---
tag: D4Q8VM7826
kind: concept
name: Riccati difference equation
msc: [39A20, 39A06]
specializes: [T8YDEASD1H]
sources:
  - url: https://en.wikipedia.org/wiki/Rational_difference_equation
    locator: Riccati difference equation
---

## Summary

The Riccati difference equation is the first-order [[rational difference equation|T8YDEASD1H]] $x_{n+1} = (a x_n + b)/(c x_n + d)$ with $c \ne 0$.
It is the iteration of a [[Möbius transformation|0SG74WBFA2]].
A substitution turns it into a linear recurrence of order two, so it can be solved in closed form.
It is the discrete analogue of the [[Riccati equation|0VT2QK6X6Q]] for differential equations, which is linearized by a similar substitution.

## Explanations

### L1 human

Each new term is a fraction with $x_n$ in the top and in the bottom.
Such a rule looks nonlinear, but it hides a linear rule.
If you write $c x_n + d$ as a ratio $y_{n+1}/y_n$, the numbers $y_n$ obey an ordinary two-step linear recurrence.
So the behaviour of the fraction rule is decided by the roots of a quadratic equation.