find solutions of Riccati Recurrence equations
problem statement or RESol
Attempts to solve Riccati recurrence equations using various substitutions.
A Riccati recurrence equation in y(k) is one of the form y⁡k+1⁢y⁡k+A⁡k⁢y⁡k+1+B⁡k⁢y⁡k=C⁡k where A(k), B(k), and C(k) are independent of y(k). If the equation is homogeneous (C⁡k=0), then we try the substitution x⁡k=1y⁡k, which makes the equation first order linear and if not, then we try y⁡k=x⁡k−B⁡k⁢x⁡k+1x⁡k+1, which makes the equation second order linear. Finally, there is the substitution y⁡k=x⁡k+1−A⁡k+1⁢x⁡kx⁡k, which makes the equation second order linear. If rsolve can solve these new equations, then we back-substitute to obtain solutions to the original problems.
If A(k) is undefined for some k, then a set of equations may be returned, giving values of y(k) for specific k as well as the general formula.
Since it calls rsolve, this procedure can be expensive; because of the back-substitution, the answers may be overly complicated.
See the help page for LREtools[REcreate] for the definition of the format of a problem.
prob ≔ REcreate⁡y⁡k+1⁢y⁡k+2⁢y⁡k+1−3⁢y⁡k=1,y⁡k,y⁡0=12
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