Tham khảo tài liệu 'lập trình c# all chap "numerical recipes in c" part 142', công nghệ thông tin phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Reduction ofa Symmetric Matrix to Tridiagonal Form 469 for j i 1 j n j if d j p p d k j if k i d k d i d i p for j 1 j n j p v j i v j i v j k v j k p CITED REFERENCES AND FURTHER READING Golub . and Van Loan . 1989 Matrix Computations 2nd ed. Baltimore Johns Hopkins University Press . Smith . et al. 1976 Matrix Eigensystem Routines EISPACK Guide 2nd ed. vol. 6 of Lecture Notes in Computer Science New York Springer-Verlag . 1 Wilkinson . and Reinsch C. 1971 Linear Algebra vol. II of Handbook forAutomatic Computation New York Springer-Verlag . 2 of a Symmetric Matrix to Tridiagonal Form Givens and Householder Reductions As already mentioned the optimum strategy for finding eigenvalues and eigenvectors is first to reduce the matrix to a simple form only then beginning an iterative procedure. For symmetric matrices the preferred simple form is tridiagonal. The Givens reduction is a modification of the Jacobi method. Instead of trying to reduce the matrix all the way to diagonal form we are content to stop when the matrix is tridiagonal. This allows the procedure to be carried out in a finite number of steps unlike the Jacobi method which requires iteration to convergence. Givens Method For the Givens method we choose the rotation angle in equation so as to zero an element that is not at one of the four corners . not app apq or aqq in equation . Specifically we first choose P23 to annihilate a31 and by symmetry a13 . Then we choose P24 to annihilate a41. In general we choose the sequence Sample page from NUMERICAL RECIPES IN C THE ART OF SCIENTIFIC COMPUTING ISBN 0-521-43108-5 P23 P24 . P2 i P34 P3 Pn - 1 H where Pjk annihilates ak j-1. The method works because elements such as a rp and a rq with r p r q are linear combinations of the old quantities arp and arq by 470 Chapter 11. Eigensystems equation . Thus if arp and arq have already been set to zero they remain zero as the reduction proceeds. Evidently of .
