Re: Method to Solve 2 unknowns 2 Equations


In article <1140111503.428113.315890@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Atreides" <devrim.erdem@xxxxxxxxx> writes:
Hi All,

It has been a long time since I opened the cover of my numerical method
books. So this be a
re-starters questions. Please forgive beforehand :)

I am trying to understand an algorithm which uses some kind of
numerical method to solve an equation. There are two equations :

t = f( v ) and v = g( t,v )

The functions don't include any trigonometric functions or unreal
numbers but are quite crowded with many divisions, square roots.

The algorithm simply sets v = 0 and evaluates

for 10 steps:
t = f (v) and then
v = g(t,v)

I have observed depending on the constants in functions f and g, the
method can solve t and v properly. But sometimes v and t never
converge.

Can anybody tell me what kind of numerical method is lying behind this
algorithm ?

Thanks a lot !

MDE


t=f(v) and v=g(t,v) implies
v=g(f(v),v) def=H(v)
this is called a fixed point problem
you do
v(k+1)=H(v(k))=g(f(v(k),v(k))
and a sufficient criterion for this to work is
|H'(v)| = |(d/dt)g(t,v)*d/(dv)f(v)+(d/dv)g(t,v))| <1
in a region R in the (t,v) plane and that and that
for any v of a point (t,v) in this region t,H(v) is also in R.
("contractivity and selfmapping")
this naturaly depends on the properties of f and g.

hth
peter
.

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