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Consider the following rational system with three equations.
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Each of these equations is of the form (J o F)(x) = H(x) with {J[i], H[i]} known. We want to find C, such that a solution F(x), F' <> 0 exists, and compute F. Check the degrees to determine the complexity of the task.
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So the first equation is rational in x, with degree 15 in the numerator and degree 15 in the denominator, then degrees 10,10 for the second rational equation, then 12,12. The solution for this problem is:
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FunctionDecomposition can quickly solve problems like the previous one. As a more involved and difficult problem, consider the following system of three equations, depending on {F(x), x, C} and in addition depending on two variables {a, b}, on which the solutions for both unknowns C and F actually depend, as shown after this large input.
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The presence of these two extra symbolic parameters {a, b} makes the computation run an order of magnitude slower. Also, this system is intrinsically more difficult than the previous one as it is clear from the higher powers of x entering the numerators and denominators of each equation.
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The system also depends on relatively high powers of the symbols a, b and the unknown C.
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A solution for this system is given by:
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