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Chapter 7 Numerical Methods for the Solution of

Systems of Equations

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

7.1 Linear Algebra Review

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Theorem 7.1 and Corollary 7.1

- Singular v.s. nonsingular

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

7.2 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.

Theorem 7.2

Example 7.1

Example 7.1 (con.)

Partial Pivoting

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?

Discussion

- Using the naive algorithm
- Using the pivoting algorithm

incorrect

correct

7.3 Operation Counts

- You can trace Algorithms 7.1 and 7.2 to evaluate

the computational time.

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

The LU Factorization

Example 7.2

Example 7.2 (con.)

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.

The LU Factorization Algorithms 7.5 and 7.6

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Example 7.3

Example 7.3 (con.)

L

U

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.

Example 7.4

Next page

Example 7.4 (con.)

We need to keep track of the row interchanges.

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.

7.5 Perturbation, Conditioning, and Stability

Example 7.5

7.5.1 Vector and Matrix Norms

- For example
- Infinity norm
- Euclidean 2-norm

Matrix Norm

- The properties of matrix norm (1)

(2) - For example
- The matrix infinity norm
- The matrix 2-norm

Example 7.6

7.5.2 The Condition Number and Perturbations

Definition 7.3 and Theorem 7.3

AA-1 I

Theorem 7.4

Theorems 7.5 and 7.6

Theorem 7.7

Definition 7.4

- An example Example 7.7

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Theorem 7.9

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.

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.

Estimating the Condition Number

Estimate the condition number

Example 7.8

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.

Theorem 7.11 and Algorithm 7.10

Example 7.9

Example 7.9 (con.)

compare

7.6 SPD Matrices and The Cholesky Decomposition

- SPD matrices symmetric, positive definite

matrices - You can prove this theorem using induction method.

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

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.

Example 7.10

Example 7.10 (con.)

Splitting Methods (details see Chapter 9)

Theorem 7.13

Definition 7.6

Theorem 7.14

- Conclusion

Example of Splitting Methods-- Jacobi Iteration

- Jacobi iteration
- In this method, matrix M D.

Example 7.12

Example 7.12 (con.)

Example of Splitting Methods-- Gauss-Seidel

Iteration

- Gauss-Seidel Iteration
- In this method, matrix M L.

Example 7.13

Theorem 7.15

Example of Splitting Methods-- SOR Iteration

- SOR successive over-relaxation iteration

Example 7.14

Theorem 7.16