# NLsolve always gives the initial guess as the final answer

**URL:** <https://discourse.julialang.org/t/nlsolve-always-gives-the-initial-guess-as-the-final-answer/30807>\
**Category:** Modelling & Simulations\
**Created:** [November 6, 2019, 9:59pm UTC](https://discourse.julialang.org/t/nlsolve-always-gives-the-initial-guess-as-the-final-answer/30807 "2019-11-06T21:59:25Z")\
**Posts on this page:** 5\
**Page:** 1

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**Author:** ![huchiayu0517](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/huchiayu0517/32/8202_2.png) [@huchiayu0517](https://discourse.julialang.org/u/huchiayu0517)\
**Post date:** [November 6, 2019, 9:59pm UTC](https://discourse.julialang.org/t/nlsolve-always-gives-the-initial-guess-as-the-final-answer/30807/1 "2019-11-06T21:59:25Z")

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Hi there! I’m trying to solve a nonlinear system of equations using the `NLsolve` package. However, I kept getting the same answer as my initial guess as if the solver did not do any iteration at all. I did not use Jacobian as in my real applications an analytic Jobabian is not available. An example code:

```julia
kk = [2e-17, 8e-12, 3e-17]
initial_x = [0.1, 0.0, 0.8]

using NLsolve

function f!(F, x)
    rate = zeros(3)
    rate[1] = kk[1] * x[1] 
    rate[2] = kk[2] * x[2]^2 
    rate[3] = kk[3] * x[3] 
    
    F[1] = 0.5*rate[3] - rate[1]
    F[2] = rate[1] - rate[2]
    F[3] = rate[1] + rate[2] - rate[3]
end

sol = nlsolve(f!, initial_x, ftol=1e-20)

```

with the following outputs:

Results of Nonlinear Solver Algorithm

- Algorithm: Trust-region with dogleg and autoscaling
- Starting Point: [0.5, 0.0, 0.0]
- Zero: [0.5, 0.0, 0.0]
- Inf-norm of residuals: 0.000000
- Iterations: 1000
- Convergence: false
  - |x - x’| \< 0.0e+00: false
  - |f(x)| \< 1.0e-20: false

- Function Calls (f): 1
- Jacobian Calls (df/dx): 1

It seems that there has been 1000 iterations but somehow the solution (`sol.zero`) is still the same as the initial guess. I played around with the `ftol` parameter but that did not help.

On the other hand, if I used `DifferentialEquations` and integrated towards a steady-state solution, I got the solution I expected analytically:

```julia
using DifferentialEquations

function fode!(F, x, p, t)
    rate = zeros(3)
    rate[1] = kk[1] * x[1] 
    rate[2] = kk[2] * x[2]^2 
    rate[3] = kk[3] * x[3] 
    
    F[1] = 0.5*rate[3] - rate[1]
    F[2] = rate[1] - rate[2]
    F[3] = rate[1] + rate[2] - rate[3]
end

tend = 1e17
tspan = (0.0, tend)
prob = ODEProblem(fode!, initial_x, tspan)
sol = solve(prob);
#sol = solve(prob, reltol=1e-7, abstol=1e-7);
xf = sol.u[end]

```

What did I do wrong with `NLsolve `?

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<div class="post-metadata">

**Author:** ![longemen3000](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/longemen3000/32/7298_2.png) [@longemen3000](https://discourse.julialang.org/u/longemen3000)\
**Post date:** [November 6, 2019, 11:38pm UTC](https://discourse.julialang.org/t/nlsolve-always-gives-the-initial-guess-as-the-final-answer/30807/2 "2019-11-06T23:38:48Z")

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whats the correct answer?, the inicial vector `[0.1, 0.0, 0.8]` has any restrictions?

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**Author:** ![huchiayu0517](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/huchiayu0517/32/8202_2.png) [@huchiayu0517](https://discourse.julialang.org/u/huchiayu0517)\
**Post date:** [November 7, 2019, 12:18am UTC](https://discourse.julialang.org/t/nlsolve-always-gives-the-initial-guess-as-the-final-answer/30807/3 "2019-11-07T00:18:40Z")

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The correct answer is roughly xf ~ [0.3, 0, 0.4] where xf[2] is a small number. With `DifferentialEquations`, I got [0.2997403048544782, 0.0008656504849742739, 0.3996537398060098] which is pretty close. The element of the vector must be in the range of [0, 0.5], [0, 1] and [0, 1] for x[1], x[2] and x[3] respectively, and there is another constraint that 2\*x[1] + x[2] + x[3] = 1.0, which should remain true if it is true initially. Is there any way to add constraints to the solver (e.g. specifying the allowed range of the solution)?

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**Author:** ![amrods](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/amrods/32/2543_2.png) [@amrods](https://discourse.julialang.org/u/amrods)\
**Post date:** [November 7, 2019, 2:33am UTC](https://discourse.julialang.org/t/nlsolve-always-gives-the-initial-guess-as-the-final-answer/30807/4 "2019-11-07T02:33:23Z")

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Are these equations independent?

You are trying to solve the system:

```julia
0.5*(kk[3]*x[3]) - (kk[1]*x[1]) == 0
(kk[1]*x[1]) - (kk[2]*x[2]^2) == 0
(kk[1]*x[1]) + (kk[2]*x[2]^2) - (kk[3]*x[3]) == 0

```

which simplifies to:

```julia
2*kk[1]*x[1] - kk[3]*x[3] == 0
kk[1]*x[1] - kk[2]*x[2]^2 == 0
kk[1]*x[1] + kk[2]*x[2]^2 - kk[3]*x[3] == 0

```

equation 1 minus equation 2 yields equation 3.

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<div class="post-metadata">

**Author:** ![huchiayu0517](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/huchiayu0517/32/8202_2.png) [@huchiayu0517](https://discourse.julialang.org/u/huchiayu0517)\
**Post date:** [November 11, 2019, 8:54am UTC](https://discourse.julialang.org/t/nlsolve-always-gives-the-initial-guess-as-the-final-answer/30807/6 "2019-11-11T08:54:39Z")

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Thanks for the remark! As you pointed out, the three equations are not independent and so the Jacobian is singular, which is probably why `NLsolve` didn’t work. Once I replaced the third equation with `F[3] = 2*x[1] + x[2] + x[3] - 1.0`, I got the same solutions with what I got from `DifferentialEquations`.