## Computational Mathematics – Danilina, Dubrovskaya, Kvasha, Smirnov

In this post, we will see the book Computational Mathematics by N. I. Danilina; N. S. Dubrovskaya; O. P. Kvasha; G. L. Smirnov.

# About the book

The rapid development of computer engineering in recent times has led to ail expansion of application of mathematics. Quantitative methods have been introduced into practically every sphere of human activity. The use of computers in the economy requires skilled specialists who have a command of the methods of computa­tional mathematics.

Computational mathematics is one of the principal disciplines necessary for the preparation of specialists for various branches of economy. By studying it students acquire theoretical knowledge and practical skill to solve various applied problems with the aid of mathemati­cal models and numerical methods that are realized on a computer.

This study aid assumes that the reader is aware of the elementary concepts of higher mathematics, i.e. contin­uity, the derivative and the integral. It covers three large divisions of mathematics: “Algebraic Methods” (Ch. 2-6), “Numerical Methods of Analysis” (Ch. 1, 7, 8) and “Numerical Methods of Solving Differential Equa­tions” (Ch. 1), 10).
The theoretical material presented is illustrated by nu­merous examples. Each chapter is concluded by exer­cises for independent work.

The book was translated from Russian by Irene Aleksanova and was published in 1988 by Mir.

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# Contents

Preface 5

Introduction 11

Chapter 1 Elementary Theory of Errors 14

1.1. Exact and Approximate Numbers. Sources and Classification of Errors 14
1.2. Decimal Notation and Rounding Off Numbers 16
1.3. Absolute and Relative Errors 17
1.4. Valid Significant Digits 20
1.5. The Connection Between the Number of Valid Digits and the Error of the Number 22
1.6. The Errors of a Sum and a Difference 23
1.7. The Error of a Product. The Number of Valid Digits in a Product 27
1.8. The Error of a Quotient. The Number of Valid Digits of a Quotient 32
1.9. The Errors of a Power and a Root 35
1.10. The Rules of Calculating Digits 36
Exercises 38

Chapter 2 Matrix Algebra and Some Data from the Theory of Linear Vector Spaces 39

2.1. Matrices and Vectors. Principal Operations Involving Matrices and Vectors 39
2.2. Transpose of a Matrix 46
2.3. The Determinant of a Matrix. The Properties of the Determinant and the Rules of Its Calculation 48
2.4. The Inverse Matrix 57
2.5. Solving Matrix Equations 63
2.6. Triangular Matrices. Expansion of a Matrix in a Product of Two Triangular Matrices 66
2.7. Inversion of a Matrix by Expanding It in a Product of Two Triangular Matrices 72
2.8 Step Matrices and Operations Involving them 78
2.9. Inversion of a Matrix by Partitioning it into Blocks 82
2.10. The Absolute Value and the Norm of a Matrix 89
2.11. The Rank of a Matrix and the Methods of Its Calculation 91
2.12. The Concept of a Linear (Vector) Space. The Linear Dependence of Vectors 95
2.13. The Basis of Space 98
2.14. The Transformation of the Coordinates of a Vector upon a Change in the Basis 104
Exercises 105

Chapter 3. Solving Systems of Linear Equations 108

3.1. Systems of Linear Equations 108
3.2. The Kronecker-Capelli Theorem 109
3.3. Cramer’s Rule for n Linear Equations in Unknowns 111
3.4. Solving Arbitrary Systems of Linear Equations 114
3.5. Homogeneous Systems of Linear Equations 118
3.6. Basic Elimination Procedure 120
3.7. Solving Systems of Linear Equations by the Gauss Elimination Method 124
3.8. Calculating Determinants by the Gauss Elimination Method 136
3.9. The Gaussian Elimination for Inversion of a Matrix 138
3.10. Cholesky’s Method 142
3.11 The Iterative Method (the Method of Successive Approximations) 148
3.12. The Conditions for Convergence of an Iterative Process 153
3.13. Estimation of the Error of the Approximate Process of the Iterative Method 154
3.14. Seidel’s Method. The Conditions for convergence of Seidel’s Process 156
3.15. Estimation of the Errors of Seidel’s Process 159
3.16. Reducing a System of Linear Equations to a Form Convenient for Iterations 160
Exercises 162

Chapter 4. Calculating the Values of Elementary Functions 165

4.1. Calculating the Values of Algebraic Polynomials 165
4.2. Calculating the Values of Analytic Functions 170
4.3. The Iterative Method of Calculating the Value of a Function 176
Exercises 178

