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CS 205 – Class 12
Readings
: Same as last
Covered in class
: All
1. finding the Aorthogonal directions with GramSchmidt
a. given a vector V
k
, construct
k
s
by subtracting out the “Aoverlap” of V
k
with
1
s
to
1
k
s
so that
0
k
i
s
As
for i=1,k1
b. we define
j
k
j
j
j
j
k
k
k
s
As
s
As
V
V
s
1
1
i. note that
i
j
k
j
j
j
j
k
i
k
i
k
As
s
As
s
As
V
As
V
As
s
1
1
and then all the terms in the sum vanish
except for one leaving
0
i
i
i
i
i
k
i
k
i
k
As
s
As
s
As
V
As
V
As
s
as desired
c. for
i
k
,
1
1
k
k
j
k
i
k
i
j
i
k
i
j
j
j
V
As
s r
V r
s r
V r
s
As
, where the summation vanishes because the residual
at step i is orthogonal to all the previous search directions
i. when k=i this leads to
k
k
k
k
s r
V r
and
k
k
k
k
k
k
k
k
k
s r
V r
s
As
s
As
(we’ll use this below)
ii. when k < i, 0
k
i
V r
, i.e. the residual is orthogonal to all the previous
k
V
as well (we’ll use
this below)
2. Each new direction V is chosen in the steepest decent fashion, i.e.
(
)
k
k
k
V
f x
r
.
a.
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This document was uploaded on 05/25/2011.
 Spring '07

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