TAILIEUCHUNG - Lập Trình C# all Chap "NUMERICAL RECIPES IN C" part 62

Tham khảo tài liệu 'lập trình c# all chap "numerical recipes in c" part 62', công nghệ thông tin phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Singular Value Decomposition 59 Singular Value Decomposition There exists a very powerful set of techniques for dealing with sets of equations or matrices that are either singular or else numerically very close to singular. In many cases where Gaussian elimination and LU decomposition fail to give satisfactory results this set of techniques known as singular value decomposition or SVD will diagnose for you precisely what the problem is. In some cases SVD will not only diagnose the problem it will also solve it in the sense of giving you a useful numerical answer although as we shall see not necessarily the answer that you thought you should get. SVD is also the method of choice for solving most linear least-squares problems. We will outline the relevant theory in this section but defer detailed discussion of the use of SVD in this application to Chapter 15 whose subject is the parametric modeling of data. SVD methods are based on the following theorem of linear algebra whose proof is beyond our scope Any M x N matrix A whose number of rows M is greater than or equal to its number of columns N can be written as the product of an M x N column-orthogonal matrix U an N x N diagonal matrix W with positive or zero elements the singular values and the transpose of an N x N orthogonal matrix V. The various shapes of these matrices will be made clearer by the following tableau The matrices U and V are each orthogonal in the sense that their columns are orthonormal M UikUin Skn i 1 N VjkVjn Skn j 1 1 k N 1 n N 1 k N 1 n N Sample page from NUMERICAL RECIPES IN C THE ART OF SCIENTIFIC COMPUTING ISBN 0-521-43108-5 60 Chapter2. Solution ofLinearAlgebraic Equations or as a tableau i f f 1 ttT U U VT V J J J 1 1 J Since V is square it is also row-orthonormal V VT 1. The SVD decomposition can also be carried out when M N. In this case the singular values wj for j M 1 . N are all zero and the corresponding columns of U are also zero. Equation then .

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