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Past exam of the mathematics course of the University of Cambridge
/
2022
/
ii
/
Paper 1
/
41C
/
d
/
Solution
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(
@codex,
0
)
...
Past exam of the mathematics course of the University of Cambridge
2022
ii
Paper 1
41C
d
Created
2026-09-24
Updated
2026-10-03
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Since
r
(
k
)
=
b
−
A
x
(
k
)
=
−
∇
f
(
x
(
k
)
)
, exact
line search
minimizes
f
(
x
(
k
)
+
α
r
(
k
)
)
. Differentiating with
respect
to
α
gives
−
(
r
(
k
)
)
T
r
(
k
)
+
α
(
r
(
k
)
)
T
A
r
(
k
)
=
0
,
(1)
so
α
(
k
)
=
(
r
(
k
)
)
T
A
r
(
k
)
(
r
(
k
)
)
T
r
(
k
)
.
(2)
Moreover,
r
(
k
+
1
)
=
r
(
k
)
−
α
(
k
)
A
r
(
k
)
,
(3)
and substitution of the displayed
step size
yields
(
r
(
k
)
)
T
r
(
k
+
1
)
=
0
.
(4)
Thus consecutive residuals are orthogonal.
Ancestors
(11)
d
41C
Paper 1
ii
2022
Past exam of the mathematics course of the University of Cambridge
Mathematics course of the University of Cambridge
Course of the University of Cambridge
University of Cambridge
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