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This title in other editions

Applied Analysis


Applied Analysis Cover


Synopses & Reviews

Publisher Comments:

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.


Classic work on analysis and design of finite processes for approximating solutions of analytical problems. Features algebraic equations, matrices, harmonic analysis, quadrature methods, and much more.

Table of Contents


1. Pure and applied mathematics

2. "Pure analysis, practical analysis, numerical analysis"

Chapter I


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


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.


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.


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


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


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


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



Product Details

Lanczos, Cornelius
Dover Publications
New York :
Mathematical Analysis
Algebra - General
Mathematical Physics
General Mathematics
Edition Description:
Trade Paper
Dover Books on Mathematics
Series Volume:
no. 13
Publication Date:
Grade Level:
8.5 x 5.38 in 1.33 lb

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Related Subjects

Science and Mathematics » Mathematics » Analysis General
Science and Mathematics » Mathematics » Applied
Science and Mathematics » Mathematics » Calculus » General
Science and Mathematics » Mathematics » Foundations and Logic
Science and Mathematics » Mathematics » Logic and Philosophy
Science and Mathematics » Mathematics » Real Analysis

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Classic work on analysis and design of finite processes for approximating solutions of analytical problems. Features algebraic equations, matrices, harmonic analysis, quadrature methods, and much more.

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