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<A NAME=SOLVE>

<TITLE>SOLVE</TITLE></A>
<b><a href=r37_idx.html>INDEX</a></b><p><p>



<B>SOLVE</B> _ _ _  _ _ _  _ _ _  _ _ _ <B>operator</B><P>
<P>
 
 <P>
<P>
The <em>solve</em> operator solves a single algebraic 
<A HREF=r37_0045.html>equation</A> or a 
system of simultaneous equations. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>solve</em>(&lt;expression&gt; [ , &lt;kernel&gt;]) or 
<P>
<P>
<em>solve</em>({&lt;expression&gt;,...} [ ,{ &lt;kernel&gt; ,...}] ) 
<P>
<P>
 <P>
<P>
If the number of equations equals the number of distinct kernels, the 
optional kernel argument(s) may be omitted. &lt;expression&gt; is either a 
scalar expression or an 
<A HREF=r37_0045.html>equation</A>. 
When more than one expression is given, 
the 
<A HREF=r37_0053.html>list</A> of expressions is surrounded by curly braces. 
The optional list 
of 
<A HREF=r37_0002.html>kernel</A>s follows, also in curly braces. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<P><PRE><TT>
sss := solve(x^2 + 7); 

  Unknown: X
  SSS := {X= - SQRT(7)*I,
          X=SQRT(7)*I}


rhs first sss; 

  - SQRT(7)*I 


solve(sin(x^2*y),y); 

     2*ARBINT(1)*PI
  {Y=---------------
            2
           X
     PI*(2*ARBINT(1) + 1)
   Y=--------------------}
               2
              X


off allbranch; 

solve(sin(x**2*y),y); 

  {Y=0} 


solve({3x + 5y = -4,2*x + y = -10},{x,y});
 


         22   46
  {{X= - --,Y=--}} 
         7    7


solve({x + a*y + z,2x + 5},{x,y});
 


         5      2*Z - 5
  {{X= - -,Y= - -------}} 
         2        2*A


ab := (x+2)^2*(x^6 + 17x + 1);
 

         8      7      6       3       2
  AB := X  + 4*X  + 4*X  + 17*X  + 69*X  + 72*X + 4 


www := solve(ab,x); 

  {X=ROOT F(X6 + 17*X + 1),X=-2} 
         O            


root_multiplicities; 

  {1,2}

</TT></PRE><P>Results of the <em>solve</em> operator are returned as 
<A HREF=r37_0045.html>equation</A><em>s</em> 
in a 
<A HREF=r37_0053.html>list</A>. 
You can use the usual list access methods (
<A HREF=r37_0046.html>first</A>, 

<A HREF=r37_0063.html>second</A>, 
<A HREF=r37_0066.html>third</A>, 
<A HREF=r37_0057.html>rest</A> and 
<A HREF=r37_0169.html>part</A>) to 
extract the desired equation, and then use the operators 
<A HREF=r37_0175.html>rhs</A> and 

<A HREF=r37_0158.html>lhs</A> to access the right-hand or left-hand expression o
f the 
equation. When <em>solve</em> is unable to solve an equation, it returns the 
unsolved part as the argument of <em>root_of</em>, with the variable renamed 
to avoid confusion, as shown in the last example above. 
<P>
<P>
For one equation, <em>solve</em> uses square-free factorization, roots of 
unity, and the known inverses of the 
<A HREF=r37_0088.html>log</A>, 
<A HREF=r37_0259.html>sin</A>, 

<A HREF=r37_0249.html>cos</A>, 
<A HREF=r37_0236.html>acos</A>, 
<A HREF=r37_0244.html>asin</A>, and 
exponentiation operators. The quadratic, cubic and quartic formulas are 
used if necessary, but these are applied only when the switch 

<A HREF=r37_0292.html>fullroots</A> is set on; otherwise or when no closed form 
is available 
the result is returned as 

<A HREF=r37_0176.html>root_of</A> expression. The switch 
<A HREF=r37_0336.html>trigform</A> 
determines which type of cubic and quartic formula is used. 
The multiplicity of each solution is given in a list as 
the system variable 
<A HREF=r37_0017.html>root_multiplicities</A>. For systems of 
simultaneous linear equations, matrix inversion is used. For nonlinear 
systems, the Groebner basis method is used. 
<P>
<P>
Linear equation system solving is influenced by the switch 
<A HREF=r37_0276.html>cramer</A>. 
<P>
<P>
Singular systems can be solved when the switch 
<A HREF=r37_0332.html>solvesingular</A> is 
on, which is the default setting. An empty list is returned the system of 
equations is inconsistent. For a linear inconsistent system with parameters 
the variable 
<A HREF=r37_0016.html>requirements</A> constraints 
conditions for the system to become consistent. 
<P>
<P>
For a solvable linear and polynomial system with parameters 
the variable 
<A HREF=r37_0005.html>assumptions</A> 
contains a list side relations for the parameters: the solution is 
valid only as long as none of these expressions is zero. 
<P>
<P>
If the switch 
<A HREF=r37_0339.html>varopt</A> is on (default), the system rearranges the 
variable sequence for minimal computation time. Without <em>varopt</em> 
the user supplied variable sequence is maintained. 
<P>
<P>
If the solution has free variables (dimension of the solution is greater 
than zero), these are represented by 
<A HREF=r37_0139.html>arbcomplex</A> expressions 
as long as the switch 
<A HREF=r37_0268.html>arbvars</A> is on (default). Without 
<em>arbvars</em> no explicit equations are generated for free variables. 
<P>
<P>
<P>
 <P> <H3> 
related: </H3>
<P>
 _ _ _ 
<A HREF=r37_0266.html>allbranch</A>switch 
<P>
 _ _ _ 
<A HREF=r37_0268.html>arbvars</A> switch 
<P>
 _ _ _ 
<A HREF=r37_0005.html>assumptions</A> variable 
<P>
 _ _ _ 
<A HREF=r37_0292.html>fullroots</A> switch 
<P>
 _ _ _ 
<A HREF=r37_0016.html>requirements</A> variable 
<P>
 _ _ _ 
<A HREF=r37_0438.html>roots</A> operator 
<P>
 _ _ _ 
<A HREF=r37_0176.html>root_of</A> operator 
<P>
 _ _ _ 
<A HREF=r37_0336.html>trigform</A> switch 
<P>
 _ _ _ 
<A HREF=r37_0339.html>varopt</A> switch 
<P>
<P>
<P>