Collected Problems in Numerical Methods
| Date: 21 April 2011, 11:02
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CONTENTS
Chapter 1 The approximate solution of non-linear algebraic
and transcendental equations 1
1. Iterative methods for algebraic and transcendental equations of a
single variable 2
Theory
1. Newton's method 2
2. The method of false position 2
3. Dandelin's method 3
4. The direct method of iteration 3
5. Bairstow's (or Hitchcock's) method for quadratic factors . . 3
6. Graeffe root squaring (or Lobachevskiy's) method 4
Problems 1-288 5
2. The solution of systems of non-linear equations 15
Theory
7. Newton's method for two simultaneous equations 15
8. An iteration method for systems of equations 16
Problems 289-301 16
References ' 18
Chapter 2 Numerical methods in linear algebra 19
1. The solution of systems of linear equations 20
Theory
Exact methods
1. Gaussian elimination with complete pivoting 21
2. Compact Gaussian elimation without pivoting 22
3. The square-root or Cholesky's method 23
4. The method of orthogonalisation 24
^LVI
CONTENTS
Iteratiye methods
5. A simple iterative method . 25
6. The Gauss-Seidel method 25
Methods for the inversion of matrices
7. The reinforcement method 26
8. An iterative method 26
9. Inversion by partitioning 26
Problems 1-69 27
2. The calculation of eigenvalues and eigenvectors of matrices ... 45
Theory
Exact methods
10. Krylov's method 45
11. Danilevskiy's method 46
12. Faddeev's modification of Leverrier's method . .• 47
13. The interpolation method 48
Iteratiye methods
14. The classical method of Jacobi 48
15. Matrix powering 50
Problems 70-146 51
References 63
Chapter 3 Numerical solution of ordinary differential
equations 65
1. First order differential equations 65
Theory
1. The Taylor series method 66
2. Euler's method 66
3. The modified Euler method 67
4. The modified Euler-Cauchy method 67
5. The modified Euler-Cauchy method with iteration 67
6. The Runge-Kutta method 68
7. Adam's method 68
8. The method of successive approximations 70
Problems 1-25 72
2, Systems of first order differential equations 77
Theory
Problem 26 77
^LCONTENTS VII
3. Second order differential equations 79
Theory
9. Reduction to first order systems 80
10. The Runge-Kutta method 80
11. Finite difference methods 80
Problem 27 81
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