INTRODUCTION 1
1．Pure and applied mathematics 1
2．Pure analysis,practical analysis,numerical analysis 2
Chapter Ⅰ ALGEBRAIC EQUATIONS 5
1．Historical introduction 5
2．Allied fields 6
3．Cubic equations 6
4．Numerical example 8
5．Newton＇s method 10
6．Numerical example for Newton＇s method 11
7．Horner＇s scheme 12
8．The movable strip technique 13
9．The remaining roots of the cubic 15
10．Substitution of a complex number into a polynomial 16
11．Equations of fourth order 19
12．Equations of higher order 22
13．The method of moments 22
14．Synthetic division of two polynomials 24
15．Power sums and the absolutely largest root 26
16．Estimation of the largest absolute value 30
17．Scanning of the unit circle 32
18．Transformation by reciprocal radii 37
19．Roots near the imaginary axis 40
20．Multiple roots 42
21．Algebraic equations with complex coefficients 43
22．Stability analysis 44
Chapter Ⅱ MATRICES AND EIGENVALUE PROBLEMS 49
1．Historical survey 49
2．Vectors and tensors 51
3．Matrices as algebraic quantities 52
4．Eigenvalue analysis 57
5．The Hamilton-Cayley equation 60
6．Numerical example of a complete eigenvalue analysis 65
7．Algebraic treatment of the orthogonality of eigenvectors 75
8．The eigenvalue problem in geometrical interpretation 81
9．The principal axis transformation of a matrix 90
10．Skew-angular reference systems 95
11．Principal axis transformation in skew-angular systems 101
12．The invariance of matrix equations under orthogonal transformations 110
13．The inyariance of matrix equations under arbitrary linear transformations 114
14．Commutative and noncommutative matrices 117
15．Inversion of a matrix．The Gaussian elimination method 118
16．Successive orthogonalization of a matrix 123
17．Inversion of a triangular matrix 130
18．Numerical example for the successive orthogonalization of a matrix 132
19．Triangularization of a matrix 135
20．Inversion of a complex matrix 137
21．Solution of codiagonal systems 138
22．Matrix inversion by partitioning 141
23．Perturbation methods 143
24．The compatibility of linear equations 149
25．Overdetermination and the principle of least squares 156
26．Natural and artificial skewness of a linear set of equations 161
27．Orthogonalization of an arbitrary linear system 163
28．The effect of noise on the solution of large linear systems 167
Chapter Ⅲ LARGE-SCALE LINEAR SYSTEMS 171
1．Historical introduction 171
2．Polynomial operations with matrices 172
3．The p,q algorithm 175
4．The Chebyshev polynomials 178
5．Spectroscopic eigenvalue analysis 180
6．Generation of the eigenvectors 188
7．Iterative solution of large-scale linear systems 189
8．The residual test 198
9．The smallest eigenvalue of a Hermitian matrix 200
10．The smallest eigenvalue of an arbitrary matrix 203
Chapter Ⅳ HARMONIC ANALYSIS 207
1．Historical notes 207
2．Basic theorems 208
3．Least square approximations 211
4．The orthogonality of the Fourier functions 214
5．Separation of the sine and the cosine series 215
6．Differentiation of a Fourier series 219
7．Trigonometric expansion of the delta function 221
8．Extension of the trigonometric series to the nonintegrable functions 224
9．Smoothing of the Gibbs oscillations by the σ factors 225
10．General character of the σ smoothing 227
11．The method of trigonometric interpolation 229
12．Interpolation by sine functions 235
13．Interpolation by cosine functions 237
14．Harmonic analysis of equidistant data 240
15．The error of trigonometric interpolation 241
16．Interpolation by Chebyshev polynomials 245
17．The Fourier integral 248
18．The input-output relation of electric networks 255
19．Empirical determination of the input-output relation 259
20．Interpolation of the Fourier transform 263
21．Interpolatory filter analysis 264
22．Search for hidden periodicities 267
23．Separation of exponentials 272
24．The Laplace transform 280
25．Network analysis and Laplace transform 282
26．Inversion of the Laplace transform 284
27．Inversion by Legendre polynomials 285
28．Inversion by Chebyshev polynomials 288
29．Inversion by Fourier series 290
30．Inversion by Laguerre functions 292
31．Interpolation of the Laplace transform 299
Chapter Ⅴ DATA ANALYSIS 305
1．Historical introduction 305
2．Interpolation by simple differences 306
3．Interpolation by central differences 309
4．Differentiation of a tabulated function 312
5．The difficulties of a difference table 313
6．The fundamental principle of the method of least squares 315
7．Smoothing of data by fourth differences 316
8．Differentiation of an empirical function 321
9．Differentiation by integration 324
10．The second derivative of an empirical function 327
11．Smoothing in the large by Fourier analysis 331
12．Empirical determination of the cutoff frequency 336
13．Least-square polynomials 344
14．Polynomial interpolations in the large 346
15．The convergence of equidistant polynomial interpolation 352
16．Orthogonal function systems 358
17．Self-adjoint differential operators 362
18．The Sturm-Liouville differential equation 364
19．The hypergeometric series 367
20．The Jacobi polynomials 367
21．Interpolation by orthogonal polynomials 371
Chapter Ⅵ QUADRATURE METHODS 379
1．Historical notes 379
2．Quadrature by planimeters 380
3．The trapezoidal rule 380
4．Simpson＇s rule 381
5．The accuracy of Simpson＇s formula 385
6．The accuracy of the trapezoidal rule 386
7．The trapezoidal rule with end correction 386
8．Numerical examples 390
9．Approximation by polynomials of higher order 393
10．The Gaussian quadrature method 396
11．Numerical example 400
12．The error of the Gaussian quadrature 404
13．The coefficients of a quadrature formula with arbitrary zeros 407
14．Gaussian quadrature with rounded-off zeros 408
15．The use of double roots 410
16．Engineering applications of the Gaussian quadrature method 413
17．Simpson＇s formula with end correction 414
18．Quadrature involving exponentials 418
19．Quadrature by differentiation 419
20．The exponential function 425
21．Eigenvalue problems 427
22．Convergence of the quadrature based on boundary values 434
Chapter Ⅶ POWER EXPANSIONS 438
1．Historical introduction 438
2．Analytical extension by reciprocal radii 440
3．Numerical example 444
4．The convergence of the Taylor series 447
5．Rigid and flexible expansions 448
6．Expansions in orthogonal polynomials 451
7．The Chebyshev Polynomials 454
8．The shifted Chebyshev polynomials 455
9．Telescoping of a power series by successive reductions 457
10．Telescoping of a power series by rearrangement 460
11．Power expansions beyond the Taylor range 463
12．The τ method 464
13．The canonical polynomials 469
14．Examples for the τ method 474
15．Estimation of the error by the τ method 493
16．The square root of a complex number 500
17．Generalization of the τ method．The method of selected Points 504
APPENDIX:NUMERICAL TABLES 509
INDEX 531