Symmetrical Lanczos method

In numerical mathematics , the symmetric Lanczos method is a method for solving eigenvalue problems for symmetric or Hermitian matrices . It represents both a special case of the asymmetrical Lanczos method and the Arnoldi method .

The algorithm

Let a Hermitian matrix and an arbitrary start vector not equal to zero be given. Then the following algorithm creates an orthonormal basis of the Krylov subspace . This can then be used to calculate eigenvalues ​​or to solve linear systems of equations. ${\displaystyle A\in \mathbb {C} ^{n\times n}}$ ${\displaystyle r_{0}\in \mathbb {C} ^{n}}$ ${\displaystyle q_{1},..,q_{k}}$ ${\displaystyle {\mathcal {K}}={\mathcal {K}}(A,r_{0})}$

1. Set ${\displaystyle q_{0}=0}$
2. for do ${\displaystyle k=1,..,n}$
3. ${\displaystyle \beta _{k-1}=\|r_{k-1}\|}$
4. ${\displaystyle q_{k}=r_{k-1}/\beta _{k-1}}$
5. ${\displaystyle r_{k}=Aq_{k}}$
6. ${\displaystyle \alpha _{k}=\langle q_{k},r_{k}\rangle }$
7. ${\displaystyle r_{k}=r_{k}-\alpha _{k}q_{k}-\beta _{k-1}q_{k-1}}$
8. end for

literature

• Andreas Meister, Christof Vömel: Numerics of linear systems of equations. An introduction to modern procedures . 2nd edition Vieweg, Wiesbaden 2005, ISBN 3-528-13135-7 .