	*Mathematica*

		Fast Dense Numerical Linear Algebra
		High-Speed Sparse Linear Algebra
		Large-Scale Linear Systems
		Big-Number Arithmetic
		64-Bit Platform Support
		Faster MathLink
		Numerical Solving of Differential Equations
		Numerical Equation Solving
		Fitting Data
		Numeric Optimization
		`DSolve`
		`RSolve`
		`Reduce`
		`Resolve`
		`FindInstance`
		Symbolic Global Optimization
		Wider Support For Domain Specifications
		Connection Technology
		Importing and Exporting
		Statistical Plots
		Collecting Expressions During Evaluation
		New Linear Algebra Functions
		Algebraic Number Objects
		Authoring and Presentation
		Summary of New Features

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RSolve

Now a built-in Mathematica function, the newly expanded RSolve is one of the most extensive systems available today for solving recurrence relations. These include linear and nonlinear difference equations, difference algebraic equations, partial difference equations, q-difference, and divide-and-conquer equations.

Example: Solving Recurrence Equations

This gives the general solution to involving one arbitrary constant .

${{y(k) -> c_1 [CapitalGamma](k + 1)}}$

The following example solves a linear system of a difference and an algebraic equation. This second-order system has only one general constant in its solution because one degree of freedom has already been removed by the algebraic equation.

${{x(k) -> 1/4 (-1)^k c_1, y(k) -> -1/4 (-1)^k c_1}}$

RSolve can solve partial difference equations. In these cases the general solution is parameterized by general functions such as below.

${{u(k, l) -> a^(k - 1) c_1[l - k]}}$

RSolve can also solve so-called q-difference equations or divide-and-conquer equations. These are equations of the form .

This is a linear q-difference equation that gives the number of comparisons when doing a binary search.

		Documentation from The Mathematica Book Solving Recurrence Equations

		Documentation from the Reference Guide `RSolve`