Tham khảo tài liệu 'lập trình c# all chap "numerical recipes in c" part 144', công nghệ thông tin phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Hermitian Matrices 481 for i m-1 i l i f s e i b c e i e i 1 r pythag f g if r d i 1 - p e m break s f r c g r g d i 1 -p r d i -g s c b d i 1 g p s r g c r-b Next loop can be omitted if for k 1 k n k f z k i 1 z k i 1 s z k i c f z k i c z k i -s f A plane rotation as in the original QL followed by Givens rotations to restore tridiagonal form. Recover from underflow. eigenvectors not wanted Form eigenvectors. if r i l continue d l - p e l g e m while m l CITED REFERENCES AND FURTHER READING Acton . 1970 Numerical Methods That Work 1990 corrected edition Washington Mathematical Association of America pp. 331-335. 1 Wilkinson . and Reinsch C. 1971 Linear Algebra vol. II of Handbook for Automatic Computation New York Springer-Verlag . 2 Smith . et al. 1976 Matrix Eigensystem Routines EISPACK Guide 2nd ed. vol. 6 of Lecture Notes in Computer Science New York Springer-Verlag . 3 Stoer J. and Bulirsch R. 1980 Introduction to NumericalAnalysis New York Springer-Verlag . 4 Sample page from NUMERICAL RECIPES IN C THE ART OF SCIENTIFIC COMPUTING ISBN 0-521-43108-5 Hermitian Matrices The complex analog of a real symmetric matrix is a Hermitian matrix satisfying equation . Jacobi transformations can be used to find eigenvalues and eigenvectors as also can Householder reduction to tridiagonal form followed by QL iteration. Complex versions of the previous routines jacobi tred2 and tqli are quite analogous to their real counterparts. For working routines consult 1 2 . An alternative using the routines in this book is to convert the Hermitian problem to a real symmetric one If C A iB is a Hermitian matrix then the n x n complex eigenvalue problem A iB u iv A u iv 482 Chapter 11. Eigensystems is equivalent to the 2n x 2n real problem A B u u B A v A v Note that the 2n x 2n matrix in is symmetric AT A and BT B if C is Hermitian. Corresponding to a given eigenvalue A the vector v u is also an eigenvector
