Solving 100th order equation
library(PolynomF) x <- polynom() p <- x^100 - 2*x^99 + 10*x^50 + 6*x - 4000 z <- solve(p) z
[1] -1.0741267+0.0000000i -1.0739999-0.0680356i -1.0739999+0.0680356i -1.0655699-0.1354644i [5] -1.0655699+0.1354644i -1.0568677-0.2030274i -1.0568677+0.2030274i -1.0400346-0.2687815i ... [93] 1.0595174+0.2439885i 1.0746575-0.1721335i 1.0746575+0.1721335i 1.0828132-0.1065591i [97] 1.0828132+0.1065591i 1.0879363-0.0330308i 1.0879363+0.0330308i 2.0000000+0.0000000i
Now to check how good they are:
range(Mod(p(z)))
[1] 1.062855e-10 1.548112e+15
Not brilliant, but not too bad. Bill Venables CSIRO Laboratories PO Box 120, Cleveland, 4163 AUSTRALIA Office Phone (email preferred): +61 7 3826 7251 Fax (if absolutely necessary): +61 7 3826 7304 Mobile: +61 4 8819 4402 Home Phone: +61 7 3286 7700 mailto:Bill.Venables at csiro.au http://www.cmis.csiro.au/bill.venables/ -----Original Message----- From: r-help-bounces at r-project.org [mailto:r-help-bounces at r-project.org] On Behalf Of Gabor Grothendieck Sent: Sunday, 25 May 2008 10:09 AM To: Shubha Vishwanath Karanth Cc: r-help at stat.math.ethz.ch; Duncan Murdoch; Peter Dalgaard Subject: Re: [R] Solving 100th order equation Actually maybe I was premature. It does not handle the polynomial I tried it on in the example earlier in this thread but it does seem to work with the following very simple polynomials of order 100. At any rate it would not take long to try it on the real problem and see.
Solve(x^100 - 1, x)
[1] "Starting Yacas!"
expression(list(x == complex_cartesian(cos(pi/50), sin(pi/50)),
x == complex_cartesian(cos(pi/25), sin(pi/25)), x ==
complex_cartesian(cos(3 *
pi/50), sin(3 * pi/50)), x == complex_cartesian(cos(2 *
pi/25), sin(2 * pi/25)), x == complex_cartesian(cos(pi/10),
(root(5, 2) - 1)/4), x == complex_cartesian(cos(3 * pi/25),
sin(3 * pi/25)), x == complex_cartesian(cos(7 * pi/50),
sin(7 * pi/50)), x == complex_cartesian(cos(4 * pi/25),
sin(4 * pi/25)), x == complex_cartesian(cos(9 * pi/50),
sin(9 * pi/50)), x == complex_cartesian(cos(pi/5), sin(pi/5)),
x == complex_cartesian(cos(11 * pi/50), sin(11 * pi/50)),
x == complex_cartesian(cos(6 * pi/25), sin(6 * pi/25)), x ==
complex_cartesian(cos(13 * pi/50), sin(13 * pi/50)),
x == complex_cartesian(cos(7 * pi/25), sin(7 * pi/25)), x ==
complex_cartesian(cos(3 * pi/10), sin(3 * pi/10)), x ==
complex_cartesian(cos(8 * pi/25), sin(8 * pi/25)), x ==
complex_cartesian(cos(17 * pi/50), sin(17 * pi/50)),
x == complex_cartesian(cos(9 * pi/25), sin(9 * pi/25)), x ==
complex_cartesian(cos(19 * pi/50), sin(19 * pi/50)),
x == complex_cartesian((root(5, 2) - 1)/4, sin(2 * pi/5)),
x == complex_cartesian(cos(21 * pi/50), sin(21 * pi/50)),
x == complex_cartesian(cos(11 * pi/25), sin(11 * pi/25)),
x == complex_cartesian(cos(23 * pi/50), sin(23 * pi/50)),
x == complex_cartesian(cos(12 * pi/25), sin(12 * pi/25)),
x == complex_cartesian(0, 1), x == complex_cartesian(-cos(12 *
pi/25), sin(12 * pi/25)), x == complex_cartesian(-cos(23 *
