1 Chapter 7 Numerical Methods for the Solution of Systems of Equations

2 Introduction This chapter is about the techniques for solving linear and nonlinear systems of equations. Two important problems from linear algebra: – The linear systems problem: – The nonlinear systems problem:

3 7.1 Linear Algebra Review

4

5 Theorem 7.1 and Corollary 7.1 Singular v.s. nonsingular

6 Tridiagonal Matrices Upper triangular: Lower triangular: Symmetric matrices, positive definite matrices…… The concepts of independence/dependence, spanning, basis, vector space/subspace, dimension, and orthogonal/orthonormal should review……

7 7.2 Linear Systems and Gaussian Elimination In Section 2.6, the linear system can be written as a single augmented matrix: Elementary row operations to solve the linear system problems: Row equivalent: if we can manipulate from one matrix to another using only elementary row operations, then the two matrices are said to be row equivalent.

8 Theorem 7.2

9 Example 7.1

10 Example 7.1 (con.)

11 Partial Pivoting

12 The Problem of Naive Gaussian Elimination The problem of naive Gaussian elimination is the potential division by a zero pivot. For example: consider the following system The exact solution: What happens when we solve this system using the naive algorithm and the pivoting algorithm?

13 Discussion Using the naive algorithm: Using the pivoting algorithm: correct incorrect

Operation Counts You can trace Algorithms 7.1 and 7.2 to evaluate the computational time.

The LU Factorization Our goal in this section is to develop a matrix factorization that allows us save the work from the elimination step. Why don’t we just compute A -1 (to check if A is nonsingular)? – The answer is that it is not cost-effective to do so. – The total cost is (Exercise 7) What we will do is show that we can factor the matrix A into the product of a lower triangular and an upper triangular matrix:

16 The LU Factorization

17 Example 7.2

18 Example 7.2 (con.)

19 The Computational Cost The total cost of Gaussian elimination process: If we already have done the factorization, then the cost of the two solution steps: Constructing the LU factorization is surprisingly easy.

20 The LU Factorization : Algorithms 7.5 and 7.6

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22

23 Example 7.3

24 Example 7.3 (con.) L U

25 Pivoting and the LU Decomposition Can we pivoting in the LU decomposition without destroying the algorithm? – Because of the triangular structure of the LU factors, we can implement pivoting almost exactly as we did before. – The difference is that we must keep track of how the rows are interchanged in order to properly apply the forward and backward solution steps.

26 Example 7.4 Next page

27 Example 7.4 (con.) We need to keep track of the row interchanges.

28 Discussion How to deep track of the row interchanges? – Using an index array – For example: In Example 7.4, the final version of J is you can check that this is correct.

Perturbation, Conditioning, and Stability Example 7.5

Vector and Matrix Norms For example: – Infinity norm: – Euclidean 2-norm:

31 Matrix Norm The properties of matrix norm: (1) (2) For example: – The matrix infinity norm: – The matrix 2-norm:

32 Example 7.6

The Condition Number and Perturbations Note that Condition number

34 Definition 7.3 and Theorem 7.3

35 AA -1 = I

36 Theorem 7.4

37 Theorems 7.5 and 7.6

38 Theorem 7.7

39 Definition 7.4 An example: Example 7.7

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41 Theorem 7.9

42 Discussion Is Gaussian elimination with partial pivoting a stable process? – For a sufficiently accurate computer (u small enough) and a sufficiently small problem (n small enough), then Gaussian elimination with partial pivoting will produce solutions that are stable and accurate.

Estimating the Condition Number Singular matrices are perhaps something of a rarity, and all singular matrices are arbitrarily close to a nonsingular matrix. If the solution to a linear system changes a great deal when the problem changes only very slightly, then we suspect that the matrix is ill conditioned (nearly singular). The condition number is an important indicator to find the ill conditioned matrix.

44 Estimating the Condition Number Estimate the condition number

45 Example 7.8

Iterative Refinement Since Gaussian elimination can be adversely affected by rounding error, especially if the matrix is ill condition. Iterative refinement (iterative improvement) algorithm can use to improve the accuracy of a computed solution.

47 Theorem 7.11 and Algorithm 7.10

48 Example 7.9

49 Example 7.9 (con.) compare

SPD Matrices and The Cholesky Decomposition SPD matrices: symmetric, positive definite matrices You can prove this theorem using induction method.

51 The Cholesky Decomposition There are a number of different ways of actually constructing the Cholesky decomposition. All of these constructions are equivalent, because the Cholesky factorization is unique. One common scheme uses the following formulas: This is a very efficient algorithm. You can read Section 9.22 to learn more about Cholesky method. n

Iterative Method for Linear Systems: a Brief Survey If the coefficient matrix is a very large and sparse, then Gaussian elimination may not be the best way to solve the linear system problem. Why? Even though A=LU is sparse, the individual factors L and U may not be as sparse as A.

53 Example 7.10

54 Example 7.10 (con.)

55 Splitting Methods (details see Chapter 9)

56 Theorem 7.13

57 Definition 7.6

58 Theorem 7.14 Conclusion:

59 Example of Splitting Methods-- Jacobi Iteration Jacobi iteration: In this method, matrix M = D.

60 Example 7.12

61 Example 7.12 (con.)

62 Example of Splitting Methods-- Gauss-Seidel Iteration Gauss-Seidel Iteration : In this method, matrix M = L.

63 Example 7.13

64 Theorem 7.15

65 Example of Splitting Methods-- SOR Iteration SOR: successive over-relaxation iteration

66 Example 7.14

67 Theorem 7.16