Consider the iterations $$ x_0 = 1, \qquad x_1 = \frac{1}{3}, \qquad x_i = \frac{13}{3}x_{i-1} - \frac{4}{3} x_{i-2}, \qquad i=2,3,\ldots $$ The exact solution is $$ x_i = \frac{1}{3^i} $$
from numpy import zeros, float32, float64
We first try with single precision.
n = 20
x = zeros(n, dtype=float32)
y = zeros(n, dtype=float32)
x[0], y[0] = 1.0, 1.0
x[1], y[1] = 1.0/3.0, 1.0/3.0
for i in range(2,n):
x[i] = (13.0/3.0)*x[i-1] - (4.0/3.0)*x[i-2]
y[i] = y[i-1]/3.0
print "%6d %20.10e %20.10e" % (i, x[i], y[i])
2 1.1111115664e-01 1.1111111194e-01 3 3.7037219852e-02 3.7037037313e-02 4 1.2346410193e-02 1.2345679104e-02 5 4.1181510314e-03 4.1152262129e-03 6 1.3834409183e-03 1.3717421098e-03 7 5.0404260401e-04 4.5724736992e-04 8 3.3959673601e-04 1.5241578512e-04 9 7.9952902161e-04 5.0805261708e-05 10 3.0118301511e-03 1.6935087842e-05 11 1.1985225603e-02 5.6450294323e-06 12 4.7920204699e-02 1.8816764396e-06 13 1.9167391956e-01 6.2722546090e-07 14 7.6669335365e-01 2.0907515363e-07 15 3.0667726994e+00 6.9691715510e-08 16 1.2267090797e+01 2.3230571244e-08 17 4.9068363190e+01 7.7435240442e-09 18 1.9627345276e+02 2.5811746074e-09 19 7.8509381104e+02 8.6039153580e-10
The iterations start to blow up around iteration 9. Now we try with double precision.
n = 30
x = zeros(n, dtype=float64)
y = zeros(n, dtype=float64)
x[0], y[0] = 1.0, 1.0
x[1], y[1] = 1.0/3.0, 1.0/3.0
for i in range(2,n):
x[i] = (13.0/3.0)*x[i-1] - (4.0/3.0)*x[i-2]
y[i] = y[i-1]/3.0
print "%6d %20.10e %20.10e" % (i, x[i], y[i])
2 1.1111111111e-01 1.1111111111e-01 3 3.7037037037e-02 3.7037037037e-02 4 1.2345679012e-02 1.2345679012e-02 5 4.1152263374e-03 4.1152263374e-03 6 1.3717421124e-03 1.3717421125e-03 7 4.5724737062e-04 4.5724737083e-04 8 1.5241578946e-04 1.5241579028e-04 9 5.0805260180e-05 5.0805263425e-05 10 1.6935074827e-05 1.6935087808e-05 11 5.6449773443e-06 5.6450292695e-06 12 1.8814687225e-06 1.8816764232e-06 13 6.2639467164e-07 6.2722547439e-07 14 2.0575194713e-07 2.0907515813e-07 15 5.6398875392e-08 6.9691719376e-08 16 -2.9940802813e-08 2.3230573125e-08 17 -2.0494197938e-07 7.7435243751e-09 18 -8.4816084023e-07 2.5811747917e-09 19 -3.4021076685e-06 8.6039159724e-10 20 -1.3611585443e-05 2.8679719908e-10 21 -5.4447393362e-05 9.5599066360e-11 22 -2.1778992398e-04 3.1866355453e-11 23 -8.7115981276e-04 1.0622118484e-11 24 -3.4846392900e-03 3.5407061615e-12 25 -1.3938557173e-02 1.1802353872e-12 26 -5.5754228696e-02 3.9341179572e-13 27 -2.2301691478e-01 1.3113726524e-13 28 -8.9206765914e-01 4.3712421747e-14 29 -3.5682706366e+00 1.4570807249e-14
The iterations are stable for more steps but they eventually start to blow up.