Computational Mathematics – Demidov, Maron

In this post, we will see the bookComputational Mathematics by B. P. Demidovich and I. A. Maron.

About the book

The basic aim of this book is to give as far as possible a systematic and modern presentation of the most important methods and techniques of computational mathematics on the basis of the general course of higher mathematics taught in higher technical schools. The. book has been arranged so. that the basic portion constitutes a manual for the first .cycle of ·studies in approximate computations for higher technical colleges. The text contains supplementary ma- tetial Which goes beyond the scope of the ordinary college course, but the reader can select those sections which interest him and omit any extra material without loss of continuity. The chapters and sections which may be dropped out in a first reading are marked with an asterisk.

This text makes wide use of matrix calcu]us. The concepts of a vector, matrix, inverse matrix, eigenvalue and eigenvector of a matrix, etc. are workaday tools. The use of matrices offers a number of advantages in presenting the subject matter since they greatly facilitate an understanding of the development of many computations. In this sense a particular gain is achieved in the proofs of the convergence theorems of various numerical processes. Also, modern high-speed computers are nicely adapted to the performance of the basic matrix operations.
For a full comprehension of the contents of this. book, the reader should have a background of linear algebra and the theory of linear vector spaces. With the aim of making the text as self-contained as possible, the authors have included all the necessary starting material in these subjects. The appropriate chapter~ are completely independent of the basic text and can be omitted by readers who have already studied these sections.

A few words about the contents of the book. In the main it is devoted to the following problems: operations involving approximate numbers, computation of functions by means of series and iterative processes, approximate and numerical solution of algebraic ·and transcendental equations, computational methods of linear algebra, interpolation of functions, numerical differentiation and integration of functions, and the Monte Carlo method.

A great deal of attention is devoted to methods of error estimation. Nearly all processes are provided with proofs of convergence theorems, and the presentation is such that the proofs may be omitted if one wishes to confine himself to the technical aspects of the matter. In. certain case?, in order to pictorialize and lighten the presentation, the computational techniques are given as simple recipes.

The basic methods are carried to numerical applications that include computational schemes and numerical examples with de- tailed step.s of solution. To facilitate understanding the essence of the matter at hand, most of the problems are stated in simple form and are of an illustrative nature. References are given at the. end of each chapter and the complete list (in alphabetical order) is given at the end of the book.

The present text offers selected methods in computational mathematics and does not include material that involves empirical formulas, quadratic approximation of functions, approximate solutions of differential equations, etc. Likewise, the book does not include material on programming and the technical aspects of solving mathematical problems on computers. The interested reader must consult the special literature on these subjects.

The book was translated from Russian by George Yankovsky and was published by Mir in 1981.

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Contents

PREFACE 5

INTRODUCTION. GENERAL RULES OF COMPUTATIONAL WORK 15

CHAPTER 1 APPROXIMATE NUMBERS 19

1.1 Absolute and relative errors 19
1.2 Basic sources of errors 22
1.3 Scientific notation. Significant digits. The number of correct digits 23
1.4 Rounding of numbers 26
1.5 Relationship between the relative error of an approximate number and the number of correct digits 27
1.6 Tables for determining the limiting relative error from the number of correct digits and vice versa 30
1.7 The error of a sum 33
1.8 The error of a difference 35
1.9 The error of a product 37
1.10 The number of correct digits in a product 29
1.11 The error of a quotient 40
1.12 The number of correct digits in a quotient 41
1.13 The relative error of a power 41
1.14 The relative error of a root 41
1.15 Computations in which errors are not taken into exact account 42
1.16 General formula for errors 42
1.17 The inverse problem of the theory of errors 44
1.18 Accuracy in the determination of arguments from a tabulated function 48
1.19 The method of bounds 50
1.20 The notion of a probability error estimate 52

References for Chapter 1 54

CHAPTER 2 SOME FACTS FROM THE THEORY OF CONTINUED FRACTIONS 55

2.1 The definition of a continued fraction 55
2.2 Converting a continued fraction to a simple fraction and vice versa 56
2.3 Convergents 58
2.4 Nonterminating continued fractions 66
2.6 Expanding functions into continued fractions 72

