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 Incoming links: Spectral error bound for a diagonalizable fixed-point iteration

Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Digital systems, linear algebra, introduction to formal proof / 6 / a / iii / Solution 2026-10-25

For the spectral error bound for a diagonalizable fixed-point iteration, the exact solution

x

is a fixed point:

x = G x + c

. Subtracting it from the iteration gives the error recurrence

e_{n} = G e_{n - 1}

, with

e_{n} = x_{n} - x

. Multiplying by

S^{- 1}

and using

G = S D S^{- 1}

gives

f_{n} = S^{- 1} GS f_{n - 1} = D f_{n - 1} .

Iterating proves $f_{n} = D^{n} f_{0}$ , including

n = 0

because

D^{0} = I

. This argument does not assume

S

is orthogonal; it simply expresses the error in an eigenbasis.

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