Applied Analysis 9780486319261, 0486319261, 9781306324557, 1306324556

Table of contents :
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.

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