/usr/share/doc/axiom-doc/hypertex/polyroots4.xhtml

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  <div align="center"><img align="middle" src="doctitle.png"/></div>
  <hr/>
  <div align="center">Solution of Systems of Polynomial Equations</div>
  <hr/>
Given a system of equations of rational functions with exact coefficients
<pre>
     p1(x1,...,xn)
         .
         .
     pm(x1,...,xn)
</pre>
Axiom can find numeric or symbolic solutions. The system is first split 
into irreducible components, then for each component, a triangular system
of equations is found that reduces the problem to sequential solutions of
univariate polynomials resulting from substitution of partial solutions
from the previous stage.
<pre>
     q1(x1,...,xn)
         .
         .
     qm(xn)
</pre>
Symbolic solutions can be presented using "implicit" algebraic numbers
defined as roots of irreducible polynomials or in terms of radicals. Axiom
can also find approximations to the real or complex roots of a system of
polynomial equations to any user specified accuracy.

The operation <a href="dbopsolve.xhtml">solve</a> for systems is used in
a way similar to <a href="dbopsolve.xhtml">solve</a> for single equations.
Instead of a polynomial equation, one has to give a list of equations and
instead of a single variable to solve for, a list of variables. For 
solutions of single equations see
<a href="axbook/section-8.5.xhtml#subsec-8.5.2">
Solution of a Single Polynomial Equation</a>

Use the operation <a href="dbopsolve.xhtml">solve</a> if you want
implicitly presented solutions.
<ul>
 <li>
  <input type="submit" id="p1" class="subbut" 
    onclick="makeRequest('p1');"
    value="solve([3*x^2+y+1,y^2-4],[x,y])" />
  <div id="ansp1"><div></div></div>
 </li>
 <li>
  <input type="submit" id="p2" class="subbut" 
    onclick="makeRequest('p2');"
    value="solve([x=y^2-19,y=z^2+x+3,z=3*x],[x,y,z])" />
  <div id="ansp2"><div></div></div>
 </li>
</ul>
Use <a href="dbopradialsolve.xhtml">radicalSolve</a> if you want your
solutions expressed in terms of radicals.
<ul>
 <li>
  <input type="submit" id="p3" class="subbut" 
    onclick="makeRequest('p3');"
    value="radicalSolve([3*x^3+y+1,y^2-4],[x,y])" />
  <div id="ansp3"><div></div></div>
 </li>
</ul>
To get numeric solutions you only need to give the list of equations and
the precision desired. The list of variables would be redundant information
since there can be no parameters for the numerical solver.

If the precision is expressed as a floating point number you get results
expressed as floats.
<ul>
 <li>
  <input type="submit" id="p4" class="subbut" 
    onclick="makeRequest('p4');"
    value="solve([x^2*y-1,x*y^2-2],.01)" />
  <div id="ansp4"><div></div></div>
 </li>
</ul>
To get complex numeric solutions, use the operation
<a href="dbopcomplexsolve.xhtml">complexSolve</a>, which takes the same
arguments as in the real case.
<ul>
 <li>
  <input type="submit" id="p5" class="subbut" 
    onclick="makeRequest('p5');"
    value="complexSolve([x^2*y-1,x*y^2-2],1/1000)" />
  <div id="ansp5"><div></div></div>
 </li>
</ul>
It is also possible to solve systems of equations in rational functions
over the rational numbers. Note that [x=0.0,a=0.0] is not returned as
a solution since the denominator vanishes there.
<ul>
 <li>
  <input type="submit" id="p6" class="subbut" 
    onclick="makeRequest('p6');"
    value="solve([x^2/a=a,a=a*x],.001)" />
  <div id="ansp6"><div></div></div>
 </li>
</ul>
When solving equations with denominators, all solutions where the 
denominator vanishes are discarded.
<ul>
 <li>
  <input type="submit" id="p7" class="subbut" 
    onclick="makeRequest('p7');"
    value="radicalSolve([x^2/a+a+y^3-1,a*y+a+1],[x,y])" />
  <div id="ansp7"><div></div></div>
 </li>
</ul>
 </body>
</html>
axiom-hypertex-data 20120501-8 / usr / share / doc / axiom-doc / hypertex / polyroots4.xhtml