References for Chapter 2 76

CHAPTER 3 COMPUTING THE VALUES OF FUNCTIONS 77

3.1 Computing the values of a polynomial. Horner’s scheme 77
3.2 The generalized Horner scheme 80
3.3 Computing the values of rational fractions 82
3.4 Approximating the sums of numerical series 83
3.5 Computing the values of an analytic function 89
3.6 Computing the values of exponential functions 91
3.7 Computing the values of a logarithmic function 95
3.8 Computing the values of trigonometric functions 98
3.9 Computing the values of hyperbolic functions 101
3.10 Using the method of iteration for approximating the values of a function 103
3.11 Computing reciprocals 104
3.12 Computing square roots 107
3.13 Computing the reciprocal of a square root 111
3.14 Computing cube roots 112
References for Chapter 3 114

CHAPTER 4 APPROXIMATE SOLUTIONS OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS 115

4.1 Isolation of roots 115
4,2 Graphical solution of equations 119
4.3 The halving method 121
4.4 The method of proportional parts (method of chords) 122
4.5 Newton’s method (method of tangents) 127
4.6 Modified Newton method 135
4,7 Combination method 136
4.8 The method of iteration 138
4.9 The method of iteration for a system of two equations 152
4.10 Newton’s method for a system of two equations 156
4.11 Newton’s method for the case of complex roots 157

References for Chapter 4 161

CHAPTER 5 SPECIAL TECHNIQUES FOR APPROXIMATE SOLUTION OF ALGEBRAIC EQUATIONS 162

5.1 General properties of algebraic equations 162
5.2 The bounds of real roots of algebraic equations 167
5.3 The method of alternating sums 169
5,4 Newton’s method 171
5.5 The number of real roots of a polynomial 173
5.6 The theorem of Budan-Fourier 175
5.7 The underlying principle of the method of Lobachevsky-Graeffe 179
5.8 The root-squaring process 182
5.9 The Lobachevsky-Graeffe method for the case of real and distinct roots 184
5.10 The Lobachevsky-Graeffe method for the case of complex foots 187
5.11 The case of a pair of complex roots 190
5.12 The case of two pairs of complex roots 194
5.13 Bernoulli’s method 198
References for Chapter 5 202

CHAPTER 6 ACCELERATING THE CONVERGENCE OF SERIES 203

6.1 Accelerating the convergence of numerical series 203
6.2 Aecelerating the convergence of power series by the Euler-Abel methods 209
6.3 Estimates of Fourier coefficients 213
6.4 Accelerating the convergence of Fourier trigonometric series by the method of A. N. Krylov 217
6.5 Trigonometric approximation 225

References for Chapter 6 228

CHAPTER 7 MATRIX ALGEBRA 229

7.1 Basic definitions 229
7.2 Operations involving matrices 230
7.3 The transpose of a matrix 234
7.4 The.inverse matrix 236
7.5 Powers of a matrix 240
7.6 Rational functions of a matrix 241
7.7 The absolute value and norm of a matrix 242
7.8 The rank of a matrix 248
7.9 The limit of a matrix 249
7.10 Series of matrices 251
7.11 Partitioned matrices 256
7.12 Matrix inversion by partitioning 260
7.13 Triangular matrices 265
7.14 Elementary transformations of matrices 268
7.15 Computation of determinants 269

References for Chapter 7 272

CHAPTER 8 SOLVING SYSTEMS OF LINEAR EQUATIONS 273

8.1 A general description of methods of solving systems of linear equations 273
8.2 Solution by inversion of matrices, Cramer’s rule 273
8.3 The Gaussian method 277
8.4 Improving roots 284 287 288 290 293 296 300 307 309 311 313 316 321 322 322 324 327
8.5 The method of principal elements 287
8.6 Use of the Gaussian method in computing determinants 288
8.7 Inversion of matrices by the Gaussian method 290
8.8 Square-root method 293
8.9 The scheme of Khaletsky 296
8.10 The method of iteration 300
8.11 Reducing a linear system to a form convenient for iteration 307
8.12 The Seidel method 309
8.13 The case of a normal system 311
8.14 The method of relaxation 313
8.15 Correcting elements of an approximate inverse matrix 316

