# Find\_zero to find a solution x as a function of y?
**URL:** https://discourse.julialang.org/t/find-zero-to-find-a-solution-x-as-a-function-of-y/91628
**Category:** General Usage
**Tags:** question, roots
**Created:** [December 14, 2022, 3:02am UTC](https://discourse.julialang.org/t/find-zero-to-find-a-solution-x-as-a-function-of-y/91628 "2022-12-14T03:02:21Z")
**Posts on this page:** 11
**Page:** 1
### Author:  [@kwyyoo](https://discourse.julialang.org/u/kwyyoo)
#### Post date: [December 14, 2022, 3:02am UTC](https://discourse.julialang.org/t/find-zero-to-find-a-solution-x-as-a-function-of-y/91628/1 "2022-12-14T03:02:21Z")
Hi. This is my first thread topic and question ever posting here.
I have the following function and the my goal is to find a solution of x to make it zero.
```julia
f(x, y) = (x^0.5 - y)/(1-x) - x^(-0.5);
```
Can I find the solution x^\star as a function of y such that x^\star = g(y)? I need this to plug x^\star = g(y) into another equation of y to maximize so that the final solution x^\star can be found afterwards.
I guess “find\_zero” function from [Roots.jl](https://juliamath.github.io/Roots.jl/dev/reference/#The-find_zero-and-find_zeros-functions) cannot work in this case since the function is not univariate.
---
### Author:  [@cvanaret](https://discourse.julialang.org/u/cvanaret)
#### Post date: [December 14, 2022, 3:34am UTC](https://discourse.julialang.org/t/find-zero-to-find-a-solution-x-as-a-function-of-y/91628/2 "2022-12-14T03:34:24Z")
Set the equation f(x, y) = 0, multiply by 1-x, reorder, let u = \sqrt{x}, solve the quadratic equation in u and recover x \> 0. There’s probably some conditions on y for the solution to exist.
---
### Author:  [@czylabsonasa](https://discourse.julialang.org/u/czylabsonasa)
#### Post date: [December 14, 2022, 6:25am UTC](https://discourse.julialang.org/t/find-zero-to-find-a-solution-x-as-a-function-of-y/91628/3 "2022-12-14T06:25:28Z")
as @cvanaret said, by reordering you’ll get y=\frac{2x-1}{\sqrt{x}}, \ x\>0,\ x\neq 1
but it is not a julia related question… 🙂
if you are searching for a programmatic way, check out the [Symbolics.jl](https://symbolics.juliasymbolics.org/dev/)
---
### Author:  [@DNF](https://discourse.julialang.org/u/DNF)
#### Post date: [December 14, 2022, 8:07am UTC](https://discourse.julialang.org/t/find-zero-to-find-a-solution-x-as-a-function-of-y/91628/4 "2022-12-14T08:07:58Z")
> [@kwyyoo](#):
>
> `f(x, y) = (x^0.5 - y)/(1-x) - x^(-0.5);`
Programming related: `sqrt(x)` is much, much, faster than `x^0.5`.
---
### Author:  [@cvanaret](https://discourse.julialang.org/u/cvanaret)
#### Post date: [December 14, 2022, 9:08am UTC](https://discourse.julialang.org/t/find-zero-to-find-a-solution-x-as-a-function-of-y/91628/5 "2022-12-14T09:08:54Z")
I actually suggested to solve for x, not for y 😉
I found x = \frac{(y + \sqrt{y^2 + 8})^2}{16} (we keep only the positive root because x \> 0). We also have x \neq 1, therefore you can’t have y = 1 (it implies x = 1).
---
### Author:  [@Per](https://discourse.julialang.org/u/Per)
#### Post date: [December 14, 2022, 9:24am UTC](https://discourse.julialang.org/t/find-zero-to-find-a-solution-x-as-a-function-of-y/91628/6 "2022-12-14T09:24:37Z")
> [@kwyyoo](#):
>
> I need this to plug x^\star = g(y) x⋆=g(y)x^\star = g(y) into another equation of y yy to maximize so that the final solution x^\star x⋆x^\star can be found afterwards.
