TAILIEUCHUNG - Lập Trình C# all Chap "NUMERICAL RECIPES IN C" part 61

Tham khảo tài liệu 'lập trình c# all chap "numerical recipes in c" part 61', công nghệ thông tin phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Iterative Improvement ofa Solution to Linear Equations 55 Figure . Iterative improvement of the solution to A x b. The first guess x 6x is multiplied by A to produce b 6b. The known vector b is subtracted giving 6b. The linear set with this right-hand side is inverted giving 6x. This is subtracted from the first guess giving an improved solution x. Iterative Improvement of a Solution to Linear Equations Obviously it is not easy to obtain greater precision for the solution of a linear set than the precision of your computer s floating-point word. Unfortunately for large sets of linear equations it is not always easy to obtain precision equal to or even comparable to the computer s limit. In direct methods of solution roundoff errors accumulate and they are magnified to the extent that your matrix is close to singular. You can easily lose two or three significant figures for matrices which you thought were far from singular. If this happens to you there is a neat trick to restore the full machine precision called iterative improvement of the solution. The theory is very straightforward see Figure Suppose that a vector x is the exact solution of the linear set A x b You don t however know x. You only know some slightly wrong solution x Sx where Sx is the unknown error. When multiplied by the matrix A your slightly wrong solution gives a product slightly discrepant from the desired right-hand side b namely A x 5x b 5b Subtracting from gives A 5x 5b Sample page from NUMERICAL RECIPES IN C THE ART OF SCIENTIFIC COMPUTING ISBN 0-521-43108-5 56 Chapter2. Solution ofLinearAlgebraic Equations But can also be solved trivially for 5b. Substituting this into gives A Sx A x Sx b In this equation the whole right-hand side is known since x Sx is the wrong solution that you want to improve. It is essential to calculate the right-hand side in double precision since there will be a lot of cancellation in the .

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