References for Chapter 8 321

CHAPTER 9 THE CONVERGENCE OF ITERATION PROCESSES FOR SYSTEMS OF LINEAR EQUATIONS 322

9.1 Sufficient conditions for the convergence of the iteration process 322
9.2 An estimate of the error of approximations in the iteration process 324
9.3 First sufficient condition for convergence of the Seidel process 327
9.4 Estimating the error of approximations in the Seidel process by the m-norm 330
9.5 Second sufficient condition for convergence of the Seidel process 330
9.6 Estimating the error of approximations in the Seidel process by the i-norm 332
9.7 Third sufficient condition for convergence of the Seidel proces 333

References for Chapter 9 335

CHAPTER 10 ESSENTIALS OF THE THEORY OF LINEAR VECTOR SPACES 336

10.1 The concept of a linear vector space 336
10.2 The linear dependence of vectors 337
10.3 The scalar product of vectors 343
10,4 Orthogonal systems of vectors 345
10.5 Transformations of the coordinates of a vector under changes in the basis 348
10.6 Orthogonal matrices 350
10.7 Orthogonalization of matrices 351
10.8 Applying orthogonalization methods to the solution of systems of linear equations 358
10.9 The solution space of a homogeneous system 364
10.10 Linear transformations of variables 367
10.11 Inverse transformation 373
10.12 Eigenvectors and eigenvalues of a matrix 375
10.13 Similar matrices 380
10.14 Bilinear form of a matrix 384
10.15 Properties of symmetric matrices 384
10.16 Properties of matrices with real elements 389

References for Chapter 19 393

CHAPTER 11 ADDITIONAL FACTS ABOUT THE CONVERGENCE OF ITERATION PROCESSES FOR SYSTEMS OF LINEAR EQUATIONS 394

11.1 The convergence of matrix power series 394
11.2 The Cayley-Hamilton theorem 397
11.3 Necessary and sufficient conditions for the convergence of the process of iteration for a system of linear equations 398
11.4 Necessary and sufficient conditions for the convergence of the Seidel process for a system of linear equations 400
11.5 Convergence of the Seidel process for a normal system 403
11.6 Methods for effectively checking the conditions of convergence 405

References for Chapter 11 409

CHAPTER 12 FINDING THE EIGENVALUES AND EIGENVECTORS OF A MATRIX 410

12.1 Introductory remarks 410
12.2 Expansion of secular determinants 410
12,3 The method of Danilevsky 412
12.4 Exceptional cases in the Danilevsky method 418
12.5 Computation of eigenvectors by the Danilevsky method 420
12.6 The method of Krylov 421
12.7 Computation of eigenvectors by the Krylov method 424
12.8 Leverrier’s method 426
12.9 On the method of undetermined coefficients 428
12.10 A comparison of different methods of expanding a secular determinant 429
12.11 Finding the numerically largesi eigenvalue of a matrix and the corresponding eigenvector 430
12.12 The method of ‘scalar products for finding the first eigenvalue of a real matrix 436
12.13 Finding the second eigenvalue of a matrix and the second eigenvector 439
12.14 The method of exhaustion 443
12.15 Finding the EES and eigenvectors of a positive definite symmetric matrix 445
12.16 Using the coefficients of the characteristic polynomial of a matrix for matrix inversion 450
12.17 The method of Lyusternik for accelerating the convergence of the iteration process in the solution of a system of linear equations 453

