Given the quintic $$Ax^5+Bx^4+Cx^3+Dx^2+Ex+F =0$$, how would one go about solving this with a computer at disposal.
I intend to use this in an application I am writing. I only needs to go to 4 decimal places of accuracy.
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If the coefficients are known, use an optimized Newton's Method. If you're trying to find a general formula (like the quadratic formula) there is none. – JasonM Aug 04 '16 at 05:18
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2@Jason, there is a general formula, but it requires transcendental functions not usually studied in high school. Zach, practically speaking: you might consider using a modified Newton-Raphson method. I personally like (Weierstrass-)Durand-Kerner, and if your computing environment supports complex arithmetic, it is relatively easy to implement. – J. M. ain't a mathematician Aug 04 '16 at 05:20
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@J.M. Interesting, I didn't know this. I'd be interested in a link to this (I'm only an amateur). Either way, if it's transcendental then it isn't "like the quadratic formula" which is inherently an algebraic expression. – JasonM Aug 04 '16 at 05:23
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@Jason, then by your standards, casus irreducibilis of the cubic equation would not be algebraic either, since an expression that uses only real numbers will inevitably need to use trigonometric functions. (If you insist on radicals in that case, then you cannot avoid using complex numbers in your expressions even if the roots are all real.) As for the quintics: look up Bring radicals. – J. M. ain't a mathematician Aug 04 '16 at 05:27
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@J.M. Very interesting! I never thought about that – JasonM Aug 04 '16 at 05:31
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Could you point me in the direction of implementing a Newton-Raphson method? I looked at the Wikipedia page for but it was confusing. – Zach Hilman Aug 04 '16 at 12:59
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1@JasonM: See this post that summarizes related links. – Tito Piezas III Aug 05 '16 at 22:17