pi/50), sin(23 * pi/50)), x == complex_cartesian(-cos(11 *
pi/25), sin(11 * pi/25)), x == complex_cartesian(-cos(21 *
pi/50), sin(21 * pi/50)), x == complex_cartesian(-((root(5,
2) - 1)/4), sin(2 * pi/5)), x == complex_cartesian(-cos(19 *
pi/50), sin(19 * pi/50)), x == complex_cartesian(-cos(9 *
pi/25), sin(9 * pi/25)), x == complex_cartesian(-cos(17 *
pi/50), sin(17 * pi/50)), x == complex_cartesian(-cos(8 *
pi/25), sin(8 * pi/25)), x == complex_cartesian(-cos(3 *
pi/10), sin(3 * pi/10)), x == complex_cartesian(-cos(7 *
pi/25), sin(7 * pi/25)), x == complex_cartesian(-cos(13 *
pi/50), sin(13 * pi/50)), x == complex_cartesian(-cos(6 *
pi/25), sin(6 * pi/25)), x == complex_cartesian(-cos(11 *
pi/50), sin(11 * pi/50)), x == complex_cartesian(-cos(pi/5),
sin(pi/5)), x == complex_cartesian(-cos(9 * pi/50), sin(9 *
pi/50)), x == complex_cartesian(-cos(4 * pi/25), sin(4 *
pi/25)), x == complex_cartesian(-cos(7 * pi/50), sin(7 *
pi/50)), x == complex_cartesian(-cos(3 * pi/25), sin(3 *
pi/25)), x == complex_cartesian(-cos(pi/10), (root(5,
2) - 1)/4), x == complex_cartesian(-cos(2 * pi/25), sin(2 *
pi/25)), x == complex_cartesian(-cos(3 * pi/50), sin(3 *
pi/50)), x == complex_cartesian(-cos(pi/25), sin(pi/25)),
x == complex_cartesian(-cos(pi/50), sin(pi/50)), x == -1,
x == complex_cartesian(-cos(pi/50), -sin(pi/50)), x ==
complex_cartesian(-cos(pi/25),
-sin(pi/25)), x == complex_cartesian(-cos(3 * pi/50),
-sin(3 * pi/50)), x == complex_cartesian(-cos(2 * pi/25),
-sin(2 * pi/25)), x == complex_cartesian(-cos(pi/10),
-((root(5, 2) - 1)/4)), x == complex_cartesian(-cos(3 *
pi/25), -sin(3 * pi/25)), x == complex_cartesian(-cos(7 *
pi/50), -sin(7 * pi/50)), x == complex_cartesian(-cos(4 *
pi/25), -sin(4 * pi/25)), x == complex_cartesian(-cos(9 *
pi/50), -sin(9 * pi/50)), x == complex_cartesian(-cos(pi/5),
-sin(pi/5)), x == complex_cartesian(-cos(11 * pi/50),
-sin(11 * pi/50)), x == complex_cartesian(-cos(6 * pi/25),
-sin(6 * pi/25)), x == complex_cartesian(-cos(13 * pi/50),
-sin(13 * pi/50)), x == complex_cartesian(-cos(7 * pi/25),
-sin(7 * pi/25)), x == complex_cartesian(-cos(3 * pi/10),
-sin(3 * pi/10)), x == complex_cartesian(-cos(8 * pi/25),
-sin(8 * pi/25)), x == complex_cartesian(-cos(17 * pi/50),
-sin(17 * pi/50)), x == complex_cartesian(-cos(9 * pi/25),
-sin(9 * pi/25)), x == complex_cartesian(-cos(19 * pi/50),
-sin(19 * pi/50)), x == complex_cartesian(-((root(5,
2) - 1)/4), -sin(2 * pi/5)), x == complex_cartesian(-cos(21 *
pi/50), -sin(21 * pi/50)), x == complex_cartesian(-cos(11 *
pi/25), -sin(11 * pi/25)), x == complex_cartesian(-cos(23 *
pi/50), -sin(23 * pi/50)), x == complex_cartesian(-cos(12 *
pi/25), -sin(12 * pi/25)), x == complex_cartesian(0,
-1), x == complex_cartesian(cos(12 * pi/25), -sin(12 *
pi/25)), x == complex_cartesian(cos(23 * pi/50), -sin(23 *
pi/50)), x == complex_cartesian(cos(11 * pi/25), -sin(11 *
pi/25)), x == complex_cartesian(cos(21 * pi/50), -sin(21 *
pi/50)), x == complex_cartesian((root(5, 2) - 1)/4, -sin(2 *
pi/5)), x == complex_cartesian(cos(19 * pi/50), -sin(19 *
