Lecture Notes Course à MA 190 Numerical Mathematics, First ...

which may be writtenwhereR = R 1 + R 2 ,R 1 = f ′ (x)−f(x + h) − f(x − h), R 2 =2hf(x + h) − f(x − h)−2hf(x + h) − f(x − h).2hIf f has three continuous derivatives, we make use of Taylor expansion to findthat there is a constant c and a number h 1 such thatR 1 ≤ ch 2 , h ≤ h 1 ,and using the triangular inequality we establishThus the total error R satisfiesR 2 ≤ ɛ h .R ≤ ch 2 + ɛ h .The constant c is generally not known. It depends on the function f as wellas on x but not on h. We note that R 1 , the first part of the error bound canbe made arbitrarily small by taking h sufficiently small, but the second partR 2 grows indefinitely when h decreases. Hence there is a positive value of hwhich renders the bound for R a minimum but this value can as a rule not becalculated.1.4 Linear spacesWe recall that a set of numbers S such that the operations of addition, subtraction,multiplication and division are defined and satisfying the laws we are usedto from real numbers is called a set of scalars. Other examples of scalars arethe complex numbers and the rational numbers.Definition 1.4.1 E is called a vector space (or linear space) over the set ofscalars F , if vector addition and multiplication by scalars are defined, satisfyingthe laws we remember from the familiar example of R n , the space of orderedn-tuplesExample 1.4.2 Let S be a set, E the set of real-valued functions defined on S.We next define linear combinations as follows(af 1 + bf 2 )(s) = af 1 (s) + bf 2 (s),where a, b are scalars. Thus f 1 , f 2 are elements of E (vectors) and the newvector (af 1 + bf 2 ), the linear combination of f 1 and f 2 is defined by using thelaws of real numbers to evaluate the right hand side.8