Mixing
Is there a nice algorithm for testing the equality of two nested radicals?
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I'm talking about a computer algorithm in the realm of symbolic computation. I also need to be able to write the code myself as opposed to using a third-party tool.
One way I thought of was to find any polynomials $A(x)$ and $B(x)$ of which nested radicals $alpha$ and $beta$, respectively, are roots. This is easier than finding minimal polynomials, I assume.
Find their GCD $G(x)$ and make sure it isn't $1$.
Then $frac{G(x)}{x-alpha} = frac{G(x)}{x-beta} implies alpha = beta$.
This is right, right?
The expressions can get huge if this is done naively. Is there a neater way?
nested-radicals symbolic-computation
asked Jan 17 at 1:42
MackTuesdayMackTuesday
723316
$endgroup$
add a comment |
$begingroup$
I'm talking about a computer algorithm in the realm of symbolic computation. I also need to be able to write the code myself as opposed to using a third-party tool.
One way I thought of was to find any polynomials $A(x)$ and $B(x)$ of which nested radicals $alpha$ and $beta$, respectively, are roots. This is easier than finding minimal polynomials, I assume.
Find their GCD $G(x)$ and make sure it isn't $1$.
Then $frac{G(x)}{x-alpha} = frac{G(x)}{x-beta} implies alpha = beta$.
This is right, right?
The expressions can get huge if this is done naively. Is there a neater way?
nested-radicals symbolic-computation
asked Jan 17 at 1:42
MackTuesdayMackTuesday
723316
$endgroup$
$begingroup$
Why not a simple[alpha] == [beta]in Mathematica, which will give aTrueorFalse?
$endgroup$
– David G. Stork
Jan 17 at 1:44
$begingroup$
That's not exactly an "algorithm", @David, although it's a pretty useful technique.
$endgroup$
– John Hughes
Jan 17 at 1:48
$begingroup$
It's part of some software I'm writing. I can't necessarily rely on someone else's code. I've also edited the question for clarity.
$endgroup$
– MackTuesday
Jan 17 at 1:48
$begingroup$
There's a bit of literature on "denesting radicals". (I don't know this, but ...) Perhaps there's a way to wrangle nested radicals into a "canonical form" that would make them straightforward to compare.
$endgroup$
– Blue
Jan 17 at 2:31
add a comment |
$begingroup$
I'm talking about a computer algorithm in the realm of symbolic computation. I also need to be able to write the code myself as opposed to using a third-party tool.
One way I thought of was to find any polynomials $A(x)$ and $B(x)$ of which nested radicals $alpha$ and $beta$, respectively, are roots. This is easier than finding minimal polynomials, I assume.
Find their GCD $G(x)$ and make sure it isn't $1$.
Then $frac{G(x)}{x-alpha} = frac{G(x)}{x-beta} implies alpha = beta$.
This is right, right?
The expressions can get huge if this is done naively. Is there a neater way?
nested-radicals symbolic-computation
asked Jan 17 at 1:42
MackTuesdayMackTuesday
723316
$endgroup$
I'm talking about a computer algorithm in the realm of symbolic computation. I also need to be able to write the code myself as opposed to using a third-party tool.
One way I thought of was to find any polynomials $A(x)$ and $B(x)$ of which nested radicals $alpha$ and $beta$, respectively, are roots. This is easier than finding minimal polynomials, I assume.
Find their GCD $G(x)$ and make sure it isn't $1$.
Then $frac{G(x)}{x-alpha} = frac{G(x)}{x-beta} implies alpha = beta$.
This is right, right?
The expressions can get huge if this is done naively. Is there a neater way?
nested-radicals symbolic-computation
nested-radicals symbolic-computation
asked Jan 17 at 1:42
MackTuesdayMackTuesday
723316
asked Jan 17 at 1:42
MackTuesdayMackTuesday
723316
edited Jan 17 at 1:50
MackTuesday
asked Jan 17 at 1:42
MackTuesdayMackTuesday
723316
asked Jan 17 at 1:42
MackTuesdayMackTuesday
723316
asked Jan 17 at 1:42
MackTuesdayMackTuesday
723316
723316
$begingroup$
Why not a simple[alpha] == [beta]in Mathematica, which will give aTrueorFalse?
