Continuation Definition/Meaning:
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
ƒ(x, a) = F(x) + (α
- 1) Fx(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 for the next
problem.
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