Chapter 19: In this chapter, we introduce new examples of norms, with special attention to submultiplicative norms on matrices. These norms are well-adapted to applications involving power series of matrices and iterative numerical algorithms. We use them to prove a formula for the spectral radius that is the key to a fundamental theorem on positive matrices in the next chapter.

The solution of partial differential equations in two and three-dimensions using stencil iteration (Jacobi’s method) is discussed and illustrated for Laplace’s equation. A very simple kernel gives about a factor of 100 speed-up compared to the host CPU.The very slow convergence of the Jacobi method can be addressed by using solutions on lower resolution grids to initialise higher resolution grids. A convergence check using the maximum change per iteration is also illustrated. Digital image processing is another example of stencil use and a number of digital image filters are shown including the Sobel filter for edge finding and the median filter for noise reduction. The fast GPU-based median filter uses one thread per image pixel and is implemented using an optimal Batcher network.