The Multiprecision Computing Toolbox is the MATLAB extension for computing with arbitrary precision.

The toolbox equips MATLAB with a new multiple precision floating-point numeric type and extensive set of mathematical functions that are capable of computing with arbitrary precision.

The multiprecision numbers and matrices can be seamlessly used in place of the built-in double entities following standard MATLAB syntax rules. As a result, existing MATLAB programs can be converted to run with arbitrary precision with minimal changes to source code.

Quadruple precision computations (compliant with IEEE 754-2008) are supported as a special case.

Functionality

Toolbox provides a comprehensive library of computational routines covering the following areas:

• Real and complex numbers, full and sparse matrices, multidimensional arrays
• Elementary and special mathematical functions
• Solvers for system of linear equations (including direct and iterative sparse solvers)
• Matrix analysis functions and factorizations
• Eigenvalues and eigenvectors including generalized and large-scale problems.
• Singular value decomposition
• Solvers for system of nonlinear equations (fsolve with Levenberg-Marquardt and other trust region methods)
• Numerical integration (including adaptive quadgk and full set of Gaussian quadrature)
• Optimization and polynomials
• Ordinary differential equations solvers
• Data analysis and Fourier transform
• Number theory functions

Please visit the Function Reference page for a complete list of supported functions and User’s Manual for usage examples.

Performance

There is a common misbelief that arbitrary precision computations are very slow. Indeed, mainstream numerical software packages are largely responsible for this false perception. Their codes were written decades ago using textbook algorithms, without proper optimization nor updates for the latest hardware.

We are determined to change the situation by developing high-performance numerical libraries for computations with arbitrary precision, tuned for modern CPU architectures, multi-core parallelism and relying on recent state-of-the-art algorithms. All combined makes our toolbox order(s) of magnitude faster compared to famous competitors:

More detailed comparison with VPA: Symbolic Math Toolbox (VPA) vs. Multiprecision Computing Toolbox.

In some cases, the speed of quadruple precision computations in toolbox is comparable (or even higher) to double precision routines of MATLAB. Eigendecomposition of banded matrices is one of the examples.

Demos

Computation of eigenvalues and especially eigenvectors is one of classic problems requiring extended-precision. The main reason is that eigenproblem might be ill-conditioned and hard to compute even when matrix itself is well-conditioned with respect to inversion.

The two plots below show the eigenvalues of the Grcar matrix computed by MATLAB and by the toolbox, respectively. The eigenvalues of the Grcar matrix are displayed in black, and the eigenvalues of the transposed matrix – in red. In theory they should coincide:

Figure 1. MATLAB is unable to generate accurate results using the double precision.

Figure 2. Correct eigenvalues computed with quadruple precision by the toolbox.

Although condition number of the Grcar matrix is low, cond(A) = cond(A') = 3.61, MATLAB’s double precision routines suffer from accuracy loss. See for details Computing Eigenvalues in Extended Precision.

Applications

The toolbox removes MATLAB’s inherent limitation on computing precision, thus enabling the user to solve a variety of important problems previously impossible to handle with MATLAB:

• Allows solving of numerically unstable problems (e.g. computing eigenvalues, matrix functions, solving ill-conditioned matrices, algebraic Riccati equations, differential equations using RBF methods, finding roots, analyzing control system stability by solving Lyapunov and Sylvester equations, etc. etc.).
• Enables checking of accuracy of results by re-running existing MATLAB programs in extended precision.
• Minimizes rounding and cancellation errors in computations (e.g. summation of series with alternating signs, oscillatory integrals, etc.).
• Allows computing of reference constants and coefficients of algorithms with high accuracy.

Key Features

• Comprehensive set of mathematical functions. Multiprecision Computing Toolbox supplies arbitrary-precision analogs to the majority of the functions listed in the famous MATLAB Top 500, from simple arithmetic operations to advanced numerical algorithms (e.g. fft, eig, svd, quadgk, ode45, etc.), and adds original functionality not available in MATLAB. Please visit the Function Reference page for a complete list of supported routines.
• Easy porting of existing MATLAB programs to arbitrary precision.
  The Toolbox employs an operator overloading technique which allows easy porting of pre-existing programs to arbitrary precision – with little or no modifications to the source code. In most cases the conversion is automatic, and no extra action is required by the user. For further examples, visit User’s Manual.
• Algorithms. Multiprecision Computing Toolbox stands on the shoulders of giants – we implement only state-of-the-art algorithms designed for parallelism and high-performance. Usually we convert algorithms from LAPACK to precision-independent C++ code with various optimizations for vector and multi-core execution. Every routine in toolbox is a meta-algorithm, which analyses the input matrix and applies best suitable method for its type (symmetric/Hermitian, positive definite, triadiagonal, band, triangular, etc.). Thus, for example, we use multi-shift QR for unsymmetric eigen-decomposition whereas symmetric/Hermitian or tridiagonal problems are solved by MRRR or Divide & Conquer algorithms.
• High-performance. Our product is developed using compiled programming languages C/C++ and Assembler with only a thin M-file interface for smooth integration to MATLAB. Toolbox is specifically optimized for parallel execution on multi-core CPUs.
• Ease of access and user-friendliness.
  We provide comprehensive instructions and step-by-step installation help, making it easy to install and use the Multiprecision Computing Toolbox. Toolbox is available for 64-bit versions of GNU Linux, Microsoft Windows, and Apple Mac OS. The Quick Start User’s Manual, Function Reference, and our frequently updated Blog provide examples, tips, and announcements about our product.

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