New PDF release: Computational integration

By Arnold R. Krommer

ISBN-10: 0898713749

ISBN-13: 9780898713749

This article discusses computational integration equipment and the basic mathematical ideas they're in response to. It contains sections on one-dimensional and multi-dimensional integration formulation, and it bargains with concerns about the building of numerical integration algorithms.

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Example text

Let λχ and λ2 be two distinct real roots of the characteristic polynomial L(p); then the function (eXl* — βλ2*)/(λι — λ 2 ) is a solution of (1). If we now assume that with a change in the coefficients of L(p) the number λ 2 tends to λχ, then this solution tends (in the limit) to the function te^l\ which may naturally be assumed to be a solution of (1) whenever λχ is a double root of the polynomial L(p). Similarly, we arrive at the conjecture that, if λ is a fc-tuple root of the characteristic polynomial L(p), then all the functions ext, teu, .

Further, we shall develop the so-called method of complex amplitude, which is a convenient means for determining particular steady-state solutions and is widely applied in electrical engineering. Rather than confining ourselves to the solution of the purely mathemati­ cal problems arising from applications, we shall present here in very short dogmatic form an exposition of the theory of electrical circuits. The design of electrical circuits gives a good and important, from the engineering point of view, illustration of the mathematical methods developed in this chapter.

N. (4) If a;* = ^*(0, t = 1, . . , n, is a solution of (3) and xi = x*(0, ί = 1, . , n, is a solution of (4), then the system of functions x{ = αφ\0 + ßx\t), is a solution of (1). i = 1, . . , n, CHAPTER 2 LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS Systems of ordinary differential equations with constant coefficients constitute a large and important class of ordinary differential equations which may be solved completely with the aid of elementary functions. In view of the fact that the solution of these equations does not, in prin­ ciple, present any great difficulties, they are often considered to be of no great interest for theory, and in textbooks they are usually relegated to the position of simple exercises appended to the general theory of linear equa­ tions.

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Computational integration by Arnold R. Krommer

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