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Numerical Linear Algebra, by Willy Brandal

Carefully written undergraduate text at the appropriate level of rigor. Various algorithms are developed and compared. Care is taken with regard to which results hold in the field of reals and which hold in the complex numbers. Topics include: Gaussian elimination; LU decomposition; Cholesky decomposition; the Gauss-Seidel, power, and QR methods; Aitken acceleration; and others. Includes many exercises at the end of sections.

TABLE OF CONTENTS

1. PREFACE
   
2. CHAPTER 0 INTRODUCTION
   
   1. Two Problems, some General Considerations
      
   2. Elementary Results
      
3. CHAPTER 1 SOLVING LINEAR SYSTEMS
   
   1. Gaussian Elimination
      
   2. Counting Operations for Other Methods
      
   3. Pivoting and Scaling
      
   4. The LU Decomposition of a Matrix
      
4. CHAPTER 2 NORMS
   
   1. Complex Numbers and Complex Matrices
      
   2. Vector Norms
      
   3. Inner Products and Orthogonality
      
   4. Matrix Norms
      
   5. An Error Result
      
5. CHAPTER 3 EIGENVALUES
   
   1. Review of Eigenvalues and Eigenvectors
      
   2. Diagonalizability
      
   3. Matrix Norms and Eigenvalues
      
6. CHAPTER 4 LINEAR SYSTEMS WITH SPECIAL MATRICES
   
   1. The Cholesky Decomposition
      
   2. Convergence of Sequences of Matrices
      
   3. The Jacobi and Gauss-Seidel Method
      
   4. Convergence of the Gauss-Seidel Method
      
7. CHAPTER 5 EIGENVALUES BY THE POWER METHOD
   
   1. The Power Method
      
   2. Tie-Dominant Eigenvalues
      
   3. The Rate of Convergence
      
   4. Aitken Acceleration
      
8. CHAPTER 6 EIGENVALUES BY THE QR METHOD
   
   1. Householder Transformations
      
   2. QR Decompositions
      
   3. The QR Method
      
   4. Hessenberg Matrices
      
   5. Some Error Results
      
9. CHAPTER 7 MORE TOPICS ON THE POWER METHOD
   
   1. Using Symmetric Matrices
      
   2. Shifting Eigenvalues
      
   3. The Inverse Power Method
      
   4. Deflation
      
   5. Markov Matrices
      
10. APPENDIX: FIVE COMPUTER ASSIGNMENTS
    
    1. Gaussian Elimination Method
       
    2. Cholesky Method
       
    3. Jacobi Method and Gauss-Seidel Method
       
    4. Power Method with Aitken Acceleration
       
    5. Rayleigh Quotient Method
       
11. BIBLIOGRAPHY
    
12. INDEX OF NOTATION
    
13. INDEX OF TERMS
    

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