Krylov decomposition

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In numerical mathematics , a Krylov decomposition (after Alexei Nikolayevich Krylow ) is a matrix equation of the following form:

${\displaystyle AQ_{k}=Q_{k+1}{\underline {C}}_{k}=Q_{k}C_{k}+q_{k+1}c_{k+1,k}e_{k}^{T},}$

where is a square matrix, contains the basis vectors of a Krylow space as columns and is a (generally unreduced ) Hessenberg matrix . ${\displaystyle A\in \mathbb {C} ^{n\times n}}$ ${\displaystyle Q_{k+1}=\left(Q_{k},q_{k+1}\right)\in \mathbb {C} ^{n\times (k+1)}}$ ${\displaystyle C_{k}\in \mathbb {C} ^{k\times k}}$

Furthermore, denotes the k th canonical unit vector and is a Hessenberg matrix expanded by a line appended below , with only the last element of this line not being zero. ${\displaystyle e_{k}\in \mathbb {C} ^{k}}$ ${\displaystyle {\underline {C}}_{k}\in \mathbb {C} ^{(k+1)\times k}}$

These Krylow decompositions occur naturally in the algorithmic description of Krylow subspace methods . The term was coined by Pete Stewart .