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CS205 – Class 9
Covered in class:
All
Reading:
Shewchuk Paper on course web page
1.
Conjugate Gradient Method
– this covers more than just optimization, e.g. we’ll use it later
as an iterative solver to aid in solving pde’s
2.
Let’s go back to linear systems of equations Ax=b.
a.
Assume that A is square, symmetric, positive definite
b.
If A is dense we might use a direct solver, but for a sparse A, iterative solvers are better
as they only deal with nonzero entries
c.
Quadratic Form
1
()
2
TT
f
xx
A
x
b
x
c
=−
+
d.
If A is symmetric, positive definite then f(x) is minimized by the solution x to Ax=b!
i.
11
22
T
f
xA
x
A
x
b
A
x
b
∇=
+
−
=
−
since A is symmetric
ii.
() 0
fx
is equivalent to Ax=b
1.
this makes sense considering the scalar equivalent
2
1
2
f
xa
x
b
x
c
+
where
the line of symmetry is
/
x ba
=
which is the solution of ax=b and the
location of the maximum or minimum
iii.
The Hessian is H=A, and since A is symmetric, positive definite so is H, and a
solution to
, or Ax=b is a minimum
1.
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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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