Mathematical Tools for Physics - Department of Physics - University

11—Numerical Analysis 274 The errors in the (c) and (d) formulas are both therefore the order ofh^3. Notice that just as the err ...

11—Numerical Analysis 275 To apply Simpson’s rule, it is necessary to divide the region of integration into an even number of pi ...

11—Numerical Analysis 276 which is proportional to 3 2 x^2 − 1 2 =P 2 (x), (11.27) the Legendre polynomial of second order. Reca ...

11—Numerical Analysis 277 11.5 Differential Equations To solve the first order differential equation y′=f(x,y) y(x 0 ) =y 0 , (1 ...

11—Numerical Analysis 278 The third of these choices for example gives y=y 0 +hf(0,y 0 ) + h^2 2 [ 1 h [ f(h,y 0 )−f(0,y 0 ) ] + ...

11—Numerical Analysis 279 k 1 =hf ( 0 ,y 0 ) k 3 =hf ( h/ 2 ,y 0 +k 2 / 2 ) k 2 =hf ( h/ 2 ,y 0 +k 1 / 2 ) k 4 =hf ( h,y 0 +k 3 ...

11—Numerical Analysis 280 The next orders are 0 = ∑ k αk(−kh) +h ∑ k βk 0 = ∑ k 1 2 αk(−kh)^2 +h ∑ k βk(−kh) .. . (11.46) N= 1is ...

11—Numerical Analysis 281 the coefficients− 4 and+5ofy(−h)andy(− 2 h). Small errors are magnified by these large factors. (The c ...

11—Numerical Analysis 282 algorithm to solve this problem, it will soon deviate arbitrarily far from the desired one. The reason ...

11—Numerical Analysis 283 result — the solution is not unique. A further point: there is no requirement that all of thexiare dif ...

11—Numerical Analysis 284 y x Do this in two dimensions, fitting the given data to a straight line, and to describe the line I’l ...

11—Numerical Analysis 285 to~uwithout changing this expression. That means that for a given~vthe derivative ofD^2 as~u′changes i ...

11—Numerical Analysis 286 and thisvˆis the eigenvector having the largest eigenvalue. More generally, look at Eq. (11.57) and yo ...

11—Numerical Analysis 287 Insert the preceding series expansions into Eq. (11.60) and match the coefficients off′(0). This gives ...

11—Numerical Analysis 288 Here, to evaluate the derivative, I used the two point differentiation formula. In this equation, the ...

11—Numerical Analysis 289 Looking more closely though, the object in brackets in Eq. (11.70) has magnitude r= [ 1 + c^2 (∆t)^2 ( ...

11—Numerical Analysis 290 Problems 11.1 Show that a two point extrapolation formula is f(0)≈ 2 f(−h)−f(− 2 h) +h^2 f′′(0). 11.2 ...

11—Numerical Analysis 291 11.10 Find one and two point Gauss methods for ∫∞ 0 dxe−xf(x). (a) Solve the one point method complete ...

11—Numerical Analysis 292 11.20 Rederive the first Gauss integration formula Eq. (11.25) without assuming the symmetry of the re ...

11—Numerical Analysis 293 is, start from(x 0 ,y 0 )and fit the initial data toy=α+β(x−x 0 ) +γ(x−x 0 )^2 in order to take a step ...