Mathematical Tools for Physics
11—Numerical Analysis 333 Substitute this into the equation fory(0): y(0) = ∑N k=1 αk ∑∞ n=0 (−kh)n y(n)(0) n! +h ∑N k=1 βk ∑∞ n ...
11—Numerical Analysis 334 Example: Solvey′=y y(0) = 1 (h= 0.1) I could use Runge-Kutta to start and then switch to Adams as soon ...
11—Numerical Analysis 335 y′=yis y(0) =− 3. 6 y(−h) + 5. 2 y(− 2 h), or in terms of an index notation yn=− 3. 6 yn− 1 + 5. 2 yn− ...
11—Numerical Analysis 336 consistent with the constraint that must be kept on theα’s, ∑N k=1 αk= 1. One way is to pick all theαk ...
11—Numerical Analysis 337 have proposed a model that this data is to be represented by a linear combination of some set of funct ...
11—Numerical Analysis 338 These linear equations are easily expressed in terms of matrices. Ca=b, where Cνμ= ∑ i fν(xi)fμ(xi). ( ...
11—Numerical Analysis 339 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 340 Now to minimize this among all~uand~v I’ll first take advantage of some of the observations that I mad ...
11—Numerical Analysis 341 Minimize D^2 = ∑ wi′^2 − ∑ (~w′i.~v)^2 subject to φ=vx^2 +vy^2 −1 = 0 The independent variables arevxa ...
11—Numerical Analysis 342 and thisˆvis the eigenvector having the largest eigenvalue. More generally, look at Eq. ( 52 ) and you ...
11—Numerical Analysis 343 f(k)−f(−k) = 2kf′(0) + 1 3 k^3 f′′′(0) +··· f(3k)−f(− 3 k) = 6kf′(0) + 27 3 k^3 f′′′(0) +··· I’ll seek ...
11—Numerical Analysis 344 f′(. 5 h)≈ − 3 f(−h)−f(0) +f(h) + 3f(2h) 10 h , (59) and the variance is 2 σ^2 (α^2 +β^2 ) =σ^2 / 5 h^ ...
11—Numerical Analysis 345 In this equation, the value ofuat point (∆t,4∆x)depends on the values at (0,3∆x), (0,4∆x), and (0,5∆x) ...
11—Numerical Analysis 346 Then-fold iteration of this, therefore involves only thenthpower of the bracketed expression; that’s w ...
11—Numerical Analysis 347 By appropriate choice of∆tand∆x, this will haver≤ 1 , causing a dissipation of the wave. Another schem ...
11—Numerical Analysis 348 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 349 Show that an equal spaced integration scheme to evaluate such an integral is P ∫+h −h f(x) x dx=f(h)−f ...
11—Numerical Analysis 350 11.12 The same phenomenon caused by roundoff errors occurs in integration. For any of the integration ...
11—Numerical Analysis 351 11.20 Rederive the first Gauss integration formula Eq. ( 24 ) without assuming the symmetry of the res ...
11—Numerical Analysis 352 11.29 From the equationy′=f(x,y), one derivesy′′=fx+ffy. Derive a two point Adams type formula using t ...
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