pi/50)), x == complex_cartesian(cos(9 * pi/25), -sin(9 *
pi/25)), x == complex_cartesian(cos(17 * pi/50), -sin(17 *
pi/50)), x == complex_cartesian(cos(8 * pi/25), -sin(8 *
pi/25)), x == complex_cartesian(cos(3 * pi/10), -sin(3 *
pi/10)), x == complex_cartesian(cos(7 * pi/25), -sin(7 *
pi/25)), x == complex_cartesian(cos(13 * pi/50), -sin(13 *
pi/50)), x == complex_cartesian(cos(6 * pi/25), -sin(6 *
pi/25)), x == complex_cartesian(cos(11 * pi/50), -sin(11 *
pi/50)), x == complex_cartesian(cos(pi/5), -sin(pi/5)),
x == complex_cartesian(cos(9 * pi/50), -sin(9 * pi/50)),
x == complex_cartesian(cos(4 * pi/25), -sin(4 * pi/25)),
x == complex_cartesian(cos(7 * pi/50), -sin(7 * pi/50)),
x == complex_cartesian(cos(3 * pi/25), -sin(3 * pi/25)),
x == complex_cartesian(cos(pi/10), -((root(5, 2) - 1)/4)),
x == complex_cartesian(cos(2 * pi/25), -sin(2 * pi/25)),
x == complex_cartesian(cos(3 * pi/50), -sin(3 * pi/50)),
x == complex_cartesian(cos(pi/25), -sin(pi/25)), x ==
complex_cartesian(cos(pi/50),
-sin(pi/50)), x == 1))
On Sat, May 24, 2008 at 8:56 AM, Shubha Vishwanath Karanth
<shubhak at ambaresearch.com> wrote:
Was also wondering which theoretical method is used to solve this problem? Thanks, Shubha Karanth | Amba Research Ph +91 80 3980 8031 | Mob +91 94 4886 4510 Bangalore * Colombo * London * New York * San Jos? * Singapore * www.ambaresearch.com -----Original Message----- From: Gabor Grothendieck [mailto:ggrothendieck at gmail.com] Sent: Saturday, May 24, 2008 6:13 PM To: Peter Dalgaard Cc: Shubha Vishwanath Karanth; r-help at stat.math.ethz.ch; Duncan Murdoch Subject: Re: [R] Solving 100th order equation On Sat, May 24, 2008 at 8:31 AM, Peter Dalgaard <p.dalgaard at biostat.ku.dk> wrote:
Shubha Vishwanath Karanth wrote:
To apply uniroot I don't even know the interval values... Does numerical methods help me? Or any other method? Thanks and Regards, Shubha -----Original Message----- From: Duncan Murdoch [mailto:murdoch at stats.uwo.ca] Sent: Saturday, May 24, 2008 5:08 PM To: Shubha Vishwanath Karanth Subject: Re: [R] Solving 100th order equation Shubha Vishwanath Karanth wrote:
Hi R, I have a 100th order equation for which I need to solve the value for x. Is there a package to do this? For example my equation is: (x^100 )- (2*x^99) +(10*x^50)+.............. +(6*x ) = 4000 I have only one unknown value and that is x. How do I solve for this?
uniroot() will find one root. If you want all of them, I don't know what is available. Duncan Murdoch
polyroot() is built for this, but it stops at 48th degree polynomials, at least as currently implemented. Not sure that it (or anything else) would be stable beyond that limit. YACAS perhaps?
Unfortunately yacas does not seem to be able to handle it:
library(Ryacas)
x <- Sym("x")
Solve((x^100 )- (2*x^99) +(10*x^50)+(6*x ) - 4000 == 0, x)
[1] "Starting Yacas!" expression(list()) Simpler one works ok:
Solve(x^2 - 1, x)
expression(list(x == 1, x == -1))
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