[Top][All Lists]
[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index]
[Axiom-developer] [#12 radicalSolve fails to find all roots ?] (new)
From: |
anonymous |
Subject: |
[Axiom-developer] [#12 radicalSolve fails to find all roots ?] (new) |
Date: |
Mon, 17 Jan 2005 21:02:17 -0600 |
Re: [Axiom-developer] [Q] radicalSolve fails to find all roots ?
From: William Sit
Subject: Re: [Axiom-developer] [Q] radicalSolve fails to find all roots ?
Date: Mon, 17 Jan 2005 16:31:52 -0500
These are NOT bugs! But the following may be! Consider the equation z^n=1 for n
= 7:
(1) -> radicalSolve(z^7=2)
(1)
7+-+ +---+7+-+ 2%pi 7+-+ 2%pi
[z= \|2 , z= \|- 1 \|2 sin(----) + \|2 cos(----),
7 7
+---+7+-+ 4%pi 7+-+ 4%pi
z= \|- 1 \|2 sin(----) + \|2 cos(----),
7 7
+---+7+-+ 6%pi 7+-+ 6%pi
z= \|- 1 \|2 sin(----) + \|2 cos(----),
7 7
+---+7+-+ 8%pi 7+-+ 8%pi
z= \|- 1 \|2 sin(----) + \|2 cos(----),
7 7
+---+7+-+ 10%pi 7+-+ 10%pi
z= \|- 1 \|2 sin(-----) + \|2 cos(-----),
7 7
+---+7+-+ 12%pi 7+-+ 12%pi
z= \|- 1 \|2 sin(-----) + \|2 cos(-----)]
7 7
Type: List Equation Expression Integer
-------------- comments
Of course, these are correct solutions by Euler's Formula. A bit surprising that
radicalSolve invokes these for z^7=2 and not for z^7=1; when n is 7, these
trignometric values are not embeddable in a tower of "solvable" extensions. That
is, these are not solutions expressible in terms of radicals (of *real* numbers)
and arithmetic alone. Put another way, the regular 7-gon is not constructible by
compass and ruler alone. From:
http://mathworld.wolfram.com/ConstructiblePolygon.html
http://mathworld.wolfram.com/TrigonometryAngles.html
A necessary and sufficient condition that a regular n-gon be constructible is
that phi(n) be a power of 2, where phi(n) is the totient function (KrĂzek 2001,
p. 34).
n = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
phi= 1 1 2 2 4 2 6 4 6 4 10 4 12 6 8 8 16 6 18 8
bad= x x x x x x x
Vladimir's "not good" values are
n = 7 11 13 14 15 17 19
So if you compare the constructible regular n-gons, you can see why Axiom's
results are reasonable: radicalSolve only finds solutions that are expressible
in terms of radicals and arithmetic operations. It did not find those for n = 15
and 17 probably (I am guessing) because at the time of implementation, these
constructions were not known (at least to the programmer). On the other hand,
for n = 9, 18, the solutions are expressible in radicals only if radicals of
*complex* numbers are allowed and Axiom found those (perhaps it shouldn't?). The
expansion for (-1)^(1/7) that Vladimir gave involves radicals of complex
numbers, as theory predicts.
When Axiom cannot find solutions, it is (presumably) a PROOF that the other
solutions are NOT solvable by radicals (using *real* numbers), or at least,
there is no known proof that it is solvable at the time of implementation. (That
is why I am surprised at the above result for z^7=2).
In other words, rather than viewing the answer for z^7=1 as a bug, we should
view the answers for z^7=2, z^7=3 (and may be even z^9=1, z^18=1) as bugs!
Still, the package should be upgraded.
