MATLAB: Iteration to convergence

convergnceiterationloop

I am very very new to Matlab and this will become quite apparent, but I am working on a problem where the variable I'm solving for is also in the equation. I was told this could be accomplished in Matlab.
v=(((2*Ke)/deltaZ)*log(Ar))+ sqrt((alpha+v^2)/2)
PS. this is a Hydrogeology equation to determine velocity using heat as a tracer
%Constants
lambdaE=2.8;  %Effecitve Thermal Conductivity
p=2.7; %Density of Fluid and Sediment
c=4.2;  %Heat Capacity of Fluid and Sediment
Ke=[lambdaE/(p*c)];  %Effective Thermal Diffusivity
f=2;  %Frequency
P=(1/f);  %Period
v=.3; %Fluid Velocity (initial)
deltaPhi = 0.004;  %Phase shift between shallow and deep points [Measured in the Lab]
Ar = 0.1; %Amp ratio of shallow and deep points[Measured in the Lab]
deltaZ=.2;  %Distance between shallow and deep points
alpha=sqrt(v^4+(8*pi*Ke/P)^2);
Amp Method
%Ar=exp((deltaZ/(2*Ke))*(v-sqrt((alpha + v^2)/2)))
%VAr
v=(((2*Ke)/deltaZ)*log(Ar))+ sqrt((alpha+v^2)/2)
I really hope someone out there can take pity and give me a brief walk through how this can be solved.
-Ethan

Best Answer

I think it converges pretty quick; see below. My tweaks are commented with % *comment *
clc;clear
%%Constants
lambdaE=2.8;  %Effecitve Thermal Conductivity
p=2.7; %Density of Fluid and Sediment
c=4.2;  %Heat Capacity of Fluid and Sediment
Ke=lambdaE/(p*c);  %Effective Thermal Diffusivity
f=2;  %Frequency
P=(1/f);  %Period
deltaPhi = 0.004;  %Phase shift between shallow and deep points [Measured in the Lab]
Ar = 0.1; %Amp ratio of shallow and deep points[Measured in the Lab]
deltaZ=.2;  %Distance between shallow and deep points
Amp Method
%Ar=exp((deltaZ/(2*Ke))*(v-sqrt((alpha + v^2)/2)))
%VAr
tol = 1;
v=zeros(50,1); % *changed v to vector* 
alpha = v;     % *changed alpha to vector*
v(1)=.3;                   %Fluid Velocity (initial)
alpha(1)=sqrt(v(1)^4+(8*pi*Ke/P)^2);
c = 0;      % *counter*
while tol>.00001  % *desired accuracy*
c=c+1;
v(c+1)=(((2*Ke)/deltaZ)*log(Ar))+ ...
       sqrt((alpha(c)+v(c).^2)/2);  % * note changes to alpha & c
alpha(c+1)=sqrt(v(c+1)^4+(8*pi*Ke/P)^2); % *added*
tol = abs(v(c)-v(c+1));
end
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