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Thuật toán Algorithms (Phần 8)

Tham khảo tài liệu 'thuật toán algorithms (phần 8)', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | GAUSSLAN ELIMINATION 63 Obviously a library routine would check for this explicitly. An alternate way to proceed after forward elimination has created all zeros below the diagonal is to use precisely the same method to produce all zeros above the diagonal first make the column zero except for a N N by adding the appropriate multiple of N then do the same for the to-last column etc. That is we do partial pivoting again but on the other part of each column working backwards through the columns. After this process called Gauss- Jordan reduction is complete only diagonal elements are non-zero which yields a trivial solution. Computational errors are a prime source of concern in Gaussian elimination. As mentioned above we should be wary of situations when the magnitudes of the coefficients vastly differ. Using the largest available element in the column for partial pivoting ensures that large coefficients won t be arbitrarily created in the pivoting process but it is not always possible to avoid severe errors. For example very small coefficients turn up when two different equations have coefficients which are quite close to one another. It is actually possible to determine in advance whether such problems will cause inaccurate answers in the solution. Each matrix an associated numerical quantity called the condition number which can be used to estimate the accuracy of the computed answer. A good library subroutine for Gaussian elimination will compute the condition number of the matrix as well as the solution so that the accuracy of the solution can be lknown. Full treatment of the issues involved would be beyond the scope of this book. Gaussian elimination with partial pivoting using the largest available pivot is guaranteed to produce results with very small computational errors. There are quite carefully worked out mathematical results which show that the calculated answer is quite accurate except for ill-conditioned matrices which might be more indicative of .