Library of functions operating on nonlinear equations

This package contains functions to perform tasks such as numerically integrating a function, or solving a nonlinear algebraic equation system. The common feature of the functions in this package is that the nonlinear characteristics are passed as user definable functions.

For details about how to define and to use functions as input arguments to functions, see ModelicaReference.Classes.'function' or Section 12.4.2 (Functional Input Arguments to Functions) of the Modelica 3.4 specification.

Extends from Modelica.Icons.Package (Icon for standard packages).

Name | Description |
---|---|

Examples | Examples demonstrating the usage of the functions in package Nonlinear |

Interfaces | Interfaces for functions |

quadratureLobatto | Return the integral of an integrand function using an adaptive Lobatto rule |

solveOneNonlinearEquation | Solve f(u) = 0 in a very reliable and efficient way (f(u_min) and f(u_max) must have different signs) |

Return the integral of an integrand function using an adaptive Lobatto rule

quadratureLobatto(function f(), a, b);quadratureLobatto(function f(), a, b, tolerance=100*Modelica.Constants.eps);

Compute definite integral over function f(u,...) from u=a up to u=b using the adaptive Lobatto rule according to:

- Walter Gander:
Adaptive Quadrature - Revisited. 1998. ftp://ftp.inf.ethz.ch/pub/publications/tech-reports/3xx/306.ps

See the examples in Modelica.Math.Nonlinear.Examples.

Extends from Modelica.Icons.Function (Icon for functions).

Name | Description |
---|---|

f | Integrand function |

a | Lower limit of integration interval |

b | Upper limit of integration interval |

tolerance | Relative tolerance for integral value |

Name | Description |
---|---|

integral | Integral value |

Solve f(u) = 0 in a very reliable and efficient way (f(u_min) and f(u_max) must have different signs)

solveOneNonlinearEquation(function f(), u_min, u_max);solveOneNonlinearEquation(function f(), u_min, u_max, tolerance=100*Modelica.Constants.eps);

This function determines the solution of **one non-linear algebraic equation** "y=f(u)"
in **one unknown** "u" in a reliable way. It is one of the best numerical
algorithms for this purpose. As input, the nonlinear function f(u)
has to be given, as well as an interval u_min, u_max that
contains the solution, i.e., "f(u_min)" and "f(u_max)" must
have a different sign. The function computes a smaller interval
in which a sign change is present using the relative tolerance
"tolerance" that can be given as 4th input argument.

The interval reduction is performed using inverse quadratic interpolation (interpolating with a quadratic polynomial through the last 3 points and computing the zero). If this fails, bisection is used, which always reduces the interval by a factor of 2. The inverse quadratic interpolation method has superlinear convergence. This is roughly the same convergence rate as a globally convergent Newton method, but without the need to compute derivatives of the non-linear function. The solver function is a direct mapping of the Algol 60 procedure "zero" to Modelica, from:

- Brent R.P.:
Algorithms for Minimization without derivatives. Prentice Hall, 1973, pp. 58-59.

Download: https://maths-people.anu.edu.au/~brent/pd/rpb011i.pdf

Errata and new print: https://maths-people.anu.edu.au/~brent/pub/pub011.html

See the examples in Modelica.Math.Nonlinear.Examples.

Extends from Modelica.Icons.Function (Icon for functions).

Name | Description |
---|---|

f | Function y = f(u); u is computed so that y=0 |

u_min | Lower bound of search interval |

u_max | Upper bound of search interval |

tolerance | Relative tolerance of solution u |

Name | Description |
---|---|

u | Value of independent variable u so that f(u) = 0 |