-------------------
(1) -> radicalSolve(z^9=1,z)
(1)
+------------+
| +---+
+------------+ +---+ +-+ |- \|- 3 - 1
| +---+ (\|- 1 \|3 - 1) 3|------------
|- \|- 3 - 1 \| 2
[z= 3|------------ , z= -------------------------------,
\| 2 2
+------------+
| +---+
+---+ +-+ |- \|- 3 - 1 +----------+
(- \|- 1 \|3 - 1) 3|------------ | +---+
\| 2 |\|- 3 - 1
z= ---------------------------------, z= 3|---------- ,
2 \| 2
+----------+ +----------+
| +---+ | +---+
+---+ +-+ |\|- 3 - 1 +---+ +-+ |\|- 3 - 1
(\|- 1 \|3 - 1) 3|---------- (- \|- 1 \|3 - 1) 3|----------
\| 2 \| 2
z= -----------------------------, z= -------------------------------,
2 2
+---+ +---+
- \|- 3 - 1 \|- 3 - 1
z= ------------, z= ----------, z= 1]
2 2
Type: List Equation Expression Integer
(2) -> radicalSolve(z^7=3)
(2)
7+-+ +---+7+-+ 2%pi 7+-+ 2%pi
[z= \|3 , z= \|- 1 \|3 sin(----) + \|3 cos(----),
7 7
+---+7+-+ 4%pi 7+-+ 4%pi
z= \|- 1 \|3 sin(----) + \|3 cos(----),
7 7
+---+7+-+ 6%pi 7+-+ 6%pi
z= \|- 1 \|3 sin(----) + \|3 cos(----),
7 7
+---+7+-+ 8%pi 7+-+ 8%pi
z= \|- 1 \|3 sin(----) + \|3 cos(----),
7 7
+---+7+-+ 10%pi 7+-+ 10%pi
z= \|- 1 \|3 sin(-----) + \|3 cos(-----),
7 7
+---+7+-+ 12%pi 7+-+ 12%pi
z= \|- 1 \|3 sin(-----) + \|3 cos(-----)]
7 7
Type: List Equation Expression Integer
(3) -> radicalSolve(z^7=1.)
7
WARNING (genufact): No known algorithm to factor ? - 1.0
, trying square-free.
(3)
+-----+
[z= 1.0, z= 0.7818314824 6802980871\|- 1.0 + 0.6234898018 5873353053,
+-----+
z= 0.9749279121 8182360702\|- 1.0 - 0.2225209339 5631440428,
+-----+
z= 0.4338837391 1755812048\|- 1.0 - 0.9009688679 0241912624,
+-----+
z= - 0.4338837391 1755812046\|- 1.0 - 0.9009688679 0241912625,
+-----+
z= - 0.9749279121 8182360702\|- 1.0 - 0.2225209339 563144043,
+-----+
z= - 0.7818314824 6802980872\|- 1.0 + 0.6234898018 5873353052]
Type: List Equation Expression Float
(4) -> radicalSolve(z^6+z^5+z^4+z^3+z^2+z+1=0)
(4) []
Type: List Equation Expression Integer
William
-----------------
Vladimir Bondarenko wrote:
>
> Hi *,
>
> Any comments are highly appreciated on the following stuff.
> Thank you in advance.
>
> .....................................................................
>
> Obviously, all the roots of the equation z^7 = 1 can be expressed in
> radicals, and Mathematica can easily produce the explicit expressions
> in terms of radicals.
>
> Solve[z^7 == 1, z]
>
> {{z -> 1}, {z -> -(-1)^(1/7)}, {z -> (-1)^(2/7)}, {z -> -(-1)^(3/7)},
> {{z -> {z -> (-1)^(4/7)}, {z -> -(-1)^(5/7)}, {z -> (-1)^(6/7)}}
>
> To save the space, below the only example is given.
