Computational Physics - Department of Physics

6.8 Exercises 209 fori= 1 , 2 ,...,n. The algorithm for solving this set of equations is rather simple and re- quires two steps ...

210 6 Linear Algebra ofstream ofile; // Main program only, no other functions intmain(intargc,charargv[]) { charoutfilename; int ...

6.8 Exercises 211 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 u(x) x Numerical solution Analytical solution Fig. 6.4Numerical soluti ...

212 6 Linear Algebra made. Beyondn= 105 the relative error becomes bigger, telling us that there is no point in increasingn. For ...

Chapter 7 Eigensystems AbstractWe present here two methods for solving directly eigenvalueproblems using sim- ilarity transforma ...

214 7 Eigensystems ( A−λ(ν)I ) x(ν)= 0 , withIbeing the unity matrix. This equation provides a solution tothe problem if and onl ...

7.4 Jacobi’s method 215 The importance of a similarity transformation lies in the fact that the resulting matrix has the same ei ...

216 7 Eigensystems bii=aii,i 6 =k,i 6 =l bik=aikcosθ−ailsinθ,i 6 =k,i 6 =l bil=ailcosθ+aiksinθ,i 6 =k,i 6 =l bkk=akkcos^2 θ− 2 a ...

7.4 Jacobi’s method 217 resulting in t=−τ± √ 1 +τ^2 , andcandsare easily obtained via c= √^1 1 +t^2 , ands=tc. Choosingtto be th ...

218 7 Eigensystems We will choose the angleθin order to haveb 23 =b 32 = 0. We get the new symmetric matrix B=   a 11 a 12 c−a ...

7.4 Jacobi’s method 219 { // Setting up the eigenvector matrix for(inti = 0; i < n; i++ ){ for(intj = 0; j < n; j++ ){ if( ...

220 7 Eigensystems } doublea_kk, a_ll, a_ik, a_il, r_ik, r_il; a_kk = A[k][k]; a_ll = A[l][l]; // changing the matrix elements w ...

7.5 Similarity Transformations with Householder’s method 221 where the primed quantities represent a matrixA′of dimensionn− 1 wh ...

222 7 Eigensystems witheT={ 1 , 0 , 0 ,... 0 }. Solving the latter equation gives usuand thus the needed transforma- tionP. We d ...

7.5 Similarity Transformations with Householder’s method 223 for(j = 0;j <= l;j++){ a[j][i] = a[i][j]/h;// can be omitted if ...

224 7 Eigensystems 7.5.2 Diagonalization of a Tridiagonal Matrix via Francis’Algorithm The matrix is now transformed into tridia ...

7.6 Power Methods 225 us introduce a transformationS 1 S 1 =    cosθ 0 0 sinθ 0 0 0 0 0 0 0 0 −sinθ0 0 cosθ    Then the ...

226 7 Eigensystems then: A subsequence of(bk)converges to an eigenvector associated with the dominant eigen- value Note that t ...

7.8 Schrödinger’s Equation Through Diagonalization 227 Using the fact thatQˆQˆT=Iˆ, we can rewrite Tˆ=QˆTAˆQˆ, as QˆTˆ=AˆQˆ, and ...

228 7 Eigensystems To solve the Schrödinger equation as a matrix diagonalization problem, let us study the radial part of the Sc ...