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CS 205 – Class 12
Readings
: Same as last
Covered in class
: All
1.
Convergence and the Error
a.
The error looks like
0
jj
j
ea
s
=
∑
where the
j
s
are the search directions and
j
a
are
numerical coefficients
b.
0
kkj
j
j
k
j
k
k
k
sA
e sA a
s
a
s a
s
⋅=
⋅
=
⋅
∑∑
since the search directions are orthogonal
in A space
i.
thus
1
0
1
0
()
k
kj
j
j
k
k
kk
e
s
e
a
s
s
α
−
=
⋅+
⋅
==
⋅⋅
∑
where the summation can be added since it
is identically zero when multiplied by
k
⋅
ii.
now
11
1
2 2
21
1
k
kkk
k
ee
s
e
s
s
αα
−−
− − −
− −
−
=+
+
=
…
1.
i.e.
1
0
1
k
j
j
s
−
=
∑
iii.
thus
k
e
a
s
⋅
=
⋅
and (from above)
a
= −
c.
so the error is
0
j
es
=−
∑
and
1
1
k
j
ess
−
=
+
∑
∑
i.
after n steps the second term is equal to the first term and the error is zero
d.
multiplying this error equation by
i
⋅
gives
1
1
k
ik
j
ij
j
i j
e
s
s
−
=
−
⋅
∑
∑
i.
for i < k, there is exactly one nonzero term in each sum, and these terms cancel
ii.
thus for i < k,
0
e
and thus
0
sr
⋅
=
(we’ll use this below)
iii.
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This note was uploaded on 01/29/2008 for the course CS 205A taught by Professor Fedkiw during the Fall '07 term at Stanford.
 Fall '07
 Fedkiw

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