>
> FunctionExpand[ComplexExpand[-(-1)^(1/7)]]
>
> (1/2)*((1/3)*((1/2)*(-1 + I*Sqrt[7]) + ((-1 + I*Sqrt[3])*((1/2)*(-1 +
> I*Sqrt[7]) + (1/2)*(-1 - I*Sqrt[7])*((1/2)*(-1 + I*Sqrt[3]) +
> (1/4)*(-1 + I*Sqrt[3])^2)))/(2*(6 + (3/4)*(-1 + I*Sqrt[3])*(-1 +
> I*Sqrt[7]) + (1/2)*(-1 - I*Sqrt[7])*(1 + (3/4)*(-1 +
> I*Sqrt[3])^2))^(1/3)) + (1/4)*(-1 + I*Sqrt[3])^2*(6 + (3/4)*(-1 +
> I*Sqrt[3])*(-1 + I*Sqrt[7]) + (1/2)*(-1 - I*Sqrt[7])*(1 + (3/4)*(-1 +
> I*Sqrt[3])^2))^(1/3)) +(1/3)*((1/2)*(1 + I*Sqrt[7]) - ((-1 +
> I*Sqrt[3])^2*((1/2)*(-1 -I*Sqrt[7]) + (1/2)*(-1 +
> I*Sqrt[7])*((1/2)*(-1 + I*Sqrt[3]) + (1/4)*(-1 + I*Sqrt[3])^2)))/(4*(6
> + (3/4)*(-1 + I*Sqrt[3])*(-1 - I*Sqrt[7]) + (1/2)*(-1 + I*Sqrt[7])*(1
> + (3/4)*(-1 + I*Sqrt[3])^2))^(1/3)) -(1/2)*(-1 + I*Sqrt[3])*(6 +
> (3/4)*(-1 + I*Sqrt[3])*(-1 - I*Sqrt[7]) + (1/2)*(-1 + I*Sqrt[7])*(1 +
> (3/4)*(-1 + I*Sqrt[3])^2))^(1/3))) + (1/2)*((1/3)*((1/2)*(-1 +
> I*Sqrt[7]) + ((-1 + I*Sqrt[3])*((1/2)*(-1 + I*Sqrt[7]) + (1/2)*(-1 -
> I*Sqrt[7])*((1/2)*(-1 + I*Sqrt[3]) + (1/4)*(-1 + I*Sqrt[3])^2)))/(2*(6
> + (3/4)*(-1 + I*Sqrt[3])*(-1 + I*Sqrt[7]) + (1/2)*(-1 - I*Sqrt[7])*(1
> + (3/4)*(-1 + I*Sqrt[3])^2))^(1/3)) +(1/4)*(-1 + I*Sqrt[3])^2*(6 +
> (3/4)*(-1 + I*Sqrt[3])*(-1 + I*Sqrt[7]) + (1/2)*(-1 - I*Sqrt[7])*(1 +
> (3/4)*(-1 + I*Sqrt[3])^2))^(1/3)) + (1/3)*((1/2)*(-1 - I*Sqrt[7])
> +((-1 + I*Sqrt[3])^2*((1/2)*(-1 - I*Sqrt[7]) + (1/2)*(-1 +
> I*Sqrt[7])*((1/2)*(-1 + I*Sqrt[3]) + (1/4)*(-1 + I*Sqrt[3])^2)))/(4*(6
> + (3/4)*(-1 + I*Sqrt[3])*(-1 - I*Sqrt[7]) + (1/2)*(-1 + I*Sqrt[7])*(1
> + (3/4)*(-1 + I*Sqrt[3])^2))^(1/3)) +(1/2)*(-1 + I*Sqrt[3])*(6 +
> (3/4)*(-1 + I*Sqrt[3])*(-1 - I*Sqrt[7]) + (1/2)*(-1 + I*Sqrt[7])*(1 +
> (3/4)*(-1 + I*Sqrt[3])^2))^(1/3)))
>
> According to the AXIOM Book
>
> AXIOM Book> Use radicalSolve if you want your solutions expressed in
> AXIOM Book> terms of radicals.
>
> However, already for z^7 = 1 this is not so,
>
> -> radicalSolve(z^7=1, z)
>
> [z= 1]
>
> and the problem exists for 11, 13, 14, 15, 17, 19 etc
>
> -> for i in 1..20 repeat print([i,#radicalSolve(z^i=1,z)])
>
> [1,1]
> [2,2]
> [3,3]
> [4,4]
> [5,5]
> [6,6]
> [7,1] <-- not good
> [8,8]
> [9,9]
> [10,10]
> [11,1] <-- not good
> [12,12]
> [13,1] <-- not good
> [14,2] <-- not good
> [15,7] <-- not good
> [16,16]
> [17,1] <-- not good
> [18,18]
> [19,1] <-- not good
> [20,20]
>
> .....................................................................
>
> Best,
>
> Vladimir
>
> _______________________________________________
> Axiom-developer mailing list
> address@hidden
> http://lists.nongnu.org/mailman/listinfo/axiom-developer
--
William Sit
Department of Mathematics....Email: address@hidden
City College of New York................Tel: 212-650-5179
Convent Ave at West 138th Street........Fax: 212-862-0004
New York, NY 10031..Axiom, A Scientific Computation Sytem
USA............... http://www.nongnu.org/axiom/index.html
--
forwarded from http://page.axiom-developer.org/zope/mathaction/address@hidden
- [Axiom-developer] [#12 radicalSolve fails to find all roots ?] (new),
anonymous <=