Mathematica 9 is now available

THIS IS DOCUMENTATION FOR AN OBSOLETE PRODUCT.
SEE THE DOCUMENTATION CENTER FOR THE LATEST INFORMATION.

DOCUMENTATION CENTER SEARCH

New to Mathematica? Find your learning path »

Mathematica > Mathematics and Algorithms > Calculus > Discrete Calculus > RecurrenceTable >

Mathematica > Mathematics and Algorithms > Discrete Mathematics > Discrete Calculus > RecurrenceTable >

BUILT-IN MATHEMATICA SYMBOL

See Also »|More About »

RecurrenceTable

RecurrenceTable

generates a list of values of expr for successive n based on solving the recurrence equations eqns.

RecurrenceTable

generates a list of values of expr over the range of n values specified by nspec.

RecurrenceTable

generates an array of values of expr for successive

,

, ....

MORE INFORMATION

The eqns must be recurrence equations whose solutions over the range specified can be determined completely from the initial or boundary values given.

The eqns can involve objects of the form where i is any fixed integer.

The range specification nspec can have any of the forms used in Table.

The following options can be given:

	DependentVariables	Automatic	the list of all dependent variables
	Method	Automatic	method to use
	WorkingPrecision	Automatic	precision used in internal computations

With DependentVariables->Automatic, RecurrenceTable attempts to determine the dependent variables by analyzing the equations given.

With WorkingPrecision->Automatic, results for exact inputs are computed exactly, and for inexact inputs, the precision to use is determined adaptively at each iteration.

With WorkingPrecision->p, a fixed precision p is used for all iterations.

Basic Examples (4)

Solve an initial-value problem for a first-order difference equation:

Find the first few Fibonacci numbers:

Study the evolution for a nonlinear map of the plane:

Compute a table of Stirling numbers of the first kind:

Solve an initial-value problem for a first-order difference equation:

Out[1]=

Find the first few Fibonacci numbers:

Out[1]=

Out[2]=

Study the evolution for a nonlinear map of the plane:

Out[1]//Short=

Out[2]=

Compute a table of Stirling numbers of the first kind:

Out[1]=

Out[2]=

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

:

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:

Generalizations & Extensions (3)

Generate a subset of values from a given range:

Get only the last value from an iteration:

This is faster than when all the values are saved:

Use a vector initial condition:

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

Save only

:

Save both in order

:

Use Method->{Compiled->False} to prevent the Mathematica 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:

Applications (6)

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

:

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

:

Compare the methods for different vector fields with Manipulate:

Properties & Relations (1)

RSolve finds a symbolic solution for this difference equation:

RecurrenceTable generates a procedural solution for the same problem:

Neat Examples (1)

Visualize the smoothing of the initial data for the heat equation using the discretized version:

RSolve Table FindInstance NDSolve

New in 7

Ask a question about this page | Suggest an improvement | Leave a message for the team

Format: HTML | CDF