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.;Basic text for graduate and advanced undergraduate deals with search for roots of algebraic equations encountered in vibration and flutter problems and in those of static and dynamic stability. Other topics devoted to matrices and eigenvalue problems, large-scale linear systems, harmonic analysis and data analysis, more. 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. This is a basic text for graduate and advanced undergraduate study in those areas of mathematical analysis that are of primary concern to the engineer and the physicist, most particularly analysis and design of finite processes that approximate the solution of an analytical problem. The work comprises seven chapters: Chapter I (Algebraic Equations) deals with the search for roots of algebraic equations encountered in vibration and flutter problems and in those of static and dynamic stability. Useful computing techniques are discussed, in particular the Bernoulli method and its ramifications. Chapter II (Matrices and Eigenvalue Problems) is devoted to a systematic development of the properties of matrices, especially in the context of industrial research. Chapter III (Large-Scale Linear Systems) discusses the "spectroscopic method" of finding the real eigenvalues of large matrices and the corresponding method of solving large-scale linear equations as well as an additional treatment of a perturbation problem and other topics. Chapter IV (Harmonic Analysis) deals primarily with the interpolation aspects of the Fourier series and its flexibility in representing empirically given equidistant data. Chapter V (Data Analysis) deals with the problem of reduction of data and of obtaining the first and even second derivatives of an empirically given function — constantly encountered in tracking problems in curve-fitting problems. Two methods of smoothing are discussed: smoothing in the small and smoothing in the large. Chapter VI (Quadrature Methods) surveys a variety of quadrature methods with particular emphasis on Gaussian quadrature and its use in solving boundary value problems and eignenvalue problems associated with ordinary differential equations. Chapter VII (Power Expansions) discusses the theory of orthogonal function systems, in particular the "Chebyshev polynomials." This unique work, perennially in demand, belongs in the library of every engineer, physicist, or scientist interested in the application of mathematical analysis to engineering, physical, and other practical problems.