Computational Physics

A7 Differential equations 583 This leads to the discretised form of(A.86): ∇^2 Dψij=−ρij. (A.88) The grid on which the Laplace o ...

584 Appendix A We can change the convergence rate by another discretisation of the con- tinuum initial value problem (A.90), in ...

A7 Differential equations 585 solution which is in general unknown). In practice, a theoretical approximation of this optimal va ...

586 Appendix A k=Ah; λ=r 2 /(h·k); g=g−λk; r 2 ′=g·g; γ=r′ 2 /r 2 ; x=x+λh; h=g+γh; r 2 =r′ 2 ; END WHILE. The algorithm can be ...

A7 Differential equations 587 This equation is easily solved because it is diagonal: we have simplyL^2 independent trivial equat ...

588 Appendix A Figure A.3. Restriction and prolongation. The coarse grid points are marked by ×and the fine grid points by•. The ...

A7 Differential equations 589 Level 5 Level 4 Level 3 Level 2 Level 1 Level 1 Level 2 Level 3 Level 4 Level 5 Level 5 Figure A.4 ...

590 Appendix A How efficient is the multigrid method? The Gauss–Seidel iterations on a grid of sizeL×LrequireO(L^2 )steps, as do ...

A8 Linear algebra problems 591 The method used for this problem is straightforward. We use the fact that the rows ofAcan be inte ...

592 Appendix A We can now solve for thexistraightforwardly: first, we see that its last elementxN is equal tob′′N/a′′NN. Then th ...

A8 Linear algebra problems 593 redundant and we are allowed to put the diagonal elements ofLequal to 1. In that case we have   ...

594 Appendix A corresponding to neighbouring or identical grid points. In two dimensions with a grid constanthwe obtain ∇^2 ψ(r) ...

A8 Linear algebra problems 595 Dense matrices: the Householder method The majority of the matrices to be diagonalised in computa ...

596 Appendix A So,uuTis anN×Nmatrix, which can be written as         u 1 uT 0 ··· 0 u 2 uT 0 ··· 0 .. .0··· 0 ui− 1 ...

A8 Linear algebra problems 597 sparseness. In most cases we need only the lowest part of the eigenvalue spectrum of the discreti ...

598 Appendix A with respect to the basisψp, takes on a tridiagonal form; the eigenvalues of the p×ptridiagonal matrix obtained a ...

A9 The fast Fourier transform 599 transformfˆis also a function defined on discrete points,kn, defined by kn= 2 nπ L , n=0,...,N ...

600 Appendix A From now on, we shall use the notationf(xj)=fjandfˆ(xn)=fˆn. We assume that the number of pointsNis equal to a po ...

Exercises 601 Exercises A.1 [C] Write routines for generating the spherical Bessel functionsjlandnlaccording to the method descr ...

602 Appendix A We shall take
^2 / 2 m=1 in the following. This equation can be solved analytically (it can be mapped onto the La ...