Convergence Issues

1
Convergence Issues
http//www.math.iastate.edu/wu/math597.html
http//www.math.iastate.edu/wu/Math597HW0000/index
.htm

• Math/BCB/ComS597
• Zhijun Wu
• Department of Mathematics

2
Newtons Method
tol 0.00001 input initial x while f(x)
gt tol xc x x xc f (xc) / f
(xc) end print x, f (x)
3
y x argtan (x) ln (x2 1) / 2
x0 1.25
y argtan (x)
4
y x argtan (x) ln (x2 1) / 2
x0 1.50
y argtan (x)
The derivative does not converge to zero!
5
Global Convergence
Global Strategy 1 x xc f (xc) / f
(xc) while f (x) gt f (xc)
x ( x xc ) / 2 Note x
(x xc) / 2 xc 1/2k f (xc) / f
(xc) k 1, 2,
6
f (x) (x 4)2 3 sin 3 x
moving uphill!
7
f (x) (x 4)2 3 sin 3 x
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Global Convergence
Global Strategy 2 if
f (xc) gt 0 x
xc f (xc) / f (xc) if f (xc) lt
0 x xc f (xc) / f
(xc) while f (x) gt f (xc) x
( x xc ) / 2
9
Strategy 1
Strategy 2
10
f (x) (x 4)2 3 sin 3 x
11
f (x) (x 4)2 3 sin 3 x
Strategy 1
12
f (x) (x 4)2 3 sin 3 x
Strategy 2