Lecture Notes in Numerical Analysis with Mathematica

In this chapter, propagation of round-off errors in floating point arithmetic operations of computers numbers is presented. The notions of the conditional number, stability and complexity of algorithms are introduced and illustrated by examples. The Horners scheme for evaluation of Polynomials is given to elucidate the optimal and well-conditioned algorithms when they are implemented in a computer system like Mathematica. The chapter ends with a set of questions.

In this chapter, Lagranges and Hermits interpolation by polynomials, by trigonometric polynomials, by Chebyshevs polynomials and by generalized polynomials spanned on Chebyshevs systems of coordinates are presented. Lagranges and Newtons formulas to find the interpolating polynomials are derived and clarified. Mathematica modules are designed to determine interpolating polynomials. Fundamental theorems on interpolation with the errors bounds are stated and proved. The application of the theorems has been elucidated by examples. The Chapter ends with a set of questions.

In the chapter, the space S_m(Δ,m − 1) of piecewise polynomial splines of degree m and differentiable up to the order m − 1 is introduced. In particular, the space S₁(Δ, 0) of piecewise linear splines and the space S₃(Δ, 2) of cubic splines are determined . Theorems on interpolation by the splines are stated and proved. The space S₁₁(Δ, 0, 0) of belinear splines and the space S₃₃(Δ, 2, 2) of be-cubic splines in two variables defined on rectangular grids are presented. On triangular grids the spaces Π₁ (Δ) and Π₃ (Δ) are considered. Mathematica modules have been designed for solving problems associated with application of splines. The chapter ends with a set of questions.

In the subject of uniform approximations the fundamental theorems with their proofs are presented. Like Taylor and Weierstrass Theorems, Bernstein's and Chebyshev's polynomials, Chebyshev's series and the best uniform approximation of a function are considered . The theorems are clarified by examples. The Mathematica modules are designed for uniform approximation of one variable functions. The chapter ends with a set of questions.

This chapter is designed for the least squares method . It begins with the least squares method to determine a polynomial of degree n well fitted to m discrete points (xi, yi), i = 1, 2, ...,m, when n ≤ m. In the simplest form, the line of regression through m points is determined. The algorithm to find well fitted function to given discrete data is also presented. The Mathematica modules are designed and example illustrating the method are provided . The chapter ends with a set of questions.

Two classes of methods for numerical integration are presented : The class of Newton-Cotes methods and the class of Gauss methods. In the class of Newton-Cotes methods, the trapezoidal rule, Simpson's rule and other higher order rules like Romberg's method have been derived . The methods are clarified by examples. Mathematica modules are designed to derive Newton Cotes methods of any accuracy and to apply them for evaluation of integrals. Gauss methods are derived in general form with n Gauss-Legendres knots and in particular with n=1,2,3,4 knots. Example illustrating the methods are presented. The exact analysis of both methods of numerical integration has been carrying out. A set of questions is enclosed in the chapter.

In this chapter the equation F(x) = 0, a≤ x ≤ b, is solved by the Fix Point Iterations, Newton's Method, Secant Method and Bisection Method. The theorems on convergence and errors estimates of the methods have been stated and proved. Also, the rates of convergence of the iterative methods are determined . The methods are illustrated by a number of selected examples. The chapter ends with a set of questions.