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RecurrenceTable

 RecurrenceTable[eqns, expr, {n, nmax}] generates a list of values of expr for successive n based on solving the recurrence equations eqns. RecurrenceTable[eqns, expr, nspec]generates a list of values of expr over the range of n values specified by nspec. RecurrenceTable[eqns, expr, {n1, ...}, {n2, ...}, ...]generates an array of values of expr for successive n1, n2, ....
• 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 a[n+i] 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 , RecurrenceTable attempts to determine the dependent variables by analyzing the equations given.
• With , results for exact inputs are computed exactly, and for inexact inputs, the precision to use is determined adaptively at each iteration.
• With , a fixed precision p is used for all iterations.
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:
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Find the first few Fibonacci numbers:
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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]=
 Scope   (9)
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:
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:
 Options   (3)
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 Mathematica compiler from being used:
Results differ due to arithmetic change from optimization:
Use for the fastest iterations:
Use 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 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 for 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]:
RSolve finds a symbolic solution for this difference equation:
RecurrenceTable generates a procedural solution for the same problem:
Visualize the smoothing of the initial data for the heat equation using the discretized version:
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