Linear ordinary difference equation with exact coefficients:

Nonlinear ordinary difference equation with inexact coefficients:

System of ordinary difference equation with symbolic initial conditions:

Return only the values of x:

Iterate using exact arithmetic:

Iterate using adaptive arithmetic starting with precision 20:

The precision decreases with each iteration:

Iterate using fixed 20-digit-precision arithmetic:

Iterate using machine arithmetic:

Iterate several values at once by giving a vector initial condition:

Iterate a matrix recurrence:

Use the partial recurrence equations for binomial coefficients:

Procedural solution for a nonlinear partial difference equation:

Solve a linear difference-algebraic equation with constant coefficients:

Compare with the symbolic solution given by RSolve:

The state response and the output response of a state-space model to a sampled sinusoid:

The state response of a discrete-time system with initial conditions {1,-1}:

The output response of a two-input system:

Use DependentVariables to specify the variables when you only want to save some of them:

Save only y:

Save both in order {y,x}:

Use Method->{Compiled->False} to prevent the Wolfram Language compiler from being used:

Results differ due to arithmetic change from optimization:

Use WorkingPrecision->MachinePrecision for the fastest iterations:

Use WorkingPrecision->p for slower, but higher-precision iterations:

Exact computations have no error, but may be very slow indeed:

Study the behavior of the logistic equation for different values of the parameter r:

Implement the Cliff random number generator:

The random numbers appear to be uniformly distributed:

Compare with the parameters for the uniform distribution:

Plot the Douady rabbit fractal:

Initial condition with 250 points in each direction on the rectangle with corners and :

Iterate starting from these initial conditions:

Use ArrayPlot to show the fractal:

Find iterates from and of the map for 1000 values of :

Scale the iterates to be integers between 1 and and transpose so the rows correspond to :

Define a function that gives a rule based on the logarithm of counts of each value:

Make a sparse matrix based on applying Count to the iterates for each :

Use ArrayPlot to make the bifurcation diagram:

For , Euler's method is unconditionally unstable:

The symplectic Euler method is stable, but is very sensitive to initial conditions for large h:

Compare the methods for different vector fields with Manipulate:

Stretching and folding induced by the standard map for a line of initial conditions [more info]: