Mantissa (Mathematical Algorithms for Numerical Tasks In Space System Applications) contains various algorithms useful for dynamics simulation and 3D geometry computation.
The library error messages in exceptions are internationalized (only english and french are supported for now).
Mantissa contains a collection of algorithms, among which:
■ A small set of linear algebra classes
■ Least squares estimator (one Gauss-Newton based, one Levenberg-Marquardt based, which should even work for over-determined systems)
■ Some curve fitting classes
■ Several ordinary differentials equations integrators, either with fixed steps or ■ Adaptive stepsize control (see below)
■ Vectors and rotations in a three dimensional space
■ Algebra-related classes like rational and double polynomials
■ Various orthogonal polynomials:
■ Chebyshev
■ Hermite
■ Laguerre
■ Legendre
■ Some random numbers and vectors generation classes:
■ Robert M. Ziff four tap shift register (contributed by Bill Maier)
■ Makoto Matsumoto and Takuji Nishimura Mersenne twister
■ Generators for vectors with correlated components
■ Some basic (min, max, mean, standard deviation) statistical analysis classes
■ Some optimization algorithms using direct search methods:
■ The Nelder-Mead simplex method
■ Virginia Torczon’s multi-directional method
Mantissa is devoted to be a general purpose library, however, its most popular feature is an extensive package for Ordinary Differential Equations integration. This package is intended to be very efficient and provide a complete ODE integration framework with many practical controls while still remaining a simple to use tool.
All integrators provide dense output. This means that besides computing the state vector at discrete times, they also provide a cheap mean to get the state between the time steps.
All integrators handle multiple switching functions. This means that the integrator can be driven by discrete events (occurring when the signs of user-supplied switching functions change).
The steps are shortened as needed to ensure the events occur at step boundaries (even if the integrator is a fixed-step integrator).
When the events are triggered, integration can be stopped (this is called a G-stop facility), the state vector can be changed, or integration can simply go on. The latter case is useful to handle discontinuities in the differential equations gracefully and get accurate dense output even close to the discontinuity.
The solution of the integration problem is provided by two means. The first one is aimed towards simple use: the state vector at the end of the integration process is copied in a user-supplied array. The second one should be used when more in-depth information is needed throughout the integration process.
The user can register an object implementing the StepHandler interface into the integrator before performing integration. The user object will be called appropriately during the integration process, allowing the user to process intermediate results.
The default step handler does nothing. Mantissa also provides a special-purpose step handler that is able to store all steps and to provide transparent access to any intermediate result once the integration is over.
This object is serializable, hence a complete continuous model of the integrated function throughout the integration range can be reused later (if stored into a persistent medium like a filesystem or a database) or elsewhere (if sent to another application in a distributed system).
Some integrators (the simple ones) use fixed steps that are set at creation time. The more efficient integrators use variable steps that are handled internally in order to control the integration error with respect to a specified accuracy.
Adaptive stepsize integrators can automatically compute the initial stepsize by themselves, however the user can specify it if he prefers to retain full control over the integration or if the automatic guess is wrong.


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■ The baseline run
In the baseline run, Mantissa is based on the idea of defining a transient process by using a transition process for the state vector instead of using the state equations.
This idea has several advantages:
■ Transient processes are easier to build than steady-state ones, and the library presents a coherent set of classes for their design.
■ New algorithms can be proposed by modifying the transient process directly.
■ There is no need to hack steady-state equations.
■ The transient process can be built independently from the simulation, making a tool to model transient processes.
■ The transient process can be stored and reused to obtain simulations even for complex transient processes.
■ The steady-state equation is never needed: the simulation code is just as simple as the transient equation.
The transient process is built from the fact that the state vector changes whenever a discrete event occurs. A discrete event is specified by a switching function. The transient process is then built from a sequence of transients of functions returning the state vector. A transient function returns the state vector at the next possible discrete event occurrence.
A transient process is defined by a transient function which returns the state vector after a discrete event. It is also possible to associate a state vector at the time of a discrete event.
Mantissa adds the possibility to define a list of continuous events (or continuous processes, in short) to the transient process. A continuous event is specified by a switching function, as well as by the discrete event which causes it. To define a continuous event, a continuous function is required. This continuous function returns the state vector at an intermediate state if the events has happened.
An interesting feature of Mantissa is that any transient process with continuous events can be used as a steady-state process, that is a steady-state process with a steady-state equation. It is thus possible to obtain a steady-state simulation by calling a transient process without steady-state function. The transient process is thus always guaranteed to perform the simulation up to the final discrete events, at which point it becomes a steady-state process.
■ A transient process can be described as a differential equation:
– transient =
a0 (e0 (t)) dt + (e1 (t)) dt

