Book contents
- Frontmatter
- Contents
- Preface
- Chapter 1 Notation and elementary properties of matrices
- Chapter 2 The vibration of conservative systems having a finite number of degrees of freedom
- Chapter 3 Linear equations
- Chapter 4 Further development of the theory of conservative systems
- Chapter 5 Damped forced vibration
- Chapter 6 Continuous systems
- Chapter 7 The solution of linear equations and the inversion of matrices
- Chapter 8 Iterative methods for characteristic value problems
- Chapter 9 Direct methods for characteristic value problems
- Appendix
- Answers to examples
- Index
Chapter 9 - Direct methods for characteristic value problems
Published online by Cambridge University Press: 24 November 2009
- Frontmatter
- Contents
- Preface
- Chapter 1 Notation and elementary properties of matrices
- Chapter 2 The vibration of conservative systems having a finite number of degrees of freedom
- Chapter 3 Linear equations
- Chapter 4 Further development of the theory of conservative systems
- Chapter 5 Damped forced vibration
- Chapter 6 Continuous systems
- Chapter 7 The solution of linear equations and the inversion of matrices
- Chapter 8 Iterative methods for characteristic value problems
- Chapter 9 Direct methods for characteristic value problems
- Appendix
- Answers to examples
- Index
Summary
I should show you in a moment how to grapple with the question,
And you'd really be astonished at the force of my suggestion.
RuddigoreIntroduction
A direct method for finding characteristic values is one in which values, which are exact apart from the inevitable rounding off errors, are found after a certain finite number of steps. Direct methods may thus be contrasted with iterative methods, which proceed by means of a sequence of approximate values. In this chapter we present one direct method by which the characteristic values of a symmetric matrix may be found, and another method for finding the characteristic values of an unsymmetric matrix. Thus the chapter is not, and is not intended to be, a compendium of methods; for this the reader must look elsewhere.
Both of the methods presented depend on the reduction of the matrix to ‘triple-diagonal form’ The technique applicable to symmetric matrices is described in § 9.3, and that applicable to unsymmetric matrices in § 9.8. When a symmetric matrix has been reduced to a symmetric triple diagonal form its characteristic values may be found by using certain convenient properties of what are called ‘Sturm's sequences’. These are described in §§ 9.4 and 9.5. A method for finding the characteristic vectors of the reduced, triple-diagonal matrix, and the corresponding vectors of the original matrix, is described in §§ 9.6 and 9.7.
- Type
- Chapter
- Information
- The Matrix Analysis of Vibration , pp. 323 - 365Publisher: Cambridge University PressPrint publication year: 1979