computation by recursion

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21: 3.2 Linear Algebra

… ►When the factorization (3.2.5) is available, the accuracy of the computed solution $𝐱$ can be improved with little extra computation. … ►Let ${𝐱}^{\ast }$ denote a computed solution of the system (3.2.1), with $𝐫=𝐛-𝐀{𝐱}^{\ast }$ again denoting the residual. … ►Define the Lanczos vectors ${𝐯}_j$ and coefficients ${\alpha }_j$ and ${\beta }_j$ by ${𝐯}_0=\mathrm{𝟎}$, a normalized vector ${𝐯}_1$ (perhaps chosen randomly), ${\alpha }_1={𝐯}_1^{\mathrm{T}}𝐀{𝐯}_1$, ${\beta }_1=0$, and for $j=1,2,\mathrm{\dots },n-1$ by the recursive scheme …Its characteristic polynomial can be obtained from the recursion … ►

§3.2(vii) Computation of Eigenvalues

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