Science used to be experiments and theory, now it is experiments, theory and computations. The computational approach to understanding nature and technology is currently flowering in many fields such as physics, geophysics, astrophysics, chemistry, biology, and most engineering disciplines. This book is a gentle introduction to such computational methods where the techniques are explained through examples. It is our goal to teach principles and ideas that carry over from field to field. You will learn basic methods and how to implement them. In order to gain the most from this text, you will need prior knowledge of calculus, basic linear algebra and elementary programming.

En Ä M.b a/h2 : (1.32) (l) In comparison with (1.32), there exists a sharper error estimate of the form: En Ä h2 .b 12 ˇ ˇ a/ max ˇf 00 .x/ˇ : a6x6b (1.33) The above error estimate is obtained by using a refined representation of the error. You will find the argument in the book of Conte and de Boor [10]. Discuss the quality of the estimate (1.33) in light of the experiments of Sect. 1.4. Useful Results from Calculus In the project above, you will need some results from calculus. – An

constant. We want to compute an approximate solution of r in the time interval ranging from t D 0 to t D T: Recall that rn denotes an approximation of r.tn /, where tn D n t and t D T =N denotes the time step. Since r.tnC1 / r 0 .tn / we define the scheme r.tn / t ; rn D arn t for n > 0, where we recall that r0 is given. We get rnC1 (2.27) rnC1 D .1 C a t/rn ; (2.28) so r1 D .1 C a t/r0 ; r2 D .1 C a t/r1 D .1 C a t/2 r0 ; and so on. In general, we have rn D .1 C a t/n r0 : (2.29)

than 10 5 : Make a ranking of the schemes based on how much CPU time they need to achieve this accuracy. • Chapter 3 Systems of Ordinary Differential Equations In Chap. 2, we saw that models of the form y 0 .t/ D F .y/; y.0/ D y0 ; (3.1) can be used to model natural processes. We observed that simple versions of (3.1) can be solved analytically, and we saw that the problem can be solved adequately by using numerical methods. The purpose of the present chapter is to extend our knowledge

(5.78) t D 1, we estimate ˛ to be ˛n D p.n C 1/ p.n/ p.n/ (5.79) for n D 0; 1; 2; 3; 4 corresponding to the years from 1950 to 1954. Since these numbers are small, we multiply them by 100 and compute bn D 100 p.n C 1/ p.n/ : p.n/ The results are given in Table 5.3 below. 6 Another approach to this problem is discussed in Project 1. (5.80) 168 5 The Method of Least Squares Table 5.3 The calculated values of bn using (5.80) based on the numbers of the world’s population from 1950 to

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .419 10 A Glimpse of Parallel Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .423 10.1 Motivations for Parallel Computing .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .423 10.1.1 From the Perspective of Speed .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . .423 10.1.2 From the Perspective of Memory . . . . . . . . . . . . . . . . . . .