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Heavy-ball error propagation matrix
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Past exam of the mathematics course of the University of Cambridge
/
2023
/
ii
/
Paper 3
/
40C
/
c
/
Solution
Created
2026-09-24
Updated
2026-09-25
View more
Because
A
x
∗
=
b
, the error
e
(
k
)
=
x
∗
−
x
(
k
)
obeys
e
(
k
+
1
)
=
((
1
+
β
)
I
−
α
A
)
e
(
k
)
−
β
e
(
k
−
1
)
.
(1)
Thus the
heavy-ball error propagation matrix
is
(
e
(
k
+
1
)
e
(
k
)
)
=
M
(
e
(
k
)
e
(
k
−
1
)
)
,
M
=
(
(
1
+
β
)
I
−
α
A
I
−
β
I
0
)
.
(2)
If
A
=
diag
(
λ
1
,
…
,
λ
n
)
, permute the coordinates from
(
e
1
(
k
)
,
…
,
e
n
(
k
)
,
e
1
(
k
−
1
)
,
…
,
e
n
(
k
−
1
)
)
(3)
to
(
e
1
(
k
)
,
e
1
(
k
−
1
)
,
…
,
e
n
(
k
)
,
e
n
(
k
−
1
)
)
.
(4)
For the corresponding
permutation matrix
P
,
PM
P
T
=
diag
(
M
1
,
…
,
M
n
)
,
M
i
=
(
1
+
β
−
α
λ
i
1
−
β
0
)
.
(5)
Total
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:
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