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Titlebook: Complexity and Real Computation; Lenore Blum,Felipe Cucker,Steve Smale Textbook 1998 Springer Science+Business Media New York 1998 algorit

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书目名称Complexity and Real Computation
编辑Lenore Blum,Felipe Cucker,Steve Smale
视频video
概述Unique work on this core topic * Written by internationally recognised specialists in mathematics and computing * Provides the basics for numerous practical industrial applications, e.g. AI, robotics,
图书封面Titlebook: Complexity and Real Computation;  Lenore Blum,Felipe Cucker,Steve Smale Textbook 1998 Springer Science+Business Media New York 1998 algorit
描述Computational complexity theory provides a framework for understanding the cost of solving computational problems, as measured by the requirement for resources such as time and space. The objects of study are algorithms defined within a formal model of computation. Upper bounds on the computational complexity of a problem are usually derived by constructing and analyzing specific algorithms. Meaningful lower bounds on computational complexity are harder to come by, and are not available for most problems of interest. The dominant approach in complexity theory is to consider algorithms as oper­ ating on finite strings of symbols from a finite alphabet. Such strings may represent various discrete objects such as integers or algebraic expressions, but cannot rep­ resent real or complex numbers, unless the numbers are rounded to approximate values from a discrete set. A major concern of the theory is the number of com­ putation steps required to solve a problem, as a function of the length of the input string.
出版日期Textbook 1998
关键词algorithms; complexity; fundamental theorem; linear optimization; theoretical computer science
版次1
doihttps://doi.org/10.1007/978-1-4612-0701-6
isbn_softcover978-1-4612-6873-4
isbn_ebook978-1-4612-0701-6
copyrightSpringer Science+Business Media New York 1998
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Newton’s Methodr a polynomial of one complex variable we cannot decide if Newton’s method will converge to a root of the polynomial on a given input. In this chapter we begin a more comprehensive study of Newton’s method. We introduce quantities α, β, and γ which play an important role in analyzing the complexity
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Bézout’s Theoremex polynomial equations in .-unknowns. It is the goal of this chapter to prove Bézout’s Theorem. In Chapter 16 we use Bézout’s Theorem as a tool to derive geometric upper bounds on the number of connected components of semi-algebraic sets and complexity-theoretic lower bounds on some problems such a
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Linear Programming Section 15.1 we show that inputs for rational machines can be supposed to be given by pairs of integers without substantially altering the complexity of the considered problem. In Section 15.2 we define an auxiliary problem which is a modification of the linear programming optimization problem and
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The Class NP and NP-Complete Problemsy a solution that may be presented to us. Just plug the purported solution into the polynomial and evaluate it. Is this verification tractable in our model of computation? An affirmative answer will depend on the underlying mathematical properties of the ring or field, as well as our measure of complexity, and is at the core of the notion of NP.
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