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INTRODUCTION1

1．Pure and applied mathematics1

2．Pure analysis,practical analysis,numerical analysis2

Chapter Ⅰ ALGEBRAIC EQUATIONS5

1．Historical introduction5

2．Allied fields6

3．Cubic equations6

4．Numerical example8

5．Newton＇s method10

6．Numerical example for Newton＇s method11

7．Horner＇s scheme12

8．The movable strip technique13

9．The remaining roots of the cubic15

10．Substitution of a complex number into a polynomial16

11．Equations of fourth order19

12．Equations of higher order22

13．The method of moments22

14．Synthetic division of two polynomials24

15．Power sums and the absolutely largest root26

16．Estimation of the largest absolute value30

17．Scanning of the unit circle32

18．Transformation by reciprocal radii37

19．Roots near the imaginary axis40

20．Multiple roots42

21．Algebraic equations with complex coefficients43

22．Stability analysis44

Chapter Ⅱ MATRICES AND EIGENVALUE PROBLEMS49

1．Historical survey49

2．Vectors and tensors51

3．Matrices as algebraic quantities52

4．Eigenvalue analysis57

5．The Hamilton-Cayley equation60

6．Numerical example of a complete eigenvalue analysis65

7．Algebraic treatment of the orthogonality of eigenvectors75

8．The eigenvalue problem in geometrical interpretation81

9．The principal axis transformation of a matrix90

10．Skew-angular reference systems95

11．Principal axis transformation in skew-angular systems101

12．The invariance of matrix equations under orthogonal transformations110

13．The inyariance of matrix equations under arbitrary linear transformations114

14．Commutative and noncommutative matrices117

15．Inversion of a matrix．The Gaussian elimination method118

16．Successive orthogonalization of a matrix123

17．Inversion of a triangular matrix130

18．Numerical example for the successive orthogonalization of a matrix132

19．Triangularization of a matrix135

20．Inversion of a complex matrix137

21．Solution of codiagonal systems138

22．Matrix inversion by partitioning141

23．Perturbation methods143

24．The compatibility of linear equations149

25．Overdetermination and the principle of least squares156

26．Natural and artificial skewness of a linear set of equations161

27．Orthogonalization of an arbitrary linear system163

28．The effect of noise on the solution of large linear systems167

Chapter Ⅲ LARGE-SCALE LINEAR SYSTEMS171

1．Historical introduction171

2．Polynomial operations with matrices172

3．The p,q algorithm175

4．The Chebyshev polynomials178

5．Spectroscopic eigenvalue analysis180

6．Generation of the eigenvectors188

7．Iterative solution of large-scale linear systems189

8．The residual test198

9．The smallest eigenvalue of a Hermitian matrix200

10．The smallest eigenvalue of an arbitrary matrix203

Chapter Ⅳ HARMONIC ANALYSIS207

1．Historical notes207

2．Basic theorems208

3．Least square approximations211

4．The orthogonality of the Fourier functions214

5．Separation of the sine and the cosine series215

6．Differentiation of a Fourier series219

7．Trigonometric expansion of the delta function221

8．Extension of the trigonometric series to the nonintegrable functions224

9．Smoothing of the Gibbs oscillations by the σ factors225

10．General character of the σ smoothing227

11．The method of trigonometric interpolation229

12．Interpolation by sine functions235

13．Interpolation by cosine functions237

14．Harmonic analysis of equidistant data240

15．The error of trigonometric interpolation241

16．Interpolation by Chebyshev polynomials245

17．The Fourier integral248

18．The input-output relation of electric networks255

19．Empirical determination of the input-output relation259

20．Interpolation of the Fourier transform263

21．Interpolatory filter analysis264

22．Search for hidden periodicities267

23．Separation of exponentials272

24．The Laplace transform280

25．Network analysis and Laplace transform282

26．Inversion of the Laplace transform284

27．Inversion by Legendre polynomials285

28．Inversion by Chebyshev polynomials288

29．Inversion by Fourier series290

30．Inversion by Laguerre functions292

31．Interpolation of the Laplace transform299

Chapter Ⅴ DATA ANALYSIS305

1．Historical introduction305

2．Interpolation by simple differences306

3．Interpolation by central differences309

4．Differentiation of a tabulated function312

5．The difficulties of a difference table313

6．The fundamental principle of the method of least squares315

7．Smoothing of data by fourth differences316

8．Differentiation of an empirical function321

9．Differentiation by integration324

10．The second derivative of an empirical function327

11．Smoothing in the large by Fourier analysis331

12．Empirical determination of the cutoff frequency336

13．Least-square polynomials344

14．Polynomial interpolations in the large346

15．The convergence of equidistant polynomial interpolation352

16．Orthogonal function systems358

17．Self-adjoint differential operators362

18．The Sturm-Liouville differential equation364

19．The hypergeometric series367

20．The Jacobi polynomials367

21．Interpolation by orthogonal polynomials371

Chapter Ⅵ QUADRATURE METHODS379

1．Historical notes379

2．Quadrature by planimeters380

3．The trapezoidal rule380

4．Simpson＇s rule381

5．The accuracy of Simpson＇s formula385

6．The accuracy of the trapezoidal rule386

7．The trapezoidal rule with end correction386

8．Numerical examples390

9．Approximation by polynomials of higher order393

10．The Gaussian quadrature method396

11．Numerical example400

12．The error of the Gaussian quadrature404

13．The coefficients of a quadrature formula with arbitrary zeros407

14．Gaussian quadrature with rounded-off zeros408

15．The use of double roots410

16．Engineering applications of the Gaussian quadrature method413

17．Simpson＇s formula with end correction414

18．Quadrature involving exponentials418

19．Quadrature by differentiation419

20．The exponential function425

21．Eigenvalue problems427

22．Convergence of the quadrature based on boundary values434

Chapter Ⅶ POWER EXPANSIONS438

1．Historical introduction438

2．Analytical extension by reciprocal radii440

3．Numerical example444

4．The convergence of the Taylor series447

5．Rigid and flexible expansions448

6．Expansions in orthogonal polynomials451

7．The Chebyshev Polynomials454

8．The shifted Chebyshev polynomials455

9．Telescoping of a power series by successive reductions457

10．Telescoping of a power series by rearrangement460

11．Power expansions beyond the Taylor range463

12．The τ method464

13．The canonical polynomials469

14．Examples for the τ method474

15．Estimation of the error by the τ method493

16．The square root of a complex number500

17．Generalization of the τ method．The method of selected Points504

APPENDIX:NUMERICAL TABLES509

INDEX531