Lanczos and biorthogonalization methods for eigenvalues and eigenvectors of matrices

Transcript

1 Lanzos and iorthogonalization methods for eigenvalues and eigenvetors of matries rolem formulation Many prolems are redued to solving the following system: x x where is an unknown numer А a matrix n n Every non-zero vetor x whih satisfies this equality is alled eigenvetor of matrix А orresponding to an eigenvalue he set of all eigenvalues of А is alled matrix spetrum of matrix he eigenvalues of А are roots of the algerai equation n n n Det E p pn whih is alled harateristi equation of matrix А and is marked y n he prolem is redued to: Finding the oeffiients of the harateristi equation n Finding the roots of the harateristi equation ie the eigenvalues of matrix А Finding the non-zero solutions of the homogeneous system x x for every found ie the eigenvetors of А Diretly finding the oeffiients of the harateristi equation its roots and the solution of the system for every usually leads to a large numer of alulations and rounding off errors and that is why it isn t used when solving pratial prolems Other methods are used: Lanzos method Krylov Danilevsky Jaoi power method whih are onsidered in details in literature

2 Lanzos method Biorthogonalization method Neessary formulas I Formulas for harateristi polynomial n y the Lanzos method for an aritrary matrix n n proess of orthogonalization: where If m m m m m m m i i i i / i m m m n m Ρm / Ρn : m Ρ Ρ Ρ Ρ m m m m m i i i From theory we know that: harateristi polynomial of А max m n hen n m is the If then for i m m i II roess of iortogonalization: We hoose two non-zero initial vetors and onstrut the sequene of orthogonal vetors: m and m i m i i m m m m m m m m m m m m β β β m m m m m m i m i m i m i m i m i i m m i β i i i

3 m / : m m m m m m m m m Q / : n m m m n Q β Q β Q m m m m m m m Q Q β III Formulas for the orresponding eigenvetor: m m x m x x m m k k he system from whih we find i : k k m m m k m m m m m m m k m k m m Example symmetri matrix А is given Its initial vetor is prolem finding the eigenvalues and eigenvetors y using the Lanzos method solve the Solution: In order to simplify the notations we mark We alulate onseutively

4 from 8 9 / / Ρ 6 Here we have sine А is symmetri 7 / 5/ 4 n For the eigenvetor х we seek in the form of: x he values of we find using the system: x x We equalize the oeffiients efore : : : : or from III he homogeneous system has an infinite numer of solutions Let 6 5 hen x x -4 nalogially we find the eigenvetors for 4 x - 6 x

5 he normalized eigenvetors х are: Example With the help of iorthogonalization method find the eigenvalues and the eigenvetors of the matrix: А a - solved we reah and the prolem is partially solved analogially to m m m m m mm mm- m harateristi polynomial he roots of are: 4 he eigenvetors of А analogially to Example are sought in the form: x and are also a solution to the system: Ах х

6 for 4 : One solution of the homogeneous system is: 6 7 from where х - Т the normalized vetor х / 5-/ 5 Т For - analogially is found х - Т or Example Using iorthogonalization method find the eigenvalues and the eigenvetors of the matrix 4 5 a the divisor of the haratersti polynomial is found 9 6 solved β β 5-6 6

7 9 When с the polynomial is a divisor of the harateristi polynomial he roots of are 6 4? We use 4 Sp Sp a a a a44 4 Eigenvetors: For : 6 he system has an infinite numer of solutions and one of them is: x ~ For 6 the result is the following system: Solutions are: x ~ Example 4 Find without iorthogonalization method Р х for the matrix nonsymmetri: using the initial vetor с Т Solution: he results are entered into the tale: m i i mi i -

8 he roots of Р are: - For from system III: х Т for х -- Т for - х -4 Т Referenes: B Boyanov H Semerdzhiev Numerial methods Ed lovdiv University Bl Sendov V opov Numerial methods part I Ed Nauka i Izkustvo Sofia 976 uthor: Lua opova lupop@uni-plovdivg