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CHAPTER 2

CHAPTER 2. Solution of linear Systems. AX=B. 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

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CHAPTER 2

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  1. CHAPTER 2 Solution of linear Systems AX=B

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

  4. Algorithm of forward substitution Inputan n×n triangular matrix A with aii≠0, an n×1 matrix B. Outputann×1 matrix X (the solution set). Step 1 Set . Step 2 For , set Step 3 Output 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

  7. 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:

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

  9. Gaussian elimination with pivoting 0 0 0 partial pivoting

  10. Triangular Factorization 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.

  11. Rewriting techniques of Guassian elimination

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

  13. Doolittle’s Method

  14. Doolittle’s Method

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

  16. is not a lower-triangular. Example can not be factorizeded into LU directly. permutation matrix

  17. 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 triangularLwith 1’s on the diagonal and an upper triangularU such that A=LU.

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

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

  20. (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 . 2.4 Iterative Methods for Linear Systems • 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 .

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

  22. For example, converges to 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.

  23. Norm of the matrix 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 :

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

  25. Jacobi Iterative Method

  26. (ii) Gauss-Seidel Iterative Mehtod

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

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

  29. Guass-Seidel:

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

  31. Theorem Definition Thespectral radius of a matrix A is defined by

  32. Example

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