Applied Analysis

Cover
 Title Page
 Copyright Page
 Dedication
 Preface
 Contents
 Bibliography
 Introduction
 1. Pure and applied mathematics
 2. Pure analysis, practical analysis, numerical analysis
 Chapter I: Algebraic Equations
 1. Historical introduction
 2. Allied fields
 3. Cubic equations
 4. Numerical example
 5. Newton\'s method
 6. Numerical example for Newton\'s method
 7. Homer\'s scheme
 8. The movable strip technique
 9. The remaining roots of the cubic
 10. Substitution of a complex number into a polynomial
 11. Equations of fourth order
 12. Equations of higher order
 13. The method of moments. 14. Synthetic division of two polynomials15. Power sums and the absolutely largest root
 16. Estimation of the largest absolute value
 17. Scanning of the unit circle
 18. Transformation by reciprocal radii
 19. Roots near the imaginary axis
 20. Multiple roots
 21. Algebraic equations with complex coefficients
 22. Stability analysis
 Chapter II: Matrices and Eigenvalue Problems
 1. Historical survey
 2. Vectors and tensors
 3. Matrices as algebraic quantities
 4. Eigenvalue analysis
 5. The Hamilton-Cayley equation
 6. Numerical example of a complete eigenvalue analysis. 7. Algebraic treatment of the orthogonality of eigenvectors8. The eigenvalue problem in geometrical interpretation
 9. The principal axis transformation of a matrix
 10. Skew-angular reference systems
 11. Principal axis transformation in skew-angular systems
 12. The invariance of matrix equations under orthogonal transformations
 13. The invariance of matrix equations under arbitrary linear transformations
 14. Commutative and noncommutative matrices
 15. Inversion of a matrix. The Gaussian elimination method
 16. Successive orthogonalization of a matrix. 17. Inversion of a triangular matrix18. Numerical example for the successive orthogonalization of a matrix
 19. Triangularization of a matrix
 20. Inversion of a complex matrix
 21. Solution of codiagonal systems
 22. Matrix inversion by partitioning
 23. Perturbation methods
 24. The compatibility of linear equations
 25. Overdetermination and the principle of least squares
 26. Natural and artificial skewness of a linear set of equations
 27. Orthogonalization of an arbitrary linear system
 28. The effect of noise on the solution of large linear systems. Chapter III: Large-Scale Linear Systems1. Historical introduction
 2. Polynomial operations with matrices
 3. The p, q algorithm
 4. The Chebyshev polynomials
 5. Spectroscopic eigenvalue analysis
 6. Generation of the eigenvectors
 7. Iterative solution of large-scale linear systems
 8. The residual test
 9. The smallest eigenvalue of a Hermitian matrix
 10. The smallest eigenvalue of an arbitrary matrix
 Chapter IV: Harmonic Analysis
 2. Basic theorems
 3. Least square approximations
 4. The orthogonality of the Fourier functions
 5. Separation of the sine and the cosine series.