Modelica.Math.Nonlinear

Information

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 the Modelica Language Specification, Chapter 12.4.2.

Package Content

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)

Modelica.Math.Nonlinear.quadratureLobatto

Information

Syntax

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

Description

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/doc/tech-reports/1998/306.ps

Example

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

Inputs

Name	Description
f	Integrand function
a	Lower limit of integration interval
b	Upper limit of integration interval
tolerance	Relative tolerance for integral value

Outputs

Name	Description
integral	integral value

Modelica.Math.Nonlinear.solveOneNonlinearEquation

Information

Syntax

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

Description

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: http://wwwmaths.anu.edu.au/~brent/pd/rpb011i.pdf
Errata and new print: http://wwwmaths.anu.edu.au/~brent/pub/pub011.html

Example

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

Inputs

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

Outputs

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