# Question ) Mathematical equivalence, but different results of fixed point?

**URL:** <https://discourse.julialang.org/t/question-mathematical-equivalence-but-different-results-of-fixed-point/65628>\
**Category:** General Usage\
**Created:** [August 1, 2021, 12:44pm UTC](https://discourse.julialang.org/t/question-mathematical-equivalence-but-different-results-of-fixed-point/65628 "2021-08-01T12:44:23Z")\
**Posts on this page:** 2\
**Page:** 1

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**Author:** ![Zerosum](https://avatars.discourse-cdn.com/v4/letter/z/b2d939/32.png) [@Zerosum](https://discourse.julialang.org/u/Zerosum)\
**Post date:** [August 1, 2021, 12:44pm UTC](https://discourse.julialang.org/t/question-mathematical-equivalence-but-different-results-of-fixed-point/65628/1 "2021-08-01T12:44:23Z")

</div>

I am working on my own project, and I proved the following recursive equation has a  
unique fixed point (due to the Banach Fixed Point Theorem).

Here is the equation with simplified notations. I couldn’t find any option to write mathematical equations in a nice way. Please let me know if this matters for your understanding.

```julia
Way1 : v(x) = -c + s*n*max(F(v(x)),v(x)) + s*m*max(G(v(x)),v(x)) + s*(1-n-m)*v(x)

```

where all the parameters are in (0,1) with n+m in (0,1). Unless there is a mathematical mistake, I believe this exacts to

```julia
Way2 : v(x) = -c/(1-s) + s/(1-s)*n*max(F(v(x))-v(x),0) + s/(1-s)*m*max(G(v(x))-v(x),0)

```

However somehow when I use **while** function to get the fixed point, whereas **Way1** attains the fixed point, **Way2** does not.

I would very much like to know if this difference is whether coming from some error in my codes or from the way iteration works differently. Just in case, I leave my real codes.

Way 1

```julia
m_A = 0.15
m_B = 0.8
maxiter = 5000
tolerance = 1e-10
v_iv = collect(1:n) ./1000 # initial condition

# setup the algorithm
v_old = v_iv
normdiff = Inf
iter = 1

while normdiff > tolerance && iter <= maxiter
    v_new = -c .+ s .* m_A .* max.(im_sur(v_old),v_old) .+ s.*m_B .* max.(na_splus(v_old),v_old) .+ s .* (1-m_A-m_B) .* v_old 
    normdiff = norm(v_new - v_old)
    v_old = v_new
    iter = iter + 1
end

```

```julia
# Way 2
c_new = -c / (1-s)
m_A_new = m_A * s / (1-s)
m_B_new = m_B * s / (1-s)

while normdiff > tolerance && iter <= maxiter
    v_new = c_new .+ m_A_new .* max.(im_sur(v_old) .- v_old , 0) .+ m_B_new .* max.(na_splus(v_old) .- v_old , 0)
    normdiff = norm(v_new - v_old)
    v_old = v_new
    iter = iter + 1
end

```

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<div class="post-metadata">

**Author:** ![Tamas\_Papp](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/tamas_papp/32/25949_2.png) [@Tamas\_Papp](https://discourse.julialang.org/u/Tamas_Papp)\
**Post date:** [August 3, 2021, 8:53am UTC](https://discourse.julialang.org/t/question-mathematical-equivalence-but-different-results-of-fixed-point/65628/2 "2021-08-03T08:53:36Z")

</div>

It would be easier to help with self-contained MWE (we don’t know what `c`, `s`, `F`, … are).

> [@Please read: make it easier to help you](https://discourse.julialang.org/t/please-read-make-it-easier-to-help-you/14757):
>
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