Can you please get back on topic?

:direction:

[QUOTE=Stargate38;374879]Can you please get back on topic?

:direction:[/QUOTE]

OK, what can *you* tell us about the solvability of this septic equation, or of high-degree polynomial equations in general?

[QUOTE=NBtarheel_33;374887]OK, what can *you* tell us about the solvability of this septic equation, or of high-degree polynomial equations in general?[/QUOTE]

Here is a more-focused question for our on-topic fetishist:
[i]
Say you desire an approximation to the real root of the OP's poly to (say) 50 decimal digits of precision. Compare the cost of obtaining such in terms of basic-hardware arithmetic ops (+,-,*, assumed to require 1 pipelined cycle each, with any more-expensive ops such /, sqrt,... given in terms of cycle-equivalents) for

[A] The '"exact" solution RDS describes in terms of theta functions (I put exact in quotes because one still needs a sufficiently precise evaluation of the theta, and those don't grow on trees);

[B] A simple Newton-Raphson iteration starting from some reasonable initial guess (based on simply plotting the poly, x0 = 0.5 or 1 will serve).[/i]

[QUOTE=ewmayer;374892]

[B] A simple Newton-Raphson iteration starting from some reasonable initial guess (based on simply plotting the poly, x0 = 0.5 or 1 will serve).[/i][/QUOTE]

One must be careful with NR on high-degree polys, especially if (at least
some of) the roots are close together.

[QUOTE=R.D. Silverman;374894]One must be careful with NR on high-degree polys, especially if (at least
some of) the roots are close together.[/QUOTE]

Not an issue in the present instance, which has just 1 real root and no points of near-zero slope - and were the poly one having degenerate (or nearly so) roots one could easily switch to an alternate higher-order iterative scheme such as [url=http://hogranch.com/mayer/nr.html#high2]Halley's[/url].

[BTW, I am using the present thread as a spur to fill in some gaps in my larnin' during my spare time (basically whenever I need a break from coding or get stumped on a debug issue); currently refreshing my acquaintance with the classical methods for cubics and quartics, after which the real fun begins, the Galois-theory approach to the quintic and beyond.]

[QUOTE=ewmayer;374897]Not an issue in the present instance, which has just 1 real root and no points of near-zero slope - and were the poly one having degenerate (or nearly so) roots one could easily switch to an alternate higher-order iterative scheme such as [url=http://hogranch.com/mayer/nr.html#high2]Halley's[/url].

[BTW, I am using the present thread as a spur to fill in some gaps in my larnin' during my spare time (basically whenever I need a break from coding or get stumped on a debug issue); currently refreshing my acquaintance with the classical methods for cubics and quartics, after which the real fun begins, the Galois-theory approach to the quintic and beyond.][/QUOTE]

Think transitively.

[QUOTE=ewmayer;374892]Say you desire an approximation to the real root of the OP's poly to (say) 50 decimal digits of precision. Compare the cost of obtaining such in terms of basic-hardware arithmetic ops (+,-,*, assumed to require 1 pipelined cycle each, with any more-expensive ops such /, sqrt,... given in terms of cycle-equivalents) for

[A] The '"exact" solution RDS describes in terms of theta functions (I put exact in quotes because one still needs a sufficiently precise evaluation of the theta, and those don't grow on trees);

[B] A simple Newton-Raphson iteration starting from some reasonable initial guess (based on simply plotting the poly, x0 = 0.5 or 1 will serve).[/QUOTE]

I expect Newton-Raphson to win there, if only because the core of a theta-function evaluation is likely to be an AGM loop which itself costs about as much as the N-R; with your cheap multiplies and costly divides, I'd be quite tempted by a pure-polynomial iteration, but my knowledge of Magma syntax isn't enough to work out what bit of homomorphism judo I need to extract the linear equations in V.1 .. V.10 from

[code]
P<x>:=PolynomialRing(Rationals());
f:=x^2*(3^5*x^5+2^4*x^4+x^3)-1;
N<theta>:=NumberField(f);
NN<xx>:=PolynomialRing(N);
V:=PolynomialRing(NN,10);
t:=V!(theta+xx);
c:=V.10 + V.1*t + V.2*t^2 + V.3*t^3 + V.4*t^4 + V.5*t^5 + V.6*t^6 + V.7*t^7 + V.8*t^8 + V.9*t^9;
Coefficients(c);
[/code]

BTW, for the OS X users out there, one useful tool I stumbled across while examining the OP's poly was this handy-dandy utility called [url=http://www.fusionmath.com/apple]Grapher[/url]. Don't know how I missed that all these years. Choose 2D mode, enter x^5+16*x^6+243*x^7-1 to the right of the y= prompt at upper left, hit <enter>, et voilà:

In that graph, for x in (-.25,+.25) the y-value looks flat at -1.0.

Could this be wrong?

[QUOTE=davar55;375033]In that graph, for x in (-.25,+.25) the y-value looks flat at -1.0.

Could this be wrong?[/QUOTE]

Could it? How about the first derivative in that interval? The second derivative? Even better - what happens to a number in that interval when you take a linear combination of its fifth, sixth, seventh powers and *subtract 1*?

I swear, cmd and Kathegetes will be the next ones to show up in this thread...:loco:

[QUOTE=NBtarheel_33;375045]Could it? How about the first derivative in that interval? The second derivative? Even better - what happens to a number in that interval when you take a linear combination of its fifth, sixth, seventh powers and *subtract 1*?

I swear, cmd and Kathegetes will be the next ones to show up in this thread...:loco:[/QUOTE]

Don't know about the insult, but my post was just a comment.
How can a seventh degree polynomial be flat over an interval?
So perhaps a larger view would show up the actual non-flat curvature?