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Induction Variable
In computer science, an induction variable is a variable that gets increased or decreased by a fixed amount on every iteration of a loop or is a linear function of another induction variable. For example, in the following loop, i and j are induction variables: for (i = 0; i < 10; ++i) Application to strength reduction A common is to recognize the existence of induction variables and replace them with simpler computations; for example, the code above could be rewritten by the compiler as follows, on the assumption that the addition of a constant will be cheaper than a multiplication. |
Computer Science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (including the design and implementation of hardware and software). Computer science is generally considered an area of academic research and distinct from computer programming. Algorithms and data structures are central to computer science. The theory of computation concerns abstract models of computation and general classes of problems that can be solved using them. The fields of cryptography and computer security involve studying the means for secure communication and for preventing security vulnerabilities. Computer graphics and computational geometry address the generation of images. Programming language theory considers different ways to describe computational processes, and database theory concerns the management of repositories ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Successor Function
In mathematics, the successor function or successor operation sends a natural number to the next one. The successor function is denoted by ''S'', so ''S''(''n'') = ''n'' +1. For example, ''S''(1) = 2 and ''S''(2) = 3. The successor function is one of the basic components used to build a primitive recursive function. Successor operations are also known as zeration in the context of a zeroth hyperoperation: H0(''a'', ''b'') = 1 + ''b''. In this context, the extension of zeration is addition, which is defined as repeated succession. Overview The successor function is part of the formal language used to state the Peano axioms, which formalise the structure of the natural numbers. In this formalisation, the successor function is a primitive operation on the natural numbers, in terms of which the standard natural numbers and addition is defined. For example, 1 is defined to be ''S''(0), and addition on natural numbers is defined recursively by: : This can be u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Function
In mathematics, the term linear function refers to two distinct but related notions: * In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For distinguishing such a linear function from the other concept, the term affine function is often used. * In linear algebra, mathematical analysis, and functional analysis, a linear function is a linear map. As a polynomial function In calculus, analytic geometry and related areas, a linear function is a polynomial of degree one or less, including the zero polynomial (the latter not being considered to have degree zero). When the function is of only one variable, it is of the form :f(x)=ax+b, where and are constants, often real numbers. The graph of such a function of one variable is a nonvertical line. is frequently referred to as the slope of the line, and as the intercept. If ''a > 0'' then the gradient is positive and t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compiler Optimization
In computing, an optimizing compiler is a compiler that tries to minimize or maximize some attributes of an executable computer program. Common requirements are to minimize a program's execution time, memory footprint, storage size, and power consumption (the last three being popular for portable computers). Compiler optimization is generally implemented using a sequence of ''optimizing transformations'', algorithms which take a program and transform it to produce a semantically equivalent output program that uses fewer resources or executes faster. It has been shown that some code optimization problems are NP-complete, or even undecidable. In practice, factors such as the programmer's willingness to wait for the compiler to complete its task place upper limits on the optimizations that a compiler might provide. Optimization is generally a very CPU- and memory-intensive process. In the past, computer memory limitations were also a major factor in limiting which optimizations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strength Reduction
In compiler construction, strength reduction is a compiler optimization where expensive operations are replaced with equivalent but less expensive operations. The classic example of strength reduction converts "strong" multiplications inside a loop into "weaker" additions – something that frequently occurs in array addressing. Examples of strength reduction include: * replacing a multiplication within a loop with an addition * replacing an exponentiation within a loop with a multiplication Code analysis Most of a program's execution time is typically spent in a small section of code (called a hot spot), and that code is often inside a loop that is executed over and over. A compiler uses methods to identify loops and recognize the characteristics of register values within those loops. For strength reduction, the compiler is interested in: *Loop invariants: the values which do not change within the body of a loop. *Induction variables: the values which are being iterated ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dependence Analysis
In compiler theory, dependence analysis produces execution-order constraints between statements/instructions. Broadly speaking, a statement ''S2'' depends on ''S1'' if ''S1'' must be executed before ''S2''. Broadly, there are two classes of dependencies--control dependencies and data dependencies. Dependence analysis determines whether it is safe to reorder or parallelize statements. Control dependencies Control dependency is a situation in which a program instruction executes if the previous instruction evaluates in a way that allows its execution. A statement ''S2'' is ''control dependent'' on ''S1'' (written S1\ \delta^c\ S2) if and only if ''S2s execution is conditionally guarded by ''S1''. ''S2'' is ''control dependent'' on ''S1'' if and only if S1 \in PDF(S2) where PDF(S) is the post dominance frontier of statement S. The following is an example of such a control dependence: S1 if x > 2 goto L1 S2 y := 3 S3 L1: z := y + 1 Here, ''S2'' only runs if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Induction
Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder: A proof by induction consists of two cases. The first, the base case, proves the statement for ''n'' = 0 without assuming any knowledge of other cases. The second case, the induction step, proves that ''if'' the statement holds for any given case ''n'' = ''k'', ''then'' it must also hold for the next case ''n'' = ''k'' + 1. These two steps establish that the statement holds for every natural number ''n''. The base case does not necessarily begin with ''n'' = 0, but often with ''n'' = 1, and possibly with any fixed natural number ''n'' = ''N'', establishing the truth of the statement for all natu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
