Tham khảo tài liệu 'lập trình c# all chap "numerical recipes in c" part 147', công nghệ thông tin phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Eigenvalues or Eigenvectors by Inverse Iteration 493 CITED REFERENCES AND FURTHER READING Wilkinson . and Reinsch C. 1971 Linear Algebra vol. II of Handbook forAutomatic Computation New York Springer-Verlag . 1 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 . 2 Improving Eigenvalues and or Finding Eigenvectors by Inverse Iteration The basic idea behind inverse iteration is quite simple. Let y be the solution of the linear system A - t 1 y b where b is a random vector and t is close to some eigenvalue A of A. Then the solution y will be close to the eigenvector corresponding to A. The procedure can be iterated Replace b by y and solve for a new y which will be even closer to the true eigenvector. We can see why this works by expanding both y and b as linear combinations of the eigenvectors Xj of A y Xj b -Xj Then gives Z j Aj - T Xj j Y- Xj j so that . Pj j T- Aj - T and pj xj y A -j Aj If t is close to An say then provided f3n is not accidentally too small y will be approximately xn up to a normalization. Moreover each iteration of this procedure gives another power of Aj - t in the denominator of . Thus the convergence is rapid for well-separated eigenvalues. Suppose at the Ath stage of iteration we are solving the equation Sample page from NUMERICAL RECIPES IN C THE ART OF SCIENTIFIC COMPUTING ISBN 0-521-43108-5 A - Tk 1 y bfc 494 Chapter 11. Eigensystems where bk and Tk are our current guesses for some eigenvector and eigenvalue of interest let s say xn and Xn . Normalize bk so that bk bk 1. The exact eigenvector and eigenvalue satisfy A xn Xnxn so A - T 1 Xn Xn - Tk Xn Since y of is an improved approximation to xn we normalize it and set bfc i Iy I We get an improved
