Tutorial:JMathLab/Equations (Nonlinear)

Nonlinear Equations

"Equation" in the following means the equation expression = 0. Equations are solved for a symbolic variable x by the function solve(expression, x). If expression is a quotient, then nominator = 0 is solved. It uses the following strategy to solve equations:

• Unordered List ItemFirst, all occurrences of the variable x in expression
• are counted, both as free variable and embedded inside functions. Example:
• In [math]\displaystyle{ x^3\cdot\sin(x)+2x^2-\sqrt{x-1} }[/math] x occurs three times: as free
• variable, in [math]\displaystyle{ \sin(x) }[/math] and in [math]\displaystyle{ \sqrt{x-1} }[/math].
• Unordered List ItemIf this count is one, then we are dealing with a
• polynomic equation, which is solved for the polynomial's main variable,
• e.g. z. This works always, if the polynomial's degree is 2 or of it is
• biquadratic, otherwise only, if the coefficients are constant. In the next
• step the solution is solved for the desired variable x. As an example:
• One can solve [math]\displaystyle{ \sin^2(x)-2\sin(x)+1=0 }[/math] for "x". It first solves
• [math]\displaystyle{ z^2-2z+1=0 }[/math] for $z$ and then [math]\displaystyle{ \sin(x)=z }[/math] for "x". Examples with free
• variables:

<jc lang="math"> syms x,b a=solve(x^2-1,x) printf('%f\n',a) a=solve(x^2-2*x*b+b^2,x) printf('%f\n',a) </jc>

An example with function variable ([math]\displaystyle{ \exp(j\cdot x) }[/math]):

<jc lang="math"> syms x a=float( solve(sin(x)^2+2*cos(x)-0.5,x) ) printf('%f\n',a) </jc>

• Unordered List ItemIf count is 2, only one case is further considered: The
• variable occurs free and inside squareroot. This squareroot is then
• isolated, the equation squared and solved. This case leads to additional
• false solutions, which have to be sorted out manually.

<jc lang="math"> syms x y=x^2+3*x-17*sqrt(3*x^2+12); a=solve(y,x) printf('%f\n',a) </jc>