Applied AnalysisPDF电子书下载
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- 作 者：
- 出 版 社：Inc.
- 出版年份：1956
- ISBN：
- 页数：539 页
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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
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