This is a nonlinear programming (NLP), where `f(x,y) == 0` is a constraint. You might want to have a look at [JuMP](https://jump.dev/JuMP.jl/stable/manual/nlp/).
---
### Author:  [@Dan](https://discourse.julialang.org/u/Dan)
#### Post date: [December 14, 2022, 10:24am UTC](https://discourse.julialang.org/t/find-zero-to-find-a-solution-x-as-a-function-of-y/91628/7 "2022-12-14T10:24:43Z")
Here is a way of using Roots to get a solution (perhaps not as performant as solving the equation algebraically):
(using DNF’s suggestion to replace `^0.5` with `sqrt`)
```julia
using Roots
f(x,y) = (sqrt(abs(x)) - y)/(1-x) - 1/sqrt(abs(x));
G(y) = x -> f(x,y)
X(y) = find_zero(G(y),0.1)
```
Now we can:
```julia
julia> X(0.2)
0.5758872343937891
julia> f(X(0.2),0.2)
0.0
```
BTW here is a Unicode chart for a specific `y`:
```julia
julia> using UnicodePlots
julia> lineplot(G(0.2); name="f(x,0.2)", ylabel="G(0.2)")
┌────────────────────────────────────────┐
4 │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ f(x,0.2)
│⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
│⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
│⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
│⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
│⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢖⣒⣒⠒⠒⠒⡗⡗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒│
│⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀⣿⠁⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠤⠤⠔⠒⠒⠒⠒⠒│
G(0.2) │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡿⠀⠀⠀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
│⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
│⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
│⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
│⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
│⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
│⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
-7 │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
└────────────────────────────────────────┘
⠀-10⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀10⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀x⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
```
Admittedly, there is a problem with the `0.1` initial guess to `find_zero`. One solution would be to use `find_zeros` from Roots package, with a known good interval. Or to use this interval in `find_zero` with a different root finding method (i.e. Bisection). Something like:
```julia
julia> X(y) = minimum((abs(x),x) for x in find_zeros(G(0.2),(-10.0,10.0)))[2]
X (generic function with 1 method)
julia> X(0.2)
0.5758872343937891
```
As for the power of Roots:
Richard Feynman said (Lectures on Physics):
> “All of the laws of physics can be contained in one equation. That equation is **U=0**.”
So root finding is pretty powerful 😉 (lots of small print here).
---
### Author:  [@czylabsonasa](https://discourse.julialang.org/u/czylabsonasa)
#### Post date: [December 14, 2022, 10:26am UTC](https://discourse.julialang.org/t/find-zero-to-find-a-solution-x-as-a-function-of-y/91628/8 "2022-12-14T10:26:17Z")
you’re right, i read it w/o too much care…and solved the easier direction.
---
### Author:  [@cvanaret](https://discourse.julialang.org/u/cvanaret)
#### Post date: [December 14, 2022, 10:31am UTC](https://discourse.julialang.org/t/find-zero-to-find-a-solution-x-as-a-function-of-y/91628/9 "2022-12-14T10:31:07Z")
> [@Dan](#):
>
> `f(x,y) = (sqrt(abs(x)) - y)/(1-x) - sqrt(abs(x));`
should be `f(x,y) = (sqrt(abs(x)) - y)/(1-x) - 1/sqrt(abs(x));` 🙂
---
### Author:  [@Dan](https://discourse.julialang.org/u/Dan)
#### Post date: [December 14, 2022, 10:51am UTC](https://discourse.julialang.org/t/find-zero-to-find-a-solution-x-as-a-function-of-y/91628/10 "2022-12-14T10:51:40Z")
Thanks, fixed this by editing answer!
---
### Author:  [@kwyyoo](https://discourse.julialang.org/u/kwyyoo)
#### Post date: [February 10, 2023, 5:45pm UTC](https://discourse.julialang.org/t/find-zero-to-find-a-solution-x-as-a-function-of-y/91628/11 "2023-02-10T17:45:18Z")
Thank you so much Dan! This worked well in my case. I wanted to solve it through the code, not algebraically. Appreciate it!