References for Chapter 12 458

CHAPTER 13 APPROXIMATE SOLUTION OF SYSTEMS OF NONLINEAR EQUATIONS 459

13.1 Newton’s method 459
13.2 General remarks on the convergence of the Newton process 465
13.3 The existence of roots of a system and the convergence of the Newton process 469
13.4 The rapidity of convergence of the Newton process 474
13.5 Uniqueness of solution 475
13.6 Stability of convergence of the Newton process under variations of the initial approximation 478
13.7 The modified Newton method 481
13.8 The method of iteration 484
13.9 The notion of a contraction mapping 487
13.10 First sufficient condition for the convergence of the process of iteration 491
13.11 Second sufficient condition for the convergence of the process of iterations 493
13.12 The method of steepest descent (gradient method) 496
13.13 The method of steepest descent for the case of a system of linear equations 501
13.14 The method of power series 504

References for Chapter 13 506

CHAPTER 14 THE INTERPOLATION OF FUNCTIONS 507

14.1 Finite differences of various orders 507
14.2 Difference table 510
14.3 Generalized power 517
14.4 Statement of the problem of interpolation 518
14.5 Newton’s first interpolation formula 519
14,6 Newton’s second interpolation formula 526
14.7 Table of central differences 530
14.8 Gaussian interpolation formulas 531
14.9 Stirling’s interpolation formula 533
14.10 Bessel’s interpolation formula 534
14.11 General description of interpolation formulas with constant interval 536
14.12 Lagrange’s interpolation formula 539
14.13 Computing Lagrangian coefficients 543
14.14 Error estimate of Lagrange’s interpolation formula 547
14.15 Error estimates of Newton’s interpolation formulas 550
14.16 Error estimates of the centrai interpolation formulas 552
14.17 On the best choice of interpolation points 553
14.18 Divided differences 554
14.19 Newton’s interpolation formula for unequally spaced values of the argument 556
14.20 Inverse interpolation for the case of equally spaced points 559
14.21 Inverse interpolation for the case of unequally spaced points 562
14.22 Finding the roots of an equation by inverse interpolation 564
14.23 The interpolation method for expanding a secular determinant 565
14.24 Interpolation of functions of two variables 567
14.25 Double differences of higher order 570
14.26 Newton’s interpolation formula for a function of two variables 571

References for Chapter 14 573

CHAPTER 15 APPROXIMATE DIFFERENTIATION 574

15.1 Statement of the problem 574
15.2 Formulas of approximate differentiation based on Newton’s first interpolation formula 575
15.3 Formulas of approximate differentiation based on Stirling’s formula 580
15.4 Formulas of numerical differentiation for equally spaced points 583
15.5 Graphical differentiation 586
15.6 On the approximate calculation of partial derivatives 588

References for Chapter 15 FBO

CHAPTER 16 APPROXIMATE INTEGRATION OF FUNCTIONS 590

16.1 General remarks 590
16.2 Newton-Cotes quadrature formulas 593
16.3 The trapezoidal formula and its remainder term 595
16.4 Simpson’s formula and its remainder term 596
16.5 Newton-Cotes formulas of higher orders 599
16.6 General trapezoidal formula (trapezoidal rule) 601
16.7 Simpson’s general formula (parabolic rule) 603
16.8 On Chebyshev’s quadrature formula 607
16.9 Gaussian quadrature formula 611
16.10 Some remarks on the accuracy of quadrature formulas 618
16.11 Richardson extrapolation 622
16.12 Bernoulli numbers 625
16.13 Euler-Maclaurin formula 628
16.14 Approximation of improper integrals 633
16.15 The method of Kantorovich for isolating singularities 635
16.16 Graphical integration 639
16.17 On cubature formulas 641
16.18 A cubature formula of Simpson type 644

References for Chapter 16 648

CHAPTER 17 THE MONTE CARLO METHOD 649

17.1 The idea of the Monte Carlo method 649
17.2 Random numbers 650
17.3 Ways of generating random numbers 653
17.4 Monte Carlo evaluation of multiple integrals 656
17.5 Solving systems of linear algebraic equations by the Monte Carlo method 666

References for Chapter 17 674

Complet list of references 675

INDEX 679

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