Reliable Evaluation of the Worst-Case Peak Gain Matrix in Multiple Precision

The worst-case peak gain (WCPG) of a linear filter is an important measure for the implementation of signal processing algorithms. It is used in the error propagation analysis for filters, thus a reliable evaluation with controlled precision is required. The WCPG is computed as an infinite sum and has matrix powers in each summand. We propose a direct formula for the lower bound on truncation order of the infinite sum in dependency of desired truncation error. Several multiprecision methods for complex matrix operations are developed and their error analysis performed. A multiprecision matrix powering method is presented. All methods yield a rigorous solution with an absolute error bounded by an a priori given value. The results are illustrated with numerical examples.