generates a list of values of expr for successive n based on solving the recurrence equations eqns.
generates a list of values of expr over the range of n values specified by nspec.
generates an array of values of expr for successive n1, n2, … .
Details and Options
- 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 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.
Examplesopen allclose all
Basic Examples (4)
Ordinary Difference Equations (6)
Linear ordinary difference equation with exact coefficients:
Nonlinear ordinary difference equation with inexact coefficients:
System of ordinary difference equation with symbolic initial conditions:
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:
Partial Difference Equations (2)
Difference-Algebraic Equations (1)
Solve a linear difference-algebraic equation with constant coefficients:
Compare with the symbolic solution given by RSolve:
Generalizations & Extensions (3)
Use DependentVariables to specify the variables when you only want to save some of them:
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:
Logistic Equations (1)
Random Number Generation (1)
Rabbit Fractal (1)
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:
Bifurcation Diagram of the Logistic Map (1)
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:
Compare Numerical Methods for ODEs (1)
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:
Standard Map (1)
Stretching and folding induced by the standard map for a line of initial conditions [more info]:
Properties & Relations (2)
RSolve finds a symbolic solution for this difference equation:
RecurrenceTable generates a procedural solution for the same problem:
Use RecurrenceFilter to filter a signal:
Obtain the same result using RecurrenceTable:
Wolfram Research (2008), RecurrenceTable, Wolfram Language function, https://reference.wolfram.com/language/ref/RecurrenceTable.html.
Wolfram Language. 2008. "RecurrenceTable." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RecurrenceTable.html.
Wolfram Language. (2008). RecurrenceTable. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RecurrenceTable.html