Many numerical methods rely on iterations, which cause so called error accumulation, and just within a few steps the numerical error makes result useless.
Typical example is the three term linear recurrence relation,
$$x_{n+1}= a_n x_n + b_n x_{n-1},$$
which can very quickly become unstable. Here $a_n$ and $b_n$ are some real or complex numbers. The three points recurrence relation is related to the finite difference equation for the general second order differential equation. But is very commonly used to obtain special functions, such as Bessel functions, associated Legendre functions, regular Coulomb wave functions, etc.
For constant coefficient, $a_n=a$ and $b_n=b$, the exact solution can be found by the zeros of the characteristic polynomial. More specifically, we search for the solution in the form
$$a_n = C r^n, $$
which gives
$$r^{n+1} = a r^n + b r^{n-1}$$
which is solved by characteristic polynomial of the form
$$r^2 - a r - b=0.$$
For $n$-point recurence relation we would have characteristic polynomial of the $n-1$ power.
For the three points recurence relation, we have thus two solutions
$$r_{1,2} = \frac{1}{2}a \pm \frac{1}{2}\sqrt{a^2+ 4 b}. $$
The generic solution is than of the form
$$ a_n = C_1 r_1^n + C_2 r_2^n. $$
If $a^2+4b=0$, and the characteristic polynomial has only one solution, the solution of the tree-point recurrence is instead:
$$ a_n = C_1 r_1^n + C_2 n r_1^n.$$
Even when coefficients are not constant, the $n$-term linear recurrence relation has exactly $n-1$ solutions. They can not be found analytically or exactly, but we know that they exist.
However, only one solution out of $n-1$ is easy to follow, namely, the one that grows the fastest in the direction of recurence.
For three-point recurence relation, we have two solutions: one is typically increasing and one is decreasing.
Hence for three point recurrence relation, this instability can be easily fixed by so-called Miller’s algorithm. If iteration in one direction is unstable, it is likely stable in the opposite direction.
Examples of three point recurrence relation include:
Bessel Functinons
$$ j_{l+1}(x) = \frac{2l+1}{x} j_l(x) - j_{l-1}(x)$$
Modified Bessel Functions
$$ j_{l+1}(x) = -\frac{2l+1}{x} j_l(x) + j_{l-1}(x)$$
Legendre Polynomials
$$(n+1) P_{n+1}(x) = (2n+1) x P_n(x)-n P_{n-1}(x)$$
Associated Legendre polynomials
$$(l-m+1) P^m_{l+1}(x)=(2l+1) x P^m_l(x) - (l+m)P^m_{l-1}$$
$$2 m x P^m_l(x) = -\sqrt{1-x^2}\left[ P^{m+1}_l(x)+(l+m)(l-m+1)P^{m-1}_l(x)\right]$$
in terms of them the spherical harmonics are given
$$Y_{l,m}(\theta,\phi)=\sqrt{\frac{(2l+1)(l-m)!}{4\pi(l+m)!}}P^m_l(\cos\theta)e^{i m\phi},$$
which are used to solve the 3D partial differential equation of the form
$$\nabla^2 \psi(\theta,\phi) + \lambda\; \psi(\theta,\phi)=0.$$
Confluent Hypergeometric Series
$$(b-n) M_{n-1} + (2n-b+x)M_n - n M_{n+1}=0$$
with solution $M_n=M(n,b; x)$.
