1.3 Robustness of Numerical Methods
Most algorithms in Advanced Numerical Methods have proven to be numerically stable either by round-off error analysis or by extensive empirical evidence. However, one should be aware that the numerical stability of an algorithm does not guarantee the accuracy of the solution if the problem is inherently ill-conditioned.
As is the case with many built-in numerical functions, the algorithms in Advanced Numerical Methods use the options WorkingPrecision and Tolerance. Typically, the default value of the option WorkingPrecision is Automatic, which further defaults to $MachinePrecision for machine-precision input and ensures that the computations are performed in the most expedient fashion using the machine arithmetic. The working precision is automatically raised for a high-precision input. The default value of the option Tolerance is Automatic, which specifies that numbers with absolute values less than, or equal to, the epsilon multiplied by a certain algorithm-dependent constant are considered to be numerical zeros.
For the iterative algorithms, the iterations are computed until one of the following conditions is achieved: convergence (up to a given tolerance) occurs; numerical saturation is detected (when precision of the arithmetic is insufficient to reach the desired tolerance); or an upper bound on the number of iterations, specified by the value of the option MaxIterations, is exceeded. By default, the maximal number of iterations is determined for each iterative algorithm individually, based on the size of the problem. The value Infinity disables the last stopping criterion, which involves potentially very expensive computations when the convergence is slow.
Common options of the numerical algorithms.
Make sure the application is loaded.
Load the collection of test examples.
This is the state-space system of a simple process control of a paper machine. This guide typically uses the control format to display results.
This solves the discrete Lyapunov equation .
Here is a residual of the solution. This and other small numerical residuals may be slightly different on your computer system (other general considerations on reproducing results on your computer, as outlined in Section 1.8 of Control System Professional, also apply).
This solves the same discrete Lyapunov equation using a higher working precision. NumberForm displays a smaller sized output.
The solution is more accurate.