sinh is either q/|q| or – q/|q|.

Formula (80) i.e. my third line can replace 3.8.1 The case of a cubic equation with real coefficients.

Third line of formula (83) i.e. my first line can replace 3.8. Solution in terms of Chebyshev radicals.

First and second line of formula (83) can be united as done in my second line.

Meanwhile I extended the formulae to x_0 and adjusted respecting Wiki-article notation i.e. x^3 + px + q = 0.

Since -q/|q| decides a sign of whole item the absolute value signs |...| are added as well.

Stap188.127.121.100 (talk) 10:58, 2 November 2010 (UTC)

Since the equations are not in that source, they are to be treated as wp:original research (see wp:SYNTH and wp:CALC), and should not be mentioned in the article. DVdm (talk) 11:07, 2 November 2010 (UTC)

I do not undertand. Formalae (80) and (83) should be inserted into (84) giving upper 3 lines but in different notation coherent to the Wikipedia article.

For example the author uses cos^(-1) instead arccos etc. Also ½q and ±⅓p are moved under square root getting (½q)^2 and (±⅓p)^3 in order a sign ambiguity to be facilitated. Stap188.127.121.100 (talk) 14:00, 2 November 2010 (UTC)

There are other reasons for which the formulas of 188.127.121.100 may not be accepted as they are: First they use semicolon to denote division, which is no more in use in mathematics. Second, they introduce several absolute values which are unnecessary (and do not appear in Weisstein's formulas). This aspect is thus original research. My suggestion is to add to the section Trigonometric method the Weisstein's hyperbolic formulas, rewritten to be coherent with current notation and trigonometric formula. I'll soon introduce this in the page.D.Lazard (talk) 14:48, 2 November 2010 (UTC)

No problem with changing a few variables, although please be very careful with the Weisstein source as the site is known to be full of errors. Over the years I have sent quite a few emails to notify them them, but I never got a reply and not one error was ever corrected. I don't really trust it. DVdm (talk)

"Stap", the concept of original research was explained to you multiple times before. Please stop proposing to insert your private musings in the article. Thank you. DVdm (talk) 15:30, 2 November 2010 (UTC)

Checking my edition of Section Solution in terms of Chebyshev radicals, I have found a flaw in this section. To explain the flaw, I have to recall that notation $C_{\frac {1}{3}}(t)$ denotes a specific root of the equation $x^{3}-3x=t$ . Without warning, the reader may suppose that, if $t$ is real, the same is true for $C_{\frac {1}{3}}(t)$ . If this were true the formula in the section would be wrong when the equation has only one real root, as we would have $t_{0}=t_{1}\,.$ . Reading carefully the page Chebyshev cube root, it appears $C_{\frac {1}{3}}(t)$ is defined as the larger real root of $x^{3}-3x=t$ if $t$ is real and positive. For the other values of $t$ in the complex plane cut by the half real line $[-\infty ,-2]\,,$ the value of $C_{\frac {1}{3}}(t)$ is defined by analytic continuation, but this value is not defined on the half line $[-\infty ,-2]\,.$ It is not difficult to define a value on this line which makes correct the formulas in Section Solution in terms of Chebyshev radicals, but this would need to rewrite the page Chebyshev cube root. I am not willing to do this as this page is apparently an original research. Therefore I'll suppress Section Solution in terms of Chebyshev radicals and leave to administrators the choice of removing the page Chebyshev cube root.D.Lazard (talk) 10:51, 4 November 2010 (UTC)

D.Lazard, I think you are doing a great job here. Whenever you find something that is, in your opinion, original research, feel free to delete it with a clear indication in the edit summary. If someone objects, they can always easily undo the removal, provided of course they do so with a proper source. Go ahead. DVdm (talk) 11:44, 5 November 2010 (UTC)

I understood that due to D. Lazard support Weisstein’s two lines are finally added in the article respecting the concept of original research. Thanks Mr. Lazard.

But rewriting them to be coherent with current notation and trigonometric formula “–” before q is missed in spite of my warning on a problem of sign ambiguities caused by:

t^3 + pt + q = 0 (2) is commencing equation in Wikipedia article but

t^3 + pt – q = 0 (15) is commencing equation in Weisstein’s work.

