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Chapter 7 Numerical Methods for the Solution of Systems of Equations

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Title: Chapter 7 Numerical Methods for the Solution of Systems of Equations


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
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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

incorrect
correct
14
7.3 Operation Counts
  • You can trace Algorithms 7.1 and 7.2 to evaluate
    the computational time.

15
7.4 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 dont 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 the above process
  • If we already have done the factorization, then
    the cost of the two solution steps
  • Constructing the LU factorization is surprisingly
    easy.
  • The LU factorization is nothing more than a very
    slight reorganization of the same Gaussian
    elimination algorithm we studied earlier in this
    chapter.

20
The LU Factorization Algorithms 7.5 and 7.6
21
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22
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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.

29
7.5 Perturbation, Conditioning, and Stability
Example 7.5
30
7.5.1 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
33
7.5.2 The Condition Number and Perturbations
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

40
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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.

43
7.5.3 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
46
7.5.4 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
50
7.6 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
52
7.7 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 ALU 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
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