By Ernst Hairer, Christian Lubich, Michel Roche

The time period differential-algebraic equation was once coined to include differential equations with constraints (differential equations on manifolds) and singular implicit differential equations. Such difficulties come up in a number of purposes, e.g. limited mechanical platforms, fluid dynamics, chemical response kinetics, simulation of electric networks, and regulate engineering. From a extra theoretical point of view, the research of differential-algebraic difficulties supplies perception into the behaviour of numerical tools for stiff usual differential equations. those lecture notes supply a self-contained and accomplished therapy of the numerical resolution of differential-algebraic platforms utilizing Runge-Kutta tools, and in addition extrapolation equipment. Readers are anticipated to have a historical past within the numerical therapy of standard differential equations. the topic is taken care of in its quite a few elements starting from the speculation in the course of the research to implementation and functions.

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5 Latency and Bandwidth Getting there is half the fun—old saying, regarding the pleasures of traveling The speed of a communications channel—whether between processor cores and memory in shared-memory platforms, or between network nodes in a cluster of machines—is measured in terms of latency, the end-to-end travel time for a single bit, and bandwidth, the number of bits per second that we can pump onto the channel. To make the notions a little more concrete, consider the San Francisco Bay Bridge, a long, multilane structure for which westbound drivers pay a toll.

1) to deal with network latency, and in GPUs (Chapter 6) to deal with memory latency. ), and many data scientists have access to computer clusters. What are the performance issues on these platforms? The next two sections 26 CHAPTER 2. 1: Symmetric Multiprocsssor System provide an overview. 1 in terms of components and, most importantly, their interconnection. What do we see? • There are processors, depicted by the Ps, in which your program is physically executed. 7 • The processors and memory banks are connected to a bus, a set of parallel wires used for communication between these computer components.

They run independently, except that they share the data of the program in common. If your program is threaded, it will be running on several of the processors at once, each thread on a different core. A key point, as we will see, is that the shared 7 These were called banks in the old days. Later the term modules became more popular, but with the recent popularity of GPUs, the word banks has come back into favor. 5. LATENCY AND BANDWIDTH 27 memory becomes the vehicle for communication between the various processes.