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PROFESSOR: Software that implements modern numerical methods has two features that aren't present in codes like ODE4 and classical Runge-Kutta. The methods in the software can estimate error and provide automatic step size control. You don't specify the step size h. You specify an accuracy you want. And the methods estimate the errors as they go along and adjust the step size accordingly. And they provide a fully accurate continuous interpolant. They don't just provide the solution at the discrete set of points. They provide a function that defines the solution everywhere in the interval. And so you can plot it, find zeroes of the function, provide a facility called event handling, and so on. Larry Shampine is an authority on the numerical solution of ordinary differential equations. He is the principal author of this textbook about solving ODEs with MATLAB. He's a, now, emeritus professor at the Southern Methodist University in Dallas. And he's been a...

The modern world has an insatiable appetite for computation, so system architects are always thinking about ways to make programs run faster. The running time of a program is the product of three terms: The number of instructions in the program, multiplied by the average number of processor cycles required to execute each instruction (CPI), multiplied by the time required for each processor cycle (t_CLK). To decrease the running time we need to decrease one or more of these terms. The number of instructions per program is determined by the ISA and by the compiler that produced the sequence of assembly language instructions to be executed. Both are fair game, but for this discussion, let's work on reducing the other two terms. As we've seen, pipelining reduces t_CLK by dividing instruction execution into a sequence of steps, each of which can complete its task in a shorter t_CLK. What about reducing CPI? In our 5-stage pipelined implementation of the Beta, we desig...