In finite element method Solution of Eigenvalue Problem by Transformation based methods such as Jacobi method.

1. Solution of
Eigenvalue Problem
Name : Kaustubh S. Garud
Roll. No. 503006
Subject : Finite Element Method
Guided by
Prof. P. N. Dhatrak

2. Methods
 Determinant based methods
 Transformation based methods
 Vector Iteration based methods

3. Transformation based method
Symmetric eigenvalue problem given by
Where [M] = I
And We have
If we scale eigenvector such that
And we arrange together individual eigenvectors as
columns of a matrix, we can write

4. Also,
Where,

5. Transformation based
methods
 Householders method
 Jacobi method
 Lanczos method
 Givens method

6. Jacobi Method
 Transformation Matrix is given as
Is orthogonal

7.  Consider eigenvlaue problem in its standard form
 [ ] is formed such way to make any one off diagonal element zero.
1T

8.  [T] is formed to make another element to zero in this stage the
previous element may become non-zero. Thus we need to perform
several iterations and compute
 After several such sweep we will get “effectively diagonalised”
matrix.

9. Jacobi method to the non
standard eigenvalue problem
To make kij and mij vanish. Let us write transformation
matrix as

10.  We can write
 From this
 Values of α and β obtained in such a way that
And

11. References
 P. Seshu, Textbook of Finite Element Analysis.
 Bathe J. K, Finite Element Procedures

12. University Questions
 Nov-Dec 2014 and May-June 2014
Explain in detail Jacobi method?

13. Thank You