diff --git a/buglist b/buglist index 3a27e26..c9f9c40 100644 --- a/buglist +++ b/buglist @@ -159,6 +159,167 @@ and the problem exists for 11, 13, 14, 15, 17, 19 etc [19,1] <-- not good [20,20] +Subject: Re: [Q] radicalSolve fails to find all roots ? + +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 (Krzek 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=2E (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 5873353= +053, + +-----+ + 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 + ========================================================================= bug 7248: differentiation bug in D(1,z) diff --git a/changelog b/changelog index 186f782..cbdc3d6 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,5 @@ +20140613 tpd src/axiom-website/patches.html 20140613.13.tpd.patch +20140613 tpd buglist bug 7249: radicalSolve(z^7=1, z) update 20140613 tpd src/axiom-website/patches.html 20140613.12.tpd.patch 20140613 tpd buglist bug 7251: integrate(1/(1+z^4), z=0..1) :: Complex Float 20140613 tpd src/axiom-website/patches.html 20140613.11.tpd.patch diff --git a/src/axiom-website/patches.html b/src/axiom-website/patches.html index 5507487..03c5f10 100644 --- a/src/axiom-website/patches.html +++ b/src/axiom-website/patches.html @@ -4444,6 +4444,8 @@ buglist bug 7072/19 D(0^z, z) closed buglist bug 7250: print(typeOf(1)::OutputForm) Value Stack Overflow 20140613.12.tpd.patch buglist bug 7251: integrate(1/(1+z^4), z=0..1) :: Complex Float +20140613.13.tpd.patch +buglist bug 7249: radicalSolve(z^7=1, z) update