+an (en (t)) dt

c0 (e0

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Mantissa is written in C++. The different features of Mantissa are located in various files in the Mantissa packages (or sub-packages). This allows to separate the different parts of the library from each other as much as possible.
Bug reports
Bug reports (security and possibly usability issues) can be sent to Bugs will get a reply only if necessary.
Obtaining the source code
The source code of the package can be obtained from the official Pobox package repository ( Once the package is downloaded, run the unpacks command to untar and unpack the source code.
To run a sample program that uses all of the Mantissa components, just type make.
The documentation
Documentation is available online at the Mantissa Wiki ( A standalone documentation viewer can be found at
The documentation is in XML format. Since the documentation is generated by the Doxygen builder, it can be configured and edited.
The Mantissa project is covered under the GNU General Public License v2 or later.
See GNU General Public License.
See the file COPYING.

Mantissa provides interfaces for dimensionally consistent data storage structures. A basic data structure for use by clients of the package is an extended data-type.
An extended data-type is a data type with additional attributes. This is a data type defined by the class DataType and derived classes like Expression. A basic extended data-type contains one datatype (a class). It also contains one or more variables. The variables will be either scalar variables or vector variables.
Matrix types
A matrix is a discrete set of elements of an extended data-type. It is a user-defined sequence of rows (indexed by one or more variables), each row representing the elements of a vector.
A row of the matrix is a vector. A vector (in general) is an extended data-type that has only one variable, the variable at which it is indexed by. A column of a matrix (in general) is either a scalar, a vector or a matrix. The size of the column of a matrix is defined by the number of rows, the number of columns is defined by the size of the vector

What’s New in the?

Mantissa’s main goal is to provide user-friendly modules for the integration of differential-equations driven by user events.
Although Mantissa is oriented toward the integration of time-discrete models, it is not limited to this. If the differential equations are time-continuous, they can be integrated either by time-discrete integrators or by time-continuous integrators. The discrete integrators integrate at discrete times to give a time-discrete model. The continuous integrators integrate until a certain accuracy is reached to give a time-continuous model.
In Mantissa, integration is performed by applying ODE integrators. Mantissa supports integrators based on the exact ODE solvers (Newton, Levenberg-Marquardt, or Gauss-Newton) and integrators based on the variable stepsize approach. Integrators based on the exact solvers have a cost proportional to the number of calls to ODE solvers, hence the Gauss-Newton-based integrators are highly recommended.
Integrators based on the variable stepsize approach can control the error much better than the exact solvers integrators, however they use more calls to the ODE solver and the cost is linear in the number of evaluations of the ODE. The variable stepsize integrators are also better at handling discontinuities.
Mantissa provides several classes of ODE integrators.
All classes use an internal mesh of parameters to discretize the domain. The degree of the approximation polynomials is automatically selected according to the differentiability of the right-hand-side functions, and then finer meshes are used to have better accuracy.
All classes can be used for integration in time. This is done using standard Runge-Kutta methods, Nordsieck methods, and rational methods.
All classes can also be used for integration in space. This is performed by using the same mesh parameter for all the different space directions.
A powerful class of integrators is provided by Mantissa. These integrators use the Exact Differential Equations library (by Takuji Nishimma and Makoto Matsumoto) to provide an extremely accurate integration. This class of integrators has been written and used extensively by the authors, and it is competitive with other ODE solvers.
The integrators are significantly more efficient than the Nordsieck methods. Even the ones based on Newton’s method are about ten times more efficient. The methods

System Requirements For Mantissa:

512 MB RAM
1024×768 resolution
DirectX: 9.0c
Windows XP
Windows Vista
Windows 7
Windows 8
Minimal Hardware:
Keyboard, Mouse, Monitor
Stereoscope: 2 monitors
A list of the components used and their manufacturer is given in the component list below.
In this project we use several components made by different manufacturers. If the model of a component is given in the component list it indicates the manufacturer.