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LREtools

 riccati
 find solutions of Riccati Recurrence equations

 Calling Sequence riccati(problem)

Parameters

 problem - problem statement or RESol

Description

 • Attempts to solve Riccati recurrence equations using various substitutions.
 • A Riccati recurrence equation in y(k) is one of the form $y\left(k+1\right)y\left(k\right)+A\left(k\right)y\left(k+1\right)+B\left(k\right)y\left(k\right)=C\left(k\right)$ where A(k), B(k), and C(k) are independent of y(k).  If the equation is homogeneous ($C\left(k\right)=0$), then we try the substitution $x\left(k\right)=\frac{1}{y\left(k\right)}$, which makes the equation first order linear and if not, then we try $y\left(k\right)=\frac{x\left(k\right)-B\left(k\right)x\left(k+1\right)}{x\left(k+1\right)}$, which makes the equation second order linear. Finally, there is the substitution $y\left(k\right)=\frac{x\left(k+1\right)-A\left(k+1\right)x\left(k\right)}{x\left(k\right)}$, 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.

Examples

 > $\mathrm{with}\left(\mathrm{LREtools}\right):$
 > $\mathrm{prob}≔\mathrm{REcreate}\left(y\left(k+1\right)y\left(k\right)+2y\left(k+1\right)-3y\left(k\right)=1,y\left(k\right),\left\{y\left(0\right)=\frac{1}{2}\right\}\right)$
 ${\mathrm{prob}}{:=}{\mathrm{RESol}}{}\left(\left\{\left({y}{}\left({k}{+}{1}\right){-}{3}\right){}{y}{}\left({k}\right){+}{2}{}{y}{}\left({k}{+}{1}\right){=}{1}\right\}{,}\left\{{y}{}\left({k}\right)\right\}{,}\left\{{y}{}\left({0}\right){=}\frac{{1}}{{2}}\right\}{,}{\mathrm{INFO}}\right)$ (1)
 > $\mathrm{riccati}\left(\mathrm{prob}\right)$
 $\frac{{1}}{{2}}{}\frac{{\left({-}{5}{-}\sqrt{{5}}\right)}^{{k}}{}\sqrt{{5}}{-}{\left({-}{5}{+}\sqrt{{5}}\right)}^{{k}}{}\sqrt{{5}}{+}{\left({-}{5}{-}\sqrt{{5}}\right)}^{{k}}{+}{\left({-}{5}{+}\sqrt{{5}}\right)}^{{k}}}{{\left({-}{5}{-}\sqrt{{5}}\right)}^{{k}}{+}{\left({-}{5}{+}\sqrt{{5}}\right)}^{{k}}}$ (2)
 > $\mathrm{riccati}\left(y\left(k+1\right)y\left(k\right)+2y\left(k+1\right)-ky\left(k\right)=0,y\left(k\right),\left\{\right\}\right)$
 ${{\mathrm{charfcn}}}_{{0}}{}\left({k}\right){}{y}{}\left({0}\right)$ (3)