/usr/share/doc/axiom-doc/hypertex/polyroots3.xhtml

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  <title>Axiom Documentation</title>
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  <div align="center"><img align="middle" src="doctitle.png"/></div>
  <hr/>
  <div align="center">Solution of a Single Polynomial Equation</div>
  <hr/>
Axiom can solve polynomial equations producing either approximate or exact
solutions. Exact solutions are either members of the ground field or can
be presented symbolically as roots of irreducible polynomials.

This returns one rational root along with an irreducible polynomial 
describing the other solutions
<ul>
 <li>
  <input type="submit" id="p1" class="subbut" 
    onclick="makeRequest('p1');"
    value="solve(x^3=8,x)" />
  <div id="ansp1"><div></div></div>
 </li>
</ul>
If you want solutions expressed in terms of radicals you would use this
instead.
<ul>
 <li>
  <input type="submit" id="p2" class="subbut" 
    onclick="makeRequest('p2');"
    value="radicalSolve(x^3=8,x)" />
  <div id="ansp2"><div></div></div>
 </li>
</ul>
The <a href="dbopsolve.xhtml">solve</a> command always returns a value but
<a href="dbopradicalsolve.xhtml">radicalSolve</a> returns only the solutions
that it is able to express in terms of radicals.

If the polynomial equation has rational coefficients you can ask for
approximations to its real roots by calling solve with a second argument
that specifies the "precision" epsilon. This means that each approximation
will be within plus or minus epsilon of the actual result.

Notice that the type of second argument controls the type of the result.
<ul>
 <li>
  <input type="submit" id="p3" class="subbut" 
    onclick="makeRequest('p3');"
    value="solve(x^4-10*x^3+35*x^2-50*x+25,.0001)" />
  <div id="ansp3"><div></div></div>
 </li>
</ul>
If you give a floating point precision you get a floating point result.
If you give the precision as a ration number you get a rational result.
<ul>
 <li>
  <input type="submit" id="p4" class="subbut" 
    onclick="makeRequest('p4');"
    value="solve(x^2-2,1/1000)" />
  <div id="ansp4"><div></div></div>
 </li>
</ul>
If you want approximate complex results you should use the command
<a href="dbopcomplexsolve.xhtml">complexSolve</a> that takes the same
precision argument epsilon.
<ul>
 <li>
  <input type="submit" id="p5" class="subbut" 
    onclick="makeRequest('p5');"
    value="complexSolve(x^3-2,.0001)" />
  <div id="ansp5"><div></div></div>
 </li>
</ul>
Each approximation will be within plus or minus epsilon of the actual result
in each of the real and imaginary parts.
<ul>
 <li>
  <input type="submit" id="p6" class="subbut" 
    onclick="makeRequest('p6');"
    value="complexSolve(x^2-2*%i+1,1/100)" />
  <div id="ansp6"><div></div></div>
 </li>
</ul>
Note that if you omit the = from the first argument Axiom generates
an equation by equating the first argument to zero. Also, when only one
variable is present in the equation, you do not need to specify the
variable to be solved for, that is, you can omit the second argument.

Axiom can also solve equations involving rational functions. Solutions
where the denominator vanishes are discarded.
<ul>
 <li>
  <input type="submit" id="p7" class="subbut" 
    onclick="makeRequest('p7');"
    value="radicalSolve(1/x^3+1/x^2+1/x=0,x)" />
  <div id="ansp7"><div></div></div>
 </li>
</ul>
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axiom-hypertex-data 20120501-8 / usr / share / doc / axiom-doc / hypertex / polyroots3.xhtml
