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.
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.
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. Horner'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 polynomials
15. 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 eigenvectors
8. The eigenvalue problem in geometrical interpretation
9. The principal axis transformation of a matrix
10. Skew-angular reference systems
11. Principal axis transformation of a matrix
12. The invariance of matrix equations under orthogonal transformations
13. The invariance of matrix equations under abitrary linear transformations
14. Commutative and noncommutative matrices
15. Inversion of a triangular matrix
16. Successive orthogonalization of a matrix
17. Inversion of a triangular matrix
18. 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. Peturbation 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 SYSTEMS
1 Historical introduction
2 Polynomial operations with matrices
3 "The p,q algorithm"
4 The Chebyshev polynomials
5 Spectroscopic eigenvalue analysis
6 Generation of the eigenvcctors
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
1. Historical notes
2. Basic theorems
3. Least square approximations
4. The orthogonality of the Fourier functions
5. Separation of the sine and the cosine series
6. Differentiation of a Fourier series
7. Trigonometric expansion of the delta function
8. Extension of the trigonometric series to the nonintegrable functions
9. Smoothing of the Gibbs oscillations by the s factors
10. General character of the s smoothing
11. The method of trigonometric interpolation
12. Interpolation by sine functions
13. Interpolation by cosine functions
14. Harmonic analysis of equidistant data
15. The error of trigonometric interpolation
16. Interpolation by Chebyshev polynomials
17. The Fourier integral
18. The input-output relation of electric networks
19. Empirial determination of the input-output relation
20. Interpolation of the Fourier transform
21. Interpolatory filter analysis
22. Search for hidden periodicities
23. Separation of exponentials
24. The Laplace transform
25. Network analysis and Laplace transform
26. Inversion of the Laplace transform
27. Inversion by Legendre polynomials
28. Inversion by Chebysev polynomials
29. Inversion by Fourier series
30. Inversion by Laguerre functions
31. Interpolation of the Laplace transform
Chapter V
DATA ANALYSIS
1. Historical introduction
2. Interpolation by simple differences
3. Interpolation by central differences
4. Differentiation of a tabulated function
5. The difficulties of a difference table
6. The fundemental principle of the method of least squares
7. Smoothing of data by fourth differences
8. Differentiation of an empirical function
9. Differentiation by integration
10. The second derivative of an empirical function
11. Smoothing in the large by Fourier analysis
12. Empirical determination of the cutoff frequency
13. Least-square polynomials
14. Polynomial interpolations in the large
15. The convergence of equidistant polynomial interpolation
16. Orthogonal function systems
17. Self-adjoint differential operators
18. The Sturm-Liouville differential equation
19. The hypergeometric series
20. The Jacobi polynomials
21. Interpolation by orthogonal polynomials
Chapter VI
QUADRATURE METHODS
1. Historical notes
2. Quadrature by planimeters
3. The trapezoidal rule
4. Simpson's rule
5. The accuracy of Simpson's formula
6. The accuracy of the trapezoidal rule
7. The trapezoidal rule with end correction
8. Numerical examples
9. Approximation by polynomials of higher order
10. The Gaussian quadrature method
11. Numerical example
12. The error of the Gaussian quadrature
13. The coefficients of a quadrature formula with arbitrary zeros
14. Gaussian quadrature with rounded-off zeros
15. The use of double roots
16. Engineering applications of the Gaussian quadrature method
17. Simpson's formula with end correction
18. Quadrature involving exponentials
19. Quadrature by differentiation
20. The exponential function
21. Eigenvalue problems
22. Convergence of the quadrature based on boundary values
Chapter VII
POWER EXPANSIONS
1. Historical introduction
2. Analytical extension by reciprocal radii
3. Numerical example
4. The convergence of the Taylor series
5. Rigid and flexible expansions
6. Expansions in orthogonal polynomials
7. The Chebyshev polynomials
8. The shifted Chebyshev polynomials
9. Telescoping of a power series by successive reductions
10. Telescoping of a power series by rearrangement
11. Power expansions beyond the Taylor range
12. The t method
13. The canonical polynomials
14. Examples of the t method
15. Estimation of the error by the t method
16. The square root of a complex number
17. Generalization of the t method. The method of selected points
APPENDIX: NUMERICAL TABLES
INDEX