$endgroup$
– David G. Stork
Jan 17 at 1:44
$begingroup$
That's not exactly an "algorithm", @David, although it's a pretty useful technique.
$endgroup$
– John Hughes
Jan 17 at 1:48
$begingroup$
It's part of some software I'm writing. I can't necessarily rely on someone else's code. I've also edited the question for clarity.
$endgroup$
– MackTuesday
Jan 17 at 1:48
$begingroup$
There's a bit of literature on "denesting radicals". (I don't know this, but ...) Perhaps there's a way to wrangle nested radicals into a "canonical form" that would make them straightforward to compare.
$endgroup$
– Blue
Jan 17 at 2:31
add a comment |
$begingroup$
Why not a simple[alpha] == [beta]in Mathematica, which will give aTrueorFalse?
$endgroup$
– David G. Stork
Jan 17 at 1:44
$begingroup$
That's not exactly an "algorithm", @David, although it's a pretty useful technique.
$endgroup$
– John Hughes
Jan 17 at 1:48
$begingroup$
It's part of some software I'm writing. I can't necessarily rely on someone else's code. I've also edited the question for clarity.
$endgroup$
– MackTuesday
Jan 17 at 1:48
$begingroup$
There's a bit of literature on "denesting radicals". (I don't know this, but ...) Perhaps there's a way to wrangle nested radicals into a "canonical form" that would make them straightforward to compare.
$endgroup$
– Blue
Jan 17 at 2:31
$begingroup$
Why not a simple
[alpha] == [beta] in Mathematica, which will give a True or False?$endgroup$
– David G. Stork
Jan 17 at 1:44
$begingroup$
Why not a simple
[alpha] == [beta] in Mathematica, which will give a True or False?$endgroup$
– David G. Stork
Jan 17 at 1:44
$begingroup$
That's not exactly an "algorithm", @David, although it's a pretty useful technique.
$endgroup$
– John Hughes
Jan 17 at 1:48
$begingroup$
That's not exactly an "algorithm", @David, although it's a pretty useful technique.
$endgroup$
– John Hughes
Jan 17 at 1:48
$begingroup$
It's part of some software I'm writing. I can't necessarily rely on someone else's code. I've also edited the question for clarity.
$endgroup$
– MackTuesday
Jan 17 at 1:48
$begingroup$
It's part of some software I'm writing. I can't necessarily rely on someone else's code. I've also edited the question for clarity.
$endgroup$
– MackTuesday
Jan 17 at 1:48
$begingroup$
There's a bit of literature on "denesting radicals". (I don't know this, but ...) Perhaps there's a way to wrangle nested radicals into a "canonical form" that would make them straightforward to compare.
$endgroup$
– Blue
Jan 17 at 2:31
$begingroup$
There's a bit of literature on "denesting radicals". (I don't know this, but ...) Perhaps there's a way to wrangle nested radicals into a "canonical form" that would make them straightforward to compare.
$endgroup$
– Blue
Jan 17 at 2:31
add a comment |
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$begingroup$
Why not a simple
[alpha] == [beta]in Mathematica, which will give aTrueorFalse?$endgroup$
– David G. Stork
Jan 17 at 1:44
$begingroup$
That's not exactly an "algorithm", @David, although it's a pretty useful technique.
$endgroup$
– John Hughes
Jan 17 at 1:48
$begingroup$
It's part of some software I'm writing. I can't necessarily rely on someone else's code. I've also edited the question for clarity.
$endgroup$
– MackTuesday
Jan 17 at 1:48
$begingroup$
There's a bit of literature on "denesting radicals". (I don't know this, but ...) Perhaps there's a way to wrangle nested radicals into a "canonical form" that would make them straightforward to compare.
$endgroup$
– Blue
Jan 17 at 2:31