1. Today’s class Boundary Value Problems Eigenvalue Problems
Numerical Methods
Lecture 18
Prof. Jinbo Bi
CSE, UConn

2. Boundary Value Problems
Auxiliary conditions
n-th order equation requires n conditions
Initial value conditions
All n conditions are specified at the same value of the independent variable
Boundary value conditions
The n conditions are specified at different values of the independent variable
Numerical Methods
Lecture 18
Prof. Jinbo Bi
CSE, UConn

3. Initial value conditions
Numerical Methods
Lecture 18
Prof. Jinbo Bi
CSE, UConn

4. Boundary value conditions
Numerical Methods
Lecture 18
Prof. Jinbo Bi
CSE, UConn

5. Types of Boundary Conditions
Simple B.C.: The value of the unknown function is given at the boundary.
Neumann B.C.: The derivative of the unknown function is given at the boundary.
Mixing B.C.: The combination of the unknown function’s value and derivative is given at the boundary.
Numerical Methods
Lecture 18
Prof. Jinbo Bi
CSE, UConn

6. Boundary Value Problems
Shooting Method
Convert the boundary value problem into an equivalent initial value problem by choosing values for all dependent variables at one boundary
Numerical Methods
Lecture 18
Prof. Jinbo Bi
CSE, UConn

7. Shooting Method Example
Convert to first-order
Guess a value for z(0)
Use RK method or any other method to solve the ODE at y=10
Numerical Methods
Lecture 18
Prof. Jinbo Bi
CSE, UConn

8. Shooting Method Example
Using RK method with an initial guess of z(0)=10 we get y(10)=
The specified condition was y(10)=200, so try again
Using RK method with an initial guess of z(0)=20 we get y(10)=
With two guesses and a linear ODE, we can interpolate to find the correct value
Numerical Methods
Lecture 18
Prof. Jinbo Bi
CSE, UConn

9. Shooting Method Example
z(0)=10
z(0)=20
z(0)=12.69
Numerical Methods
Lecture 18
Prof. Jinbo Bi
CSE, UConn

10. Shooting Method
If the ODE is non-linear, then you have to recast the problem as solving a series of root problems
Think of the guessed values as a vector V
That vector V will generate a y(yL) vector after solving the ODE
The y(yL) vector should equal the boundary value conditions at yL
Numerical Methods
Lecture 18
Prof. Jinbo Bi
CSE, UConn

11. Shooting Method
Since y is a function of V, you can create a new discrepancy function F that is dependent on V
The problem then becomes to zero the discrepancy
We don’t know F so we cannot use normal root-solving methods
But we can approach it by iteratively looking at the finite difference derivatives
Numerical Methods
Lecture 18
Prof. Jinbo Bi
CSE, UConn

12. Finite-Difference Method (FDM)
Numerical Methods
Lecture 18
Prof. Jinbo Bi
CSE, UConn

13. Finite-Difference Method (FDM)
Numerical Methods
Lecture 18
Prof. Jinbo Bi
CSE, UConn

14. Finite-Difference Method (FDM)
Numerical Methods
Lecture 18
Prof. Jinbo Bi
CSE, UConn

15. Matrix Form of Equation System
Numerical Methods
Lecture 18
Prof. Jinbo Bi
CSE, UConn

16. FDM Example
Numerical Methods
Lecture 18
Prof. Jinbo Bi
CSE, UConn

17. Eigenvalue Problems
Eigenvalue problems are a special class of boundary-value problem
Common in engineering systems with oscillating behavior
Springs
Vibrations
Numerical Methods
Lecture 18
Prof. Jinbo Bi
CSE, UConn

18. Physics of Eigenvalues & Eigenvectors
Numerical Methods
Lecture 18
Prof. Jinbo Bi
CSE, UConn

19. Physics of Eigenvalues
Numerical Methods
Lecture 18
Prof. Jinbo Bi
CSE, UConn

20. Physics of Eigenvalues
Numerical Methods
Lecture 18
Prof. Jinbo Bi
CSE, UConn

21. Steps of Solving Eigen-value Problems
Numerical Methods
Lecture 18
Prof. Jinbo Bi
CSE, UConn

22. Eigenvalue Example Numerical Methods Prof. Jinbo Bi Lecture 18
CSE, UConn

23. Eigenvalue Example Numerical Methods Prof. Jinbo Bi Lecture 18
CSE, UConn

24. Boundary value eigenvalue problems
Polynomial Method
Power Method
Numerical Methods
Lecture 18
Prof. Jinbo Bi
CSE, UConn

25. Power Method
The Power Method is an iterative procedure for determining the dominant (largest) eigenvalue of the matrix A and the corresponding eigenvector.
Example
Numerical Methods
Lecture 18
Prof. Jinbo Bi
CSE, UConn

26. Example of Power Method (Cont.)
Numerical Methods
Lecture 18
Prof. Jinbo Bi
CSE, UConn

27. Example of Power Method (Cont.)
Numerical Methods
Lecture 18
Prof. Jinbo Bi
CSE, UConn

28. Example of Power Method (Cont.)
Numerical Methods
Lecture 18
Prof. Jinbo Bi
CSE, UConn

29. Example of Power Method (Cont.)
Numerical Methods
Lecture 18
Prof. Jinbo Bi
CSE, UConn

30. Inverse Power Method
The Power Method can be modified to provide the smallest (lowest) (magnitude) eigenvalue and its eigenvector. The modified method is known as the inverse power method
The smallest eigenvalue λ in the matrix A corresponds to 1/ λ the largest eigenvalue in the inverse matrix A-1.
Numerical Methods
Lecture 18
Prof. Jinbo Bi
CSE, UConn

31. General I-P Method
The combination of Power Method and I-P method can be use to determine all the eigenvalues and their related eigenvectors.
Deflation Hotelling’s Method for symmetric matrices
First, normalize the largest eigenvector found by the sum of the squares of the elements in the eigenvector
Numerical Methods
Lecture 18
Prof. Jinbo Bi
CSE, UConn

32. General I-P Method Then create new A2 matrix
The new A2 matrix has the same eigenvalues as A1 except that the largest eigenvalue has been replaced with 0
Numerical Methods
Lecture 18
Prof. Jinbo Bi
CSE, UConn

33. Hotelling’s Method Numerical Methods Prof. Jinbo Bi Lecture 18
CSE, UConn

34. Hotelling’s Method Numerical Methods Prof. Jinbo Bi Lecture 18
CSE, UConn

35. Next class Curve Fitting Numerical Methods Prof. Jinbo Bi Lecture 18
CSE, UConn