Chapter 5. Methods of Solving Nonlinear Equations 179

5.1. Algebraic aud Transcendental Equations 179
5.2. Separating Roots 184
5.3. Computing Hoots with a Specified Accuracy. Trial and Error Method 192
5.4. Method of Chords 197
5.5. Newton’s Method of Approximation 203
5.6. Tho Combination of the Method of Chords and Newton’s Method 207
5.7. The Iterative Method 214
5.8. General Properties of Algebraic Equations. Determining the Number of Real Roots of an Algebraic Equation 223
5.9. Finding the Domains of Existence of the Roots of an Algebraic Equation 228
5.10. Horner’s Method of Approximating Real Roots of an Algebraic Equation 231

Exercises 235

Chapter 6 The Eigenvalues and Eigenvectors of a Matrix 237

6.1. The Characteristic Polynomial 237
6.2. The Method of Direct Expansion 241
6.3. Krylov’s Method of Expansion of a Characteristic Determinant 245
6.4. Using Krylov’s Method for Calculation of Eigenvectors 253
6.5. The Leverrier-Faddeev Method 254
6.6. Using the Leverrier-Faddeev Method for Calculation of Eigenvectors 258
6.7. Danilevsky’s Method 260
6.8. Using Danilevsky’s Method for Calculation of Eigenvectors 274
6.9. Using Iterative Methods to Find tho First Eigenvalue of a Matrix 277
6.10. Determining the Successive Eigenvalues and the Corresponding Eigenvectors 279
Exercises 281

Chapter 7 Interpolation and Extrapolation 283

7.1. The Function and the Methods of Its Representation 283
7.2. Mathematical Tables 285
7.3. The Approximation Theory 290
7.4. Interpolation by Polynomials 294
7.5. The Error of Interpolation Processes 297
7.6. Lagrange’s Interpolating Polynomial 202
7.7. Finite Differences 308
7.8. Stirling and Bessel Interpolating Polynomials 318
7.9. Newton’s First and Second Interpolating Polynomials 324
7.10. Divided Differences 329
7.11. Newton’s Interpolating Polynomial for an Arbitrary Net of Nodes 334
7.12. Practical Interpolation in Tables 338
7.13. Aitken’s Iterated Interpolation 330
7.14. “Optimal-Interval” Interpolation 342
7.15. Interpolation with Multiple Nodes 345
7.16. Mathematical Apparatus of Trigonometric 347
7.17. Trigonometric Interpolation 357
7.18. Numerical Methods of Determining the Fourier Coefficients 362
7.19. Backward Interpolation 366
Exercises 371

Chapter 8. Numerical Differentiation and Integration

8.1. Statement of a Problem and the Basic Formulas for Numerical Differentiation
8.2. Peculiarities of Numerical Differentiation
8.3. Statement of a Problem of Numerical Integration
8.4. Basic Quadrature Formulas
8.5. Newton-Cotes Quadrature Formulas
8.6. Quadrature Formulas of the Highest Algebraic Degree of Accuracy
8.7. Compounded Quadrature Formulas
Exercises

Chapter 9 Approximate Solution of Ordinary Differential Equations 423

9.1. Differential Equations 423
9.2. The Method of Successive Approximations (Picard’s Method) 426
9.3. Integrating Differential Equations by Means of Power Series 430
9.4. Numerical Integration of Differential Equations. Euler’s Method 434
9.5. Modifications of Euler’s Method 430
9.6. The Runge-Kutta Method 444
9.7. Adams’ Extrapolation Method 452
9.8. Milne’s Method 459
9.9. The Notion of the Boundary Value Problem for Ordinary Differential Equations 465
9.10. The Method of Finite Differences for Second-Order Linear Differential Equations 467
Exercises 469

Chapter 10 Approximate Methods of Solution of Partial Differential Equations 471

10.1. Classification of the Second-Order Differential Equations 471
10.2. Classification of Boundary Value Problems 473
10.3. Statement of the Simplest Boundary-Value Problems 477
10.4. The Method of Finite Differences. The Principal Concepts 483
10.5. Difference Schemes for Solving the Equation of Heat Conduction 494
10.6. Difference Schemes for Solving the Equation of Oscillation of a String 498

Exercises 500
Answers to Exercises 502
Index 507

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