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[Axiom-developer] [#12 radicalSolve fails to find all roots ?]
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From: |
Bill Page |
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Subject: |
[Axiom-developer] [#12 radicalSolve fails to find all roots ?] |
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Date: |
Mon, 17 Jan 2005 22:48:17 -0600 |
??changed:
-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
-[9 more lines...]
These are NOT bugs! But the following may be! Consider the equation $z^n=1$ for
$n=7$
\begin{axiom}
radicalSolve(z^7=2)
\end{axiom}
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
??changed:
-http://mathworld.wolfram.com/ConstructiblePolygon.html
-http://mathworld.wolfram.com/TrigonometryAngles.html
- http://mathworld.wolfram.com/ConstructiblePolygon.html
- http://mathworld.wolfram.com/TrigonometryAngles.html
??changed:
-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
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
??changed:
-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
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
??changed:
-for n = 9, 18, the solutions are expressible in radicals only if radicals of
for $n = 9, 18$, the solutions are expressible in radicals only if radicals of
??changed:
-expansion for (-1)^(1/7) that Vladimir gave involves radicals of complex
expansion for $(-1)^{1/7}$ that Vladimir gave involves radicals of complex
??changed:
-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!
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!
??changed:
-(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
- +----------+ +----------+
- | +---+ | +---+
-[60 more lines...]
\begin{axiom}
radicalSolve(z^9=1,z)
radicalSolve(z^7=3)
radicalSolve(z^7=1.)
radicalSolve(z^6+z^5+z^4+z^3+z^2+z+1=0)
\end{axiom}
--removed:
------------------
-
-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)}}
-[88 more lines...]
??changed:
-Obviously, all the roots of the equation z^7 = 1 can be expressed in
Obviously, all the roots of the equation $z^7 = 1$ can be expressed in
??changed:
-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 +
-[23 more lines...]
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--
Use radicalSolve if you want your solutions expressed in
terms of radicals.
??changed:
--> radicalSolve(z^7=1, z)
-
- [z= 1]
\begin{axiom}
radicalSolve(z^7=1, z)
\end{axiom}
??changed:
--> 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
-[4 more lines...]
\begin{axiom}
for i in 1..20 repeat print([i,#radicalSolve(z^i=1,z)])
\end{axiom}
::
[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
--
forwarded from http://page.axiom-developer.org/zope/mathaction/address@hidden