Predictor-corrector methods try to combine the advantages of the simplicity of explicit methods with the improved stability and accuracy of implicit methods. They achieve this by using an explicit method to predict the solution Yn+1(p) at tn+1 and then utilise f(tn+1,Yn+1(p)) as an approximation to f(tn+1,Yn+1) to correct this prediction using something similar to an implicit step.
Numerical methods were adapted for the direct solution of non-isothermal kinetic equations, and their use was illustrated by their application to the thermogravimetry of oil shale pyrolysis.
The method is based on the fact that at each iteration of an interior point algorithm it is necessary to compute the Cholesky decomposition (factorization) of a large matrix in order to find the search direction. The factorization step is the most computationally expensive step in the algorithm. Therefore it makes sense to use the same decomposition more than once before recomputing it.
At each iteration of the algorithm, Mehrotra's predictor-corrector method uses the same Cholesky decomposition to find two different directions: a predictor and a corrector.
The idea is to first compute an optimizing search direction based on a first order term (predictor). The step size that can be taken in this direction is used to evaluate how much centrality correction is needed. Then, a corrector term is computed: this contains both a centrality term and a second order term.
The complete search direction is the sum of the predictor direction and the corrector direction.
Although there is no theoretical complexity bound on it yet, Mehrotra's predictor-corrector method is widely used in practice.[2] Its corrector step uses the same Cholesky decomposition found during the predictor step in an effective way, and thus it is only marginally more expensive than a standard interior point algorithm. However, the additional overhead per iteration is usually paid off by a reduction in the number of iterations needed to reach an optimal solution. It also appears to converge very fast when close to the optimum.
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PLZZ ANY 1 SND GDB solution OV mth603 .......--
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