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zhl$@?6W(gIx3FA$x|&t-GwrTw@e>ff0Z}uVRg^F5A=)fd{l@4v9b3d07N=TgXRHxz z!EErS!t{SsneBy{!9Uack+axmfa4Iup{6><0sExvzC?N{g|3*AkHfwmJ2J^cxL-@- zT+=@Ju?Wst%DQLj=rLGkB+Z085zdK9Wl!lTGzQXB4Vd1hm8|!@90%?)2y-j)v}Hmc zwOEPCFUcc=<4QJB^x51aSNCAuzQb@#+D^F3tj!(LRZ>4Ig3dNaB=9(B1(bHK+nqZj zNh^OWtAOAdpjxm9l?D${0d9~f;zT3POlO-V4mx^bZS-%$>gY|U2cM&Gy_m7zIw%Q-osttA?pz<=Zgh0tiJd}&fT7Ukewxk{ zT44JjcUoSk-t4FHS6}ZQtB4txujSp+%eq$X9Sb9;JE7VxdpX})f61u@H)@IfUK&Qc ztpmfkX8x&YqkD-~VcMY!XK(sqnRlueB)|7FZ~lWN!4#`?{O52HArm0~8+({FNI>LK z2J$?9@FGla*AzX%f`s*5J`6=dSC zvqs_{#Lyz>;d|6!oIC3NH1jkQ`;c@4ImR;O1SV+Ng6Ipp*s{Ceg|1e2Hnu!_=gXbuum)vt zO0o5F=>&rf6EfgV?hK`PPjYbtBNwY`r)>CEMavi(q84v02%T~XJt8C|96K~dv04=v z^^9EjzjGf@8eCaP#vG+0XZY)`e!alN&?3eC1Z*G~dVl#IWFD zy=L3B%k}{WH1_g$HNTURX#bfL7Dq|`O2sa)pQcZF&-GmpLnAP0g(4MuLK*E3omM{U zBFfO)N}FR;%Ion`kbJ`7Lp@2Va}rSM&zwAdk#@5E)vlu054VVc4v#?P`JZ|l?RkD_ zhhB>;++qPt?xwtVl{Uu|(gD2KUSg^6($rFk-28(4f|F50Y_N6KqM?C3=ssfu_ |\lambda_i|$ +for $i \in 2..n$). + +We can express the seed for the eigenvector (i.e. the initial value of $v$ of +the power iteration) as a linear combination of eigenvalues: + +\[v_0 = \sum_{i=1}^n a_i x_i\] + +We can then express the result of the n-th power method as + +\[v_n = \gamma A v_{n-1} = A^n v_0 = \sum_{i=1}^n \gamma a_i \lambda_i^n x_i = +\lambda_1^n \sum_{i=1}^n \gamma a_i \left( \frac{\lambda_i}{\lambda_1} \right)^n x_i = +\gamma a_1 \lambda_1^n x_1 + \lambda_1^n \sum_{i=2}^n \gamma a_i +\left(\frac{\lambda_i}{\lambda_1}\right)^n x_i \] + +Here, $\gamma$ is just a normalization term to make $||v_n|| = 1$. $v_n$ clearly +converges to $x_1$ since all the terms in the $\sum_{i=2}^n$ contain +$\frac{\lambda_i}{\lambda_1}$, which is always less than 0 if $i > 1$ for the +sorting of eigenvalues we did before. Therefore, these terms to the power of n +converge to 0, and $\gamma$ will cancel out $a_1 \lambda_1^k$ due to the +normalization, thus making the sequence converge to $\lambda_1$. + +To see if the sequence converges linearly we use the definitions of rate of +convergence: + +\[\lim_{n \to \infty}\frac{|x_{n+1} - \lambda_1|}{|x_n - \lambda_1|^1} = \mu\] + +If this limit has a finite solution then the sequence converges linearly with +rate $\mu$. + +\[\lim_{n \to \infty}\frac{\left| a_1 \lambda_1^{n+1} x_1 + \lambda_1^{n+1} +\sum_{i=2}^n a_i \left(\frac{\lambda_i}{\lambda_1}\right)^{n+1} +x_i - \beta_{n+1} x_1\right|} +{\left| a_1 \lambda_1^n x_1 + \lambda_1^n \sum_{i=2}^n a_i +\left(\frac{\lambda_i}{\lambda_1}\right)^n x_i - \beta_n x_1\right|^1} = \mu\] + +To simplify calculations, we consider the sequence without the normalization +factor $\gamma$ that will converge to a denormalized version of $x_1$, named +$\beta x_1$. We can then simplify the $a_1\lambda_1^{i}x_1$ terms in the +sequences with $\beta_{i} x_1$ since $\beta_i$ can be set freely. + +Now we consider that if $|\lambda_2| > |\lambda_i| \forall i \in 3..n$ (since we +sorted the eigenvalues), then +$\left(\frac{\lambda_i}{\lambda_1}\right)^n$ for $i > 2$ will always converge faster to +0 than $\left(\frac{\lambda_2}{\lambda_1}\right)^n$ thus all terms other than +$i=2$ can be ignored in the limit computation. Therefore, the limit has finite +solution and the convergence rate +is + +\[\mu = \frac{\lambda_2}{\lambda_1}\] + \subsubsection{What assumptions should be made to guarantee convergence of the power method?} The first assumption to make is that the biggest eigenvalue in terms of absolute values should (let's name it $\lambda_1$) be strictly greater than all other eigenvectors, so: -$$|\lambda_1| < |\Lambda_i| \forall i \in \{2..n\}$$ +\[|\lambda_1| < |\lambda_i| \forall i \in \{2..n\}\] Also, the eigenvector \textit{guess} from which the power iteration starts must have a component in the direction of $x_i$, the eigenvector for the eigenvalue $\lambda_1$ from before. @@ -38,11 +94,11 @@ The shift and invert approach is a variant of the power method that may signific where $\alpha$ is an arbitrary constant that must be chosen wisely in order to increase the rate of convergence. Since the eigenvalues $u_i$ of B can be derived from the eigenvalues $\lambda_i$ of A, namely: -$$u_i = \frac{1}{\lambda_i - \alpha}$$ +\[u_i = \frac{1}{\lambda_i - \alpha}\] the rate of convergence of the power method on B is: -$$\left|\frac{u_2}{u_1}\right| = \left|\frac{\frac1{\lambda_2 - \alpha}}{\frac1{\lambda_1 - \alpha}}\right| = \left|\frac{\lambda_1 - \alpha}{\lambda_2 - \alpha}\right|$$ +\[\left|\frac{u_2}{u_1}\right| = \left|\frac{\frac1{\lambda_2 - \alpha}}{\frac1{\lambda_1 - \alpha}}\right| = \left|\frac{\lambda_1 - \alpha}{\lambda_2 - \alpha}\right|\] By choosing $\alpha$ close to $\lambda_1$, the convergence is sped up. To further increase the rate of convergence (up to a cubic rate), a new $\alpha$, and thus a new $B$, may be chosen for every iteration.