A sign of q isn’t same. Therefore q should be replaced by –q wherever it determines the sign of an item as approved by means of the examples:

4x^3 + 18x^2 – 81x – 220 = 0, b/3a = – 18/3/4 = – 1.5, p = – 27, q = – 143/8 = – 17.875, u = +143/432

t^3 – 27t – 17.875 = 0, t = 2*3cos(⅓arccos(+143/432)) = +5.5 and x = 5.5 – 1.5 = 4.

Formula at article as well as Weisstein’s one are working since cosine is even function: cos(n*θ) = cos(–n*θ).

4x^3 + 18x^2 – 81x – 1026 = 0, b/3a = –18/3/4 = –1.5, p = –27, q = –1755/8 = –219.375, v = +65/16 = +4.0625

t^3 – 27t – 219.375 = 0, t = –2*3cosh(⅓arccosh(+4.0625)) = –7.5 and x = –7.5 – 1.5 = –9 instead t = +7.5 and x = 7.5 – 1.5 = 6.

4x^3 + 18x^2 + 135x – 675 = 0, b/3a = –18/3/4 = –1.5, p = +27, q = –1701/8 = –212.625, w = –63/16 = –3.9375

t^3 + 27t – 212.625 = 0. t = 2*3sinh(arcsinh(–3.9375)) = –4.5 and x = –4.5 – 1.5 = –6 instead t = +4.5 and x = 4.5 – 1.5 = 3.

I have corrected the sign before the cosh formula. The two other formulas seem correct. D.Lazard (talk) 22:42, 5 November 2010 (UTC)

Last line is confusing since it refers to cosine formula only. More suitable would be:

No this last line refer syntactically only to hyperbolic formulas which are the only one which have inequalities on the right. Nevertheless the assertion is also true for cosine formula (this has been said before). D.Lazard (talk) 22:42, 5 November 2010 (UTC)

to emphasize that p is nonzero for all of three formulas mentioning case when p = 0 and t_0 = (–q)^(1/3),

I'll precise this. D.Lazard (talk) 22:42, 5 November 2010 (UTC)

to remind the users repeating three lines of 3.3.Reduction to monic trinomial: x_k = t_k – b/3a, p in terms of a, b, c, d and q in terms of a, b, c, d

I have forgotten a sentence, I had in mind. I'll adding it. D.Lazard (talk) 22:42, 5 November 2010 (UTC)

to refer to next paragraph (Factorization) that enables remaining two roots to be obtained in terms of a, b, c, d as well.

I do not agree: the interest of these formulas is to have totally real formulas for the real roots. If one want also the complex roots for hyperbolic formulas it is better to add

2ik\pi /3

to the argument of the arsinh or the arcosh.D.Lazard (talk) 22:42, 5 November 2010 (UTC)

Perhaps removed Chebyshev radicals could be mentioned within See also.

No the page on Chebyshev radical is an original research and has not to be refereed. D.Lazard (talk) 22:42, 5 November 2010 (UTC)

Stap188.127.121.100 (talk) 11:06, 5 November 2010 (UTC)

The two other formulas seem correct

but

Let (3q/2/p)sqrt(3/p) = (q/2)(3/p)^1.5 = w for p > 0 directing towards sinh substitution:

t = 2sqrt(p/3)sinh(⅓arcsinhw) that inserted into t^3 + pt + q (2) gives:

(2sqrt(p/3)sinh(⅓arcsinhw))^3 + p*2sqrt(p/3)sinh(⅓arcsinhw) + q =

2(sqrt(p/3))^3*(4sinh^3(⅓arcsinhw) + 3sinh(⅓arcsinhw)) + q =

2(sqrt(p/3))^3*sinh(3*(⅓arcsinhw)) + q = 2(sqrt(p/3))^3*w + q = 2(sqrt(p/3))^3*(q/2)(3/p)^1.5 + q =

(p/3)^(1.5 – 1.5)*q + q = (p/3)^0*q + q = q + q = 2q ≠ 0 meaning that

(–3q/2/p)sqrt(3/p) = (–q/2)(3/p)^1.5 = w is correct. Since sinh and arcsinh are uneven functions minus can be set before 2sqrt(p/3) as well.

True, but it would be original research to discuss all variants of the formulas in the article. D.Lazard (talk) 09:55, 6 November 2010 (UTC)

If one want also the complex roots for hyperbolic formulas it is better to add 2ikπ / 3 to the argument of the arsinh or the arcosh

Does it mean that something like this will be added in the article?

