# RSolveValue

RSolveValue[eqn,expr,n]

gives the value of expr determined by a symbolic solution to the ordinary difference equation eqn with independent variable n.

RSolveValue[{eqn1,eqn2,},expr,]

uses a symbolic solution for a list of difference equations.

RSolveValue[eqn,expr,{n1,n2,}]

uses a solution for the partial recurrence equation eqn.

# Details and Options  • RSolveValue[eqn,a,n] gives a solution for a as a pure function.
• The equations can involve objects of the form a[n+λ], where λ is a constant, or in general, objects of the form a[ψ[n]], a[ψ[ψ[n]], a[ψ[[ψ[n]]]], where ψ can have forms such as:
•  n+λ arithmetic difference equation μ n geometric or -difference equation μ n+λ arithmetic-geometric functional difference equation μ nα geometric-power functional difference equation linear fractional functional difference equation
• Equations such as a==val can be given to specify end conditions.
• If not enough end conditions are specified, RSolveValue will use general solutions in which undetermined constants are introduced.
• The specification aVectors[p] or aMatrices[{m,p}] can be used to indicate that the dependent variable a is a vector-valued or a matrix-valued variable, respectively. » »
• The constants introduced by RSolveValue are indexed by successive integers. The option GeneratedParameters specifies the function to apply to each index. The default is , which yields constants C, C, .
• GeneratedParameters->(Module[{C},C]&) guarantees that the constants of integration are unique, even across different invocations of RSolveValue.
• For partial recurrence equations, RSolveValue generates arbitrary functions C[n][].
• Solutions given by RSolveValue sometimes include sums that cannot be carried out explicitly by Sum. Dummy variables with local names are used in such sums.
• RSolveValue[eqn,a[Infinity],n] gives the limiting value of the solution a at Infinity.
• RSolveValue handles both ordinary difference equations and difference equations.
• RSolveValue handles differencealgebraic equations, as well as ordinary difference equations.
• RSolveValue can solve linear recurrence equations of any order with constant coefficients. It can also solve many linear equations up to second order with nonconstant coefficients, as well as many nonlinear equations.

# Examples

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## Basic Examples(4)

Solve a difference equation:

Include a boundary condition:

Get a "pure function" solution for a:

Plot the solution:

Solve a functional equation:

Obtain the value of the solution at a point:

List of values:

## Scope(45)

### Basic Uses(9)

Compute the general solution of a first-order difference equation:

Obtain a particular solution by adding an initial condition:

Plot the solution of a first-order difference equation:

Make a table of values:

Verify the solution of a difference equation by using a in the second argument:

Obtain the general solution of a higher-order difference equation:

Particular solution:

Solve a system of difference equations:

Plot their solution:

Verify the solution:

Compute the value of the solution at a point:

Compute the limiting value of the solution at Infinity:

Solve a partial difference equation:

Obtain a particular solution:

Plot the resulting solution:

Use different names for the arbitrary constants in the general solution:

### Linear Difference Equations(6)

Geometric equation:

First-order equation with variable coefficients:

A third-order constant coefficient equation:

Initial value conditions:

Plot the solution:

Second-order inhomogeneous equation:

Second-order variable coefficient equation in terms of elementary functions:

EulerCauchy equation:

In general, special functions are required to express solutions:

### Nonlinear Difference Equations(5)

Solvable logistic equations:

Riccati equations:

Solutions in terms of trigonometric and hyperbolic functions:

Higher-order equations:

Nonlinear convolution equation:

### Systems of Difference Equations(7)

Linear system with constant coefficients:

With boundary conditions:

Plot their solution:

Linear fractional systems:

Diagonal system:

Variable coefficient linear system with a polynomial solution:

Linear constant coefficient difference-algebraic system:

An index-2 system:

Solve a linear system using vector variables:

Solve a linear system using matrix variables:

### Partial Difference Equations(3)

First-order linear partial difference equation with constant coefficients:

Substitute the function Sin[2k] for the free function C:

Plot the resulting solution:

Constant coefficient linear equation of orders 2, 3, and 4:

