Numerical Computing

Task-oriented automatic solvers

Task-based Mathematica functions solve problems by automatically selecting the appropriate numerical method, even switching in mid-calculation. With hundreds of methods to choose from, this optimized algorithm selection improves speed and reliability over manual specification.

Symbolically enhanced numeric computing

With behind-the-scenes symbolic calculations, Mathematica optimizes the performance of numerical computations for time and accuracy—and makes previously unsolvable calculations directly computable. Examples include the intelligent handling of piecewise functions, discontinuities, and automatic expression transformation ahead of numerical sampling.

Results at any precision

Any number precision or number size can be used across all functions, allowing answers accurate to almost any number of digits. Internally, higher-precision calculations are often used automatically.

Unique numerical precision tracking

Mathematica automatically tracks and communicates how many digits of the result are accurate, giving almost complete protection from numerical errors, be they round-off errors or from badly conditioned systems.

Local and global optimization

Mathematica includes a full range of state-of-the-art optimization techniques, including constrained and unconstrained local optimization using conjugate gradient, interior point, and other methods; global optimization using Nelder–Mead, simulated annealing, and other methods; linear programming; traveling salesman problems; and more.

Advanced differential equation solving

Numerically solve ordinary and partial differential equations, parametric differential equations, and systems of nonlinear differential equations of any order. Mathematica's built-in methods include implicit and explicit Runge–Kutta and multistep methods, specialized methods for stiff equations, method of lines, and many more. Includes algorithms for computing derivatives of arbitrary target functions via sensitivity solutions.

Differential algebraic equations and hybrid systems

Solve differential-algebraic equations and hybrid systems with a mix of continuous and discrete time behavior. Automatic detection of discontinuous functions provides accurate solutions for discontinuous differential equations expressed in natural mathematical specification.

Integration and summation

Compute single and multidimensional numerical integrals and numerical sums and products of sequences. Many integration methods including globally adaptive subdivision, Gaussian and Clenshaw–Curtis quadrature rules, and specialized high-dimensional and oscillatory rules.

Numerical equation solving

Numerical root-finding of functions and systems of simultaneous equations are built into Mathematica. Methods include Newton, Secant, and Brent as well as specialized methods for efficient numerical solutions of systems of polynomial equations.

Linear algebra and sparse arrays

Improve speed and memory use with robust linear algebra on dense matrices using industry-standard, high-performance libraries; sparse arrays of any dimension; and numerical linear algebra on arbitrary precision and mixed symbolic-numeric matrices.