I think it is unnecessary: First it would change the mathematical level of the article, as the fact that hyperbolic functions are periodic with a complex period is not well known by non mathematicians. Second these formulas are interesting only as being totally real. For complex roots, Cardano formulas are simpler to understand and to evaluate numerically. This the reason for which the section emphasizes on trigonometric formula. D.Lazard (talk) 09:55, 6 November 2010 (UTC)

Finally I wander that either E. Weisstein’s or Wikipedia commencing equation is not well known

t^3 – 3pt –2q = 0 where p = (b^2 – 3ac)/(3a)^2 and q = ½(9abc – 2b^3 –27a^2d)/(3a)^3

that enables better shaped (fraction free) formulas diminished sign ambiguity as well (see Getting Started with LaTeX, Fractions and Roots).

Stap188.127.121.100 (talk) 07:45, 6 November 2010 (UTC)

There no sign ambiguity in the article if one follows the usual convention that the square root of a positive number is positive. I agree that t^3 – 3pt –2q = 0 would give better formulas. I know that for a long time. But a choice has to be done and the fact that so many people know the discriminant as 4p^3+27q^2 and not as -108p^3+108q^2 is a good reason to not change the choice. D.Lazard (talk) 09:55, 6 November 2010 (UTC)

We can keep positive sign before both square roots but sign of q is determinative. I am still sure that negative sign should be added as can be confirmed by complete deduction procedure for p > 0:

Starting from Equation (2), $t^{3}+pt+q=0$ , let us set $t=u\sin \theta \,.$ The idea is to choose $u$ for identifying Equation (2) with the identity

4\sinh ^{3}\theta +3\sinh \theta -\sinh(3\theta )=0\,.

In fact, choosing $u=2{\sqrt {\frac {p}{3}}}$ and dividing Equation (2) by ${\frac {u^{3}}{4}}$ we get

4\sinh ^{3}\theta +3\sinh \theta +{\frac {3q}{2p}}{\sqrt {\frac {3}{p}}}=0\,.

Combining with above identity, we get

-\sinh(3\theta )={\frac {3q}{2p}}{\sqrt {\frac {3}{p}}}

and thus the roots are

t_{0}=-2{\sqrt {\frac {p}{3}}}\sinh \left({\frac {1}{3}}\operatorname {arsinh} \left({\frac {3q}{2p}}{\sqrt {\frac {3}{p}}}\right)\right)\quad {\text{if }}\quad p>0\,.

Please pay attention to negative sign before sinh(3θ) at 6th line being in last line moved before the function due to

arcsinh(–3θ) = –arcsinh(3θ)= –σ and sinh(–σ) = –sinh(σ). An example more:

Inserting p = +3 > 0 and q = –4 < 0 into upper formula we get:

$t_{0}=-2{\sqrt {\frac {3}{3}}}\sinh \left({\frac {1}{3}}\operatorname {arsinh} \left({\frac {3*(-4)}{2*3}}{\sqrt {\frac {3}{3}}}\right)\right)=-2\sinh \left({\frac {1}{3}}\operatorname {arsinh} \left(-2\right)\right)=2\sinh \left({\frac {1}{3}}\operatorname {arsinh} 2\right)=1{\text{ satisfying }}t^{3}+3t-4=0\,.$

Last can be easily evaluated by means of scientific pocket calculator, therefore I don’t share your opinion that Cardano formulas are simpler. Stap188.127.121.100 (talk) 10:45, 8 November 2010 (UTC)

OK you are right. D.Lazard (talk) 11:20, 8 November 2010 (UTC)

In the section, Trigonometric (and hyperbolic) method I edited the Reduction to a Monic Trinomial making it an actual link. Nickalh50 (talk) 12:45, 4 February 2011 (UTC)

Numerical approximation relation

can someone explain how the numerical approximation can be applied to cubic equation as pointed out in this article. Will it result in expression similar to the one mentioned? Where should we look for it? Regards. Bhushan. —Preceding unsigned comment added by 110.225.238.233 (talk) 04:13, 21 November 2010 (UTC)

