Computational Physics

A4 Finding the optimum of a function 563 thex-axis. At the pointx 1 , the gradient is along they-axis: ∂f ∂x =0. (A.13) It now r ...

564 Appendix A Vectorshiandhi+ 1 satisfying this requirement are calledconjugate. Since we have guaranteed the gradient alonghit ...

A5 Discretisation 565 with λi= gi·gi gi·Hhi ; (A.23a) γi=− gi+ 1 ·Hhi hi·Hhi . (A.23b) For this algorithm, the following propert ...

566 Appendix A impossible since the memory required for this would exceed any reasonable bounds. Instead of representing the fun ...

A6 Numerical quadratures 567 Such an approximation is useless without an estimate of the error in the result. Iffis continuous a ...

568 Appendix A approximants can be fitted to a polynomial and the value for this polynomial at h = 0 is a very accurate approxim ...

A7 Differential equations 569 Stability. In some methods, errors in the starting values or errors due to the discrete numerical ...

570 Appendix A Stability As noted above, some differential equations are susceptible to instabilities in the numerical solution. ...

A7 Differential equations 571 This is called themidpoint rule. The fact that the error is of orderh^3 can be verified by expandi ...

572 Appendix A Verlet algorithm The Verlet algorithm is a simple method for integrating second order differential equations of t ...

A7 Differential equations 573 we may reformulate the Verlet algorithm in theleap-frogform: v(h/ 2 )=v(−h/ 2 )+hF[x( 0 ),0] (A.46 ...

574 Appendix A obtain x(h)+x(−h)− 2 x( 0 )=h^2 f( 0 )x( 0 )+ h^4 12 x(^4 )( 0 )+ h^6 360 x(^6 )( 0 )+O(h^8 ) (A.50) withx(^4 )be ...

A7 Differential equations 575 Finite difference methods The Verlet algorithm is a special example of this class of methods. Fini ...

576 Appendix A Such a table can be used to construct an interpolation polynomial forxt. First we note that x± 1 =x 0 ±δx± 1 / 2 ...

A7 Differential equations 577 Using this table, an interpolation polynomial forx ̇tcan be constructed. The Newton interpolation ...

578 Appendix A Bulirsch–Stoer method This method is similar to the Romberg method for numerical integration. Suppose the value o ...

A7 Differential equations 579 with respect toxon thejth position as follows: ∂^2 ψ(xj,t) ∂x^2 = 1 x^2 [ψj− 1 (t)+ψj+ 1 (t)− 2 ψ ...

580 Appendix A make this explicit by an inversion: ψjn+^1 = ∑ j′ ( 1 +itHD)−jj′^1 ψjn′. (A.73) As the matrix in brackets is tri ...

A7 Differential equations 581 In the Crank–Nicholson and in the implicit scheme we need to solve a tridiagonal matrix equation. ...

582 Appendix A matrix equation. We need to solve two tridiagonal matrix equations for|φ〉and|χ〉 to obtain the solution to the pro ...