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Divide-and-conquer eigenvalue algorithm: Difference between revisions

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This equation is known as the ''secular equation''. The problem has therefore been reduced to finding the roots of the [[rational function]] defined by the left-hand side of this equation.
 
All general eigenvalue algorithms must be iterative,{{Citation needed|date=April 2024}} and the divide-and-conquer algorithm is no different. Solving the [[nonlinear]]{{disambiguation needed|date=April 2024}} secular equation requires an iterative technique, such as the [[Newton's method|Newton–Raphson method]]. However, each root can be found in [[Big O notation|O]](1) iterations, each of which requires <math>\Theta(m)</math> flops (for an <math>m</math>-degree rational function), making the cost of the iterative part of this algorithm <math>\Theta(m^{2})</math>.
 
==Analysis==
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