This is explained, for equations of any degree, in the page root finding. This clearly implies that the coefficients have numerical values and the roots are obtained as numerical values. Note that the formulas, like those in the page are not well suited for numerical evaluation, as computing a cubic root is not easier nor faster (on a computer) than numerically computing a root of a general cubic equation. D.Lazard (talk) 10:29, 21 November 2010 (UTC)

error in the depressed cubic

in the article under the cardano method it starts with the depressed cubic x^3+px+q=0. the problem i noticed is starting the depressed cubic is x^3+px=q would not get the form in the article but x^3+px-q=0, but if i'm wrong then whatever 72.240.191.130 (talk) 03:35, 18 December 2010 (UTC)

I do not understand the problem: starting from the depressed cubic x^3+px+q=0 of from x^3+px-q=0 (which is equivalent with x^3+px=q) is only a matter of choice, as one easily pass from the formulas for one case to the formulas for the other by replacing q by -q in them. The choice which as been done in the article is the most usual.

By the way, the depressed form x^3+3p+2q is also equivalent and leads to simpler formulas (without denominators under the radicals). It is not convenient to use it here, as it seems that most readers are accustomed with x^3+px+q=0, and choosing another depressed form could be a unnecessary difficulty for them. D.Lazard (talk) 10:31, 18 December 2010 (UTC)

I've been studying the imaginary numbers and specifically the Del Ferro part right now; every time i see it mentioned it says that the depressed cubic was written in the form X^3+px=q, but if you say it's more confusing than the one in the article that sets it to -q. so thanks i just wanted some clarity. 72.240.191.130 (talk) 15:58, 18 December 2010 (UTC)

Error in the discriminant

As currently written the discriminant has an error.

delta = 18abcd -4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2

Consider the polynomial, f = x^2(x-1) = x^3 - x with coefficients a = 1, b =0, c = -1, d = 0.

delta = 0 -0 + 0 -4 * -1 - 0

delta = 4

However, f clearly has a root, x=0, with multiplicity 2, and a root x =1.

Plus, what is the syntax for a new line or hard carriage return. Double spacing is annoying. Nickalh50 (talk) 18:58, 2 May 2011 (UTC)

Check your poly: a=1, b=-1, c=0, d=0. Then Δ = 0 - 0 + 0 - 0 = 0. So there is a multiple root. Glrx (talk) 19:23, 2 May 2011 (UTC)

Likewise, with coefficients a = 1, b =0, c = -1, d = 0, there are three distinct roots: -1, 0 and 1.211.30.171.128 (talk) 14:53, 29 June 2011 (UTC)

Δ = 4, in that case. — Arthur Rubin (talk) 15:57, 29 June 2011 (UTC)

Negative discriminant

Should the page be amended from

If Δ < 0, then the equation has one real root and two nonreal complex conjugate roots

to

If Δ < 0, then the equation has one real root and two nonreal complex conjugate roots or one real root and a double root.

The previous case was inaccurate for some formulae, such as

f(x)=x^3-x+2/3SQRT(3)

f(2/SQRT(1/3))=0=f(1/SQRT(3)), but 18abcd-4b^3d+b^2c^2-4ac^3+27a^2d^2=0-0+0-4*1(-1)+27*4/27. The pair of roots is not nonreal. — Preceding unsigned comment added by 211.30.171.128 (talk) 15:32, 29 June 2011 (UTC)

You have a mistake in the discriminant formula:

18abcd-4b^{3}d+b^{2}c^{2}-4ac^{3}-27a^{2}d^{2}=0-0+0-4*1(-1)-27*4/27=0

— Arthur Rubin (talk) 15:55, 29 June 2011 (UTC)

This is because the whole capter is wrong! the correct version should be If Δ < 0, then the equation has three real roots If Δ > 0, then the equation has one real root and two nonreal complex conjugate roots

I can explain this, but it will extremely long, the hint here is, Z^n+Z*^n is a real number(can beproved by de Moivre thereom) --188.178.232.249 (talk) 23:12, 28 September 2011 (UTC)

The section is not wrong. Apparently you miss the fact that Δ and the quantity whose square root appears in the solution have opposite signs. D.Lazard (talk) 07:10, 29 September 2011 (UTC)

Then I would call it non-sense when your discriminate is not something can be directly used from your equation? fix the the equation or it is kinda useless when you have to do step to examine your roots?--188.178.232.249 (talk) 14:14, 1 October 2011 (UTC)