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f += e[j]*a[i][j]; } hh=f/(h+h); Form K, equation (11.2.11). for (j=1;j | 11.3 Eigenvalues and Eigenvectors ofa Tridiagonal Matrix 475 f e j a i j hh f h h for j 1 j l j f a i j e j g e j -hh f for k 1 k j k a j k - f e k g a i k Form K equation 11.2.11 . Form q and store in e overwriting p. Reduce a equation 11.2.13 . else e i a i l d i h Next statement can be omitted if eigenvectors not wanted d 1 0.0 e 1 0.0 Contents of this loop can be omitted if eigenvectors not wanted except for statement d i a i i Begin accumulation of transformation matrices. This block skipped when i 1. for i 1 i n i if d i for j 1 j l j g 0.0 for k 1 k l k g a i k a k j for k 1 k l k a k j - g a k i Use u and u H stored in a to form P-Q. d i a i i a i i 1.0 for j 1 j l j a j i a i j 0.0 This statement remains. Reset row and column of a to identity matrix for next iteration. CITED REFERENCES AND FURTHER READING Golub G.H. and Van Loan C.F. 1989 Matrix Computations 2nd ed. Baltimore Johns Hopkins University Press 5.1. 1 Smith B.T. et al. 1976 Matrix Eigensystem Routines EISPACK Guide 2nd ed. vol. 6 of Lecture Notes in Computer Science New York Springer-Verlag . Wilkinson J.H. and Reinsch C. 1971 Linear Algebra vol. II of Handbook forAutomatic Computation New York Springer-Verlag . 2 11.3 Eigenvalues and Eigenvectors of a Tridiagonal Matrix Evaluation of the Characteristic Polynomial Sample page from NUMERICAL RECIPES IN C THE ART OF SCIENTIFIC COMPUTING ISBN 0-521-43108-5 Once our original real symmetric matrix has been reduced to tridiagonal form one possible way to determine its eigenvalues is to find the roots of the characteristic polynomial pn A directly. The characteristic polynomial of a tridiagonal matrix can be evaluated for any trial value of A by an efficient recursion relation see 1 for example . The polynomials of lower degree produced during the recurrence form a 476 Chapter 11. Eigensystems Sturmian sequence that can be used to localize the eigenvalues to intervals on the real axis. A root-finding method such as bisection or Newton s method can .
