Mathematical Methods for Physics and Engineering : A Comprehensive Guide

27.4 NUMERICAL INTEGRATION has become attached to methods based on randomly generated numbers – in many ways come into their own ...

NUMERICAL METHODS averaged: t= 1 n ∑n i=1 f(ξi). (27.49) Stratified sampling Here the range ofxis broken up intoksubranges, 0=α ...

27.4 NUMERICAL INTEGRATION will have a very small variance. Further, any error in inverting the relationship betweenηandξwill no ...

NUMERICAL METHODS sampling, in both value and precision. Since we knew already thatf(x)andg(x) diverge monotonically by about 6% ...

27.4 NUMERICAL INTEGRATION x y=f(x) y=c x=ax=b Figure 27.5 A simple rectangular figure enclosing the area (shown shaded) which i ...

NUMERICAL METHODS h(ξ 1 )>ξ 2. The fraction of times that this inequality is satisfied estimates the value of the integral (w ...

27.4 NUMERICAL INTEGRATION It will be seen that, by replacing eachnmin the summation byf(x, y, z)nm, this procedure could be ext ...

NUMERICAL METHODS on (0,1) and then take as the random numberythe value ofF−^1 (ξ). We now illustrate this with a worked example ...

27.5 FINITE DIFFERENCES many values ofξifor each value ofyand is a very poor approximation if the wings of the Gaussian distribu ...

NUMERICAL METHODS derivatives beyond a particular one will vanish and there is no error in taking the differences to obtain the ...

27.6 DIFFERENTIAL EQUATIONS x h y(exact) 0. 01 0.1 0.5 1.0 1.5 2 3 0 (1) (1) (1) (1) (1) (1) (1) (1) 0.5 0.605 0.590 0.500 0 − 0 ...

NUMERICAL METHODS xy(estim.) y(exact) − 0 .5 (1.648) — 0 (1.000) (1.000) 0 .5 0.648 0.607 1 .0 0.352 0.368 1 .5 0.296 0.223 2 .0 ...

27.6 DIFFERENTIAL EQUATIONS but they may be summarised as (i) insufficiently precise approximations to the derivatives and (ii) ...

NUMERICAL METHODS xy(estim.) y(exact) 0 1.0000 1.0000 0.1 1.2346 1.2346 0.2 1.5619 1.5625 0.3 2.0331 2.0408 0.4 2.7254 2.7778 0. ...

27.6 DIFFERENTIAL EQUATIONS The forward difference estimate ofyi+1, namely yi+1=yi+h ( dy dx ) i =yi+hf(xi,yi), (27.72) would gi ...

NUMERICAL METHODS Steps (ii) and (iii) can be iterated to improve further the approximation to the average value ofdy/dx, but th ...

27.6 DIFFERENTIAL EQUATIONS We assume that this can be simulated by a form yi+1=yi+α 1 hfi+α 2 hf(xi+β 1 h, yi+β 2 hfi), (27.76) ...

NUMERICAL METHODS (ii) To orderh^4 , yi+1=yi+^16 (c 1 +2c 2 +2c 3 +c 4 ), (27.80) where c 1 =hf(xi,yi), c 2 =hf(xi+^12 h, yi+^12 ...

27.7 HIGHER-ORDER EQUATIONS 0. 2 0. 2 0. 4 0. 4 0. 6 0. 6 0. 8 0. 8 1. 0 1. 0 c y y x − 1. 0 − 0. 8 − 0. 6 − 0. 4 − 0. 2 − 0. 1 ...

NUMERICAL METHODS These can then be treated in the way indicated in the previous paragraph. The extension to more than one depen ...