2. Gaoal of Chapter 2 To develop direct or iterative methods to solve linear systems Useful Words upper/lower triangular; back/forward substitution; coefficient; matrix; augmented matrix; singular; nonsigular; elementary transformation; interchange; pivot; factorization; induction; deduction; determinant; perturbed/permutation matrix. 2.1 Upper-Triangular Linear Systems Solution for special linear systems: upper-triangular systems lower-triangular systems

3. Step 1 Set. Step 2 For, set Step 3 Output Algorithm of back substitution Input an n × n triangular matrix A with a ii ≠0, an n ×1 matrix B. Output an n×1 matrix X (the solution set).

4. Step 1 Set. Step 2 For, set Step 3 Output Algorithm of forward substitution Input an n × n triangular matrix A with a ii ≠0, an n ×1 matrix B. Output an n×1 matrix X (the solution set). Example Ex. 7, P. 45

5. 2.2 Gaussian Elimination and Pivoting One way to obtain the solution set for is to transformate the system into an triangular system equivalently. Gaussian elimination with back substitution

6. Denote the entries of as the augmented matrix

8. Example Solve the given system using Gaussian elimination without row interchanges Solution: Find the augmented matrix: Find the triangular matrix of A using the row transformations: Find the solutions of the system using the back substitution:

9. Pivoting Strategies Example Solve the linear system using Guassian method using four-digit arithmetic with rounding:

10. Gaussian elimination with pivoting partial pivoting 0 0 0

11. For, if A can be factorized into the product of two triangular matrices, the solution set of the system will be easy to be found. Triangular Factorization

12. Rewriting techniques of Guassian elimination

13. Example Solve the given system using Gaussian elimination without row interchanges Solution: Solve the given system using the LU factorization.

14. Doolittle’s Method

15. Doolittle’s Method

16. Example Solve the system using the LU factorization. (1) Find L and U (2) Solve two triangular systems

17. Examplecan not be factorizeded into LU directly. permutation matrix is not a lower-triangular.

18. Two Conclusions 1. If an n × n matrix A is nonsingular, then there must be a permutation matrix P such that PA=LU, where L is lower triangular with 1’s on the diagonal and U is upper triangular. 2. If each leading principal minor of an n×n matrix A is not zero, then there exist a lower triangular L with 1’s on the diagonal and an upper triangular U such that A=LU.

19. Elimination : Multiplications/divisions : Additions/subtractions : Computational complexity

20. Substitution : Multiplications/divisions : Additions/subtractions :

21. (II) An iterative technique to solve the linear system Ax=b starts with an initial approximation to the solution x and generates a sequence of vectors that converges to the solution x. (I)Iterative techniques are seldom used for solving linear systems of small dimension since the time required for sufficient accuracy exceeds that required for direct techniques. For large systems with a high percentage of 0 entries, these techniques are efficient in terms of both computer storage and computation. 2.4 Iterative Methods for Linear Systems

22. Norms of Vectors on Definition A vector norm on is a function,, from into with the following properties : the Euclidean norm Some Useful Norms on

23. A sequence of vectors is said to converge to a vector with respect to the norm, if, given any, there exists an integer such that, for all That is, if and only if for all i. For example, converges to

24. Definition A matrix norm on the set of all matrices is a real function,, defined on this set, satisfying for all matrices and and all real numbers : Norm of the matrix

25. Jacobi Iterative Example Solve the linear system x T c Ax=b x=Tx+c

26. Jacobi Iterative Method

27. (ii) Gauss-Seidel Iterative Mehtod

28. Example Solve the linear system using the Gauss-Seidel method. Rearrange the equations List the results Give the G-S iterative formula

29. The convergence of Jacobi and Gauss-Seidel iterative This problem is related with the matrix T ( ) Jacob:

30. Guass-Seidel:

31. Example Solution: Split A into 3 parts. Calculate the iterative matrices.

32. Theorem Definition The spectral radius of a matrix A is defined by

33. Example