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Heavy-ball error propagation matrix ( $M$ )

For

e_{k} = x^{*} - x_{k}

,

(e _{k} e _{k + 1}) = ((1 + β) I - α A I - β I 0) (e _{k - 1} e _{k}) .

If

A

is diagonal with entries

λ_{i}

, a coordinate permutation turns this matrix into blocks

(1 + β - α λ_{i} 1 - β 0) .

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- Heavy-ball rate for a two-eigenvalue diagonal quadratic Heavy-ball error propagation matrix 21

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