1.14 Solution of Equations

FriCAS also has state-of-the-art algorithms for the solution of systems of polynomial equations. When the number of equations and unknowns is the same, and you have no symbolic coefficients, you can use solve for real roots and complexSolve for complex roots. In each case, you tell FriCAS how accurate you want your result to be. All operations in the solve family return answers in the form of a list of solution sets, where each solution set is a list of equations.

A system of two equations involving a symbolic parameter t.

S(t) == [x^2-2*y^2 - t,x*y-y-5*x + 5]

_{Type: Void}

Find the real roots of S(19) with rational arithmetic, correct to within 1/1020.

solve(S(19),1/10^20)

\[\scriptstyle{ \left[ {\left[ {y=5}, \: {x=-{{80336736493669365924 189585} \over {96714065569170333976 49408}}} \right]}, \: {\left[ {y=5}, \: {x={{80336736493669365924 189585} \over {96714065569170333976 49408}}} \right]} \right]}\]

_{Type: List List Equation Polynomial Fraction Integer}

Find the complex roots of S(19) with floating point coefficients to 20 digits accuracy in the mantissa.

complexSolve(S(19),10.e-20)

\[\begin{split}\scriptsize{ \left[ {\left[ {y={5.0}}, \: {x={8.3066238629\_1807485258\_4262744905\_6951556981\_516914818}} \right]}, \\\\ \: {\left[ {y={5.0}}, \: {x=-{8.3066238629\_180748526}} \right]}, \\\\ \: {\left[ {y=-{{3.0} \ i}}, \: {x={1.0}} \right]}, \\\\ \: {\left[ {y={{3.0} \ i}}, \: {x={1.0}} \right]} \right]}\end{split}\]

_{Type: List List Equation Polynomial Complex Float}

If a system of equations has symbolic coefficients and you want a solution in radicals, try radicalSolve.

radicalSolve(S(a),[x,y])

\[\scriptsize{ \left[ {\left[ {x=-{\sqrt {{a+{50}}}}}, \: {y=5} \right]}, \: {\left[ {x={\sqrt {{a+{50}}}}}, \: {y=5} \right]}, \: {\left[ {x=1}, \: {y={{\sqrt {{-a+1}}} \over {\sqrt {2}}}} \right]}, \: {\left[ {x=1}, \: {y=-{{\sqrt {{-a+1}}} \over {\sqrt {2}}}} \right]} \right]}\]

_{Type: List List Equation Expression Integer}

For systems of equations with symbolic coefficients, you can apply solve, listing the variables that you want FriCAS to solve for. For polynomial equations, a solution cannot usually be expressed solely in terms of the other variables. Instead, the solution is presented as a triangular system of equations, where each polynomial has coefficients involving only the succeeding variables. This is analogous to converting a linear system of equations to triangular form.

A system of three equations in five variables.

eqns := [x^2 - y + z,x^2*z + x^4 - b*y, y^2 *z - a - b*x]

\[\left[ {z -y+{{x} ^ {2}}}, \: {{{{x} ^ {2}} \ z} -{b \ y}+{{x} ^ {4}}}, \: {{{{y} ^ {2}} \ z} -{b \ x} -a} \right]\]

_{Type: List Polynomial Integer}

Solve the system for unknowns [x,y,z], reducing the solution to triangular form.

solve(eqns,[x,y,z])

\[\begin{split}\scriptstyle{ \left[ {\left[ {x=-{a \over b}}, \: {y=0}, \: {z=-{{{a} ^ {2}} \over {{b} ^ {2}}}} \right]}, \\\\ \: {\left[ {x={{{{z} ^ {3}}+{2 \ b \ {{z} ^ {2}}}+{{{b} ^ {2}} \ z} -a} \over b}}, \: {y={z+b}}, \: {{{{z} ^ {6}}+{4 \ b \ {{z} ^ {5}}}+{6 \ {{b} ^ {2}} \ {{z} ^ {4}}}+{{\left( {4 \ {{b} ^ {3}}} -{2 \ a} \right)} \ {{z} ^ {3}}}+{{\left( {{b} ^ {4}} -{4 \ a \ b} \right)} \ {{z} ^ {2}}} -{2 \ a \ {{b} ^ {2}} \ z} -{{b} ^ {3}}+{{a} ^ {2}}}=0} \right]} \right]}\end{split}\]

_{Type: List List Equation Fraction Polynomial Integer}