Lectures on Numerical Methods - I. P. Mysovskih

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Biographie:
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Sommaire:
I. Numerical solution of equations.- 1. Finding an initial approximation.- 2. The secant method.- 3. The method of iterations.- 4. The method of iterations for systems of equations.- 5. Numerical evaluation of polynomials and their derivatives.- 6. Newton's method.- 7. Theorems on the convergence of Newton's method.- 8. Remarks on the practical application of Newton's method.- 9. Lobacevskii's method.- 10. Factorization methods.- Exercises for Chapter I.- II. Algebraic interpolation.- 1. Introduction.- 2. Finite differences.- 3. Divided differences.- 4. The general problem of interpolation.- 5. Interpolation of function values.- 6. The remainder term in interpolation.- 7. Interpolation at equidistant points. Newton's formulas for interpolation at the beginning and end of tables.- 8. Interpolation at equidistant points. The formulas of Gauss, Stirling, and Bessel.- 9. Inverse interpolation. Interpolation without differences.- 10. Hermite interpolation.- 11. Numerical differentiation.- Exercises for Chapter II.- III. Approximate calculation of integrals.- 1. Interpolation quadrature formulas.- 2. The simplest interpolation quadrature formulas.- 3. Numerical integration of periodic functions and the rectangular quadrature formula.- 4. Gaussian type quadrature formulas.- 5. Legendre polynomials and the Gauss formula.- 6. Other special cases of quadrature formula of the Gaussian type.- 7. A. A. Markov's quadrature formulas.- 8. ?ebySev's quadrature formula.- 9. Bernoulli numbers and polynomials.- 10. Representation of functions by means of Bernoulli polynomials.- 11. The Euler-Maclaurin formula.- 12. Concluding remarks.- Exercises for Chapter III.- IV. The numerical solution of the Cauchy problem for ordinary differential equations.- 1. Introduction.- 2. The Runge-Kutta method.- 3. On difference methods for the solution of the Cauchy problem.- 4. The Adams extrapolation method.- 5. The construction of the beginning of the table.- 6. The Adams interpolation method.- 7. Methods of Cowell type.- 8. Numerical integration of systems of equations of the first order.- 9. St?rmer's extrapolation method.- 10. St?rmer's interpolation method.- 11. Cowell's method.- 12. On the estimate of error of the Adams method.- Exercises for Chapter IV.