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# continuation

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1 A concept in programming language semantics, allowing the meaning of program constructs to be defined in terms of the effect they have on the computation remaining to be done, rather than on the current state of the computation. This is particularly useful in giving the semantics of constructs that effect the flow of control, such as GOTOs and loop exits.

2 An approach to solving a mathematical problem that involves solving a sequence of problems with different parameters; the parameters are selected so that ultimately the original problem is solved. An underlying assumption is that the solution depends continuously on the parameter. This approach is used for example on difficult problems in nonlinear equations and differential equations. For example, to solve the nonlinear equations F(x) = 0, let x(0) be a first approximation to the solution. Let α be a parameter 0≤α≤1, then define the equations F^(x,α) = F(x) + (α-1)F(x(0)) = 0 for α = 0, x(0) is a solution; for α = 1, F^(x,1) = F(x) = 0, which are the original equations. Hence by solving the sequence of problems with α given by 0 = α0 < α1 < … < αN = 1 the original problem is solved. As the calculation proceeds each solution can be used as a starting approximation in an iterative method for solving the next problem.

F(x) = 0,

F^(x,α) = F(x) + (α-1)F(x(0)) = 0

F^(x,1) = F(x) = 0,

0 = α0 < α1 < … < αN = 1

Subjects: Computing.

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