Inhomogeneous:

Variable coefficient linear equation:

### Q–Difference Equations(6)

First-order constant coefficient -difference equation:

Equivalent way of expressing the same equation:

Initial value:

Second-order equation:

Third-order:

Inhomogeneous:

Using a numeric value for :

Plot solution:

Linear varying coefficient equations:

Nonlinear equations:

Riccati equation:

A linear constant coefficient system of -difference equations:

### Functional Difference Equations(4)

Find the general solution for an arithmetic difference equation:

Verify the solution:

Solve an initial value problem for an arithmetic-geometric difference equation:

Plot the solution:

Solve a linear fractional difference equation:

Make a table of values for the solution:

Solve a geometric power difference equation:

Verify the solution:

### Limits of Recursive Sequences(5)

Compute the limit of a linear recursive sequence:

Visualize the convergence of the sequence to its limiting value:

Compute the limit of a nonlinear recursive sequence:

Visualize the convergence of the sequence to its limiting value:

Compute the limit of a trigonometric recursive sequence:

Visualize the convergence of the sequence to its limiting value:

Compute the limit of a second-order recursive sequence:

Visualize the convergence of the sequence to its limiting value:

Compute the limit of the ratio of adjacent terms, f[n+1]/f[n], for the Fibonacci sequence, which satisfies the difference equation:

Solve the difference equation to find the ratio:

Compute the limit of the ratio using DiscreteLimit:

Visualize the convergence of the sequence to its limiting value:

Alternatively, compute the limit using DiscreteLimit in RSolveValue itself:

## Generalizations & Extensions(2)

Obtain an expression for the square of the solution:

No boundary condition gives two generated parameters:

One boundary condition:

Two boundary conditions:

## Options(1)

### GeneratedParameters(1)

Use differently named constants:

Use subscripted constants:

## Applications(13)

This models the amount a[n] at year n when the interest r is paid on the principal p only:

Here the interest is paid on the current amount a[n], i.e. compound interest:

Here a[n] denotes the number of moves required in the Tower of Hanoi problem with n disks:

Here a[n] is the number of ways to tile an n×3 space with 2×1 tiles:

The number of comparisons for a binary search problem:

Number of arithmetic operations in the fast Fourier transform:

The nested radical satisfies a nonlinear difference equation:

Compute the exact value of the nested radical:

Visualize the convergence of the sequence to the exact value:

The integral satisfies the difference equation:

The integral satisfies the difference equation:

The difference equation for the series coefficients of :

The determinant of an n×n tridiagonal matrix with diagonals c, a, b satisfies:

This models the surface area s[n] in dimension n of a unit sphere:

The volume of the unit ball in dimension n:

Applying Newton's method to , or computing : Applying the Euler forward method to yields:

Solve the difference equation that describes the complexity of Karatsuba multiplication:

Compare with the complexity of schoolbook multiplication:

Efficiently compute the n term of a linear recurrence for large n:

## Properties & Relations(9)

RSolveValue returns an expression for the solution:

RSolve returns a rule for the solution:

Solutions satisfy their difference and boundary equations:

Difference equation corresponding to Sum:

Difference equation corresponding to Product:

RSolveValue finds a symbolic solution for this difference equation:

RecurrenceTable generates a procedural solution for the same problem:

FindLinearRecurrence finds the minimal linear recurrence for a list:

RSolveValue finds the sequence satisfying the recurrence:

LinearRecurrence generates the n term for a linear recurrence:

Obtain the same result using RSolveValue:

Use RecurrenceFilter to filter a signal:

Solve the corresponding difference equation using RSolveValue:

Forecast the next value for a time series based on ARProcess:

Obtain the same result using RSolveValue:

## Possible Issues(3)

The solution to this difference equation is unique as a sequence:

As a function it is only unique up to a function of period 1:

RSolveValue returns only a single branch if the solution has multiple branches: Use RSolve to get all of the solution branches:

Verify the solution when the equation involves subscripted variables:

## Neat Examples(1)

Compute the n iterate or composition of a function: