# Implementing inexact Newton solves in NonlinearSolve.jl
**URL:** https://discourse.julialang.org/t/implementing-inexact-newton-solves-in-nonlinearsolve-jl/119844
**Category:** General Usage
**Tags:** nonlinearsolve
**Created:** [September 25, 2024, 3:56am UTC](https://discourse.julialang.org/t/implementing-inexact-newton-solves-in-nonlinearsolve-jl/119844 "2024-09-25T03:56:28Z")
**Posts on this page:** 3
**Page:** 1
### Author:  [@mpf01](https://discourse.julialang.org/u/mpf01)
#### Post date: [September 25, 2024, 3:56am UTC](https://discourse.julialang.org/t/implementing-inexact-newton-solves-in-nonlinearsolve-jl/119844/1 "2024-09-25T03:56:28Z")
I’d like to implement inexact Newton solves using NonlinearSolve.jl. The output from this code snippet (shown below) suggests that, by default, each Newton equation is solved to a fixed accuracy.
```julia
nlcache = init(
prob,
reltol=1e-3, abstol=1e-3,
show_trace = Val(true),
NewtonRaphson(linsolve = KrylovJL_MINRES(verbose=10, itmax=100)),
)
for i in 1:10
step!(nlcache)
end
```
Is it possible to modify `nlcache` within the loop to adaptively specify the accuracy of each linear solve?
Here’s the output:
```julia
┌ Warning: minres! doesn't support right preconditioning.
└ @ LinearSolve ~/.julia/packages/LinearSolve/B82H3/src/iterative_wrappers.jl:275
MINRES: system of size 202
k ‖r‖ ‖Aᴴr‖ β cos sin ‖A‖ κ(A) test1 test2 timer
0 2.0e+00 0.0e+00 2.0e+00 -1.0e+00 0.0e+00 0.0e+00 0.0e+00 ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ 0.00s
10 3.0e-02 3.3e-02 8.2e-01 6.0e-01 8.0e-01 1.7e+01 6.3e+00 1.9e-03 5.2e-02 0.00s
20 1.9e-03 8.7e-03 4.4e+00 5.8e-01 8.2e-01 2.0e+01 8.5e+00 9.7e-05 1.8e-01 0.00s
30 2.3e-04 2.0e-04 8.2e-01 -5.3e-01 8.5e-01 2.4e+01 9.2e+00 9.7e-06 3.2e-02 0.00s
40 2.0e-05 2.2e-05 8.0e-01 -6.6e-01 7.5e-01 2.7e+01 9.2e+00 7.4e-07 3.0e-02 0.00s
50 1.9e-06 1.8e-06 5.8e-01 6.9e-01 7.2e-01 3.0e+01 9.2e+00 6.7e-08 2.3e-02 0.00s
60 1.5e-07 1.5e-07 8.6e-01 5.6e-01 8.3e-01 3.2e+01 9.2e+00 4.7e-09 2.6e-02 0.00s
1 2.97303732e-02 9.81027009e-01
┌ Warning: minres! doesn't support right preconditioning.
└ @ LinearSolve ~/.julia/packages/LinearSolve/B82H3/src/iterative_wrappers.jl:275
MINRES: system of size 202
k ‖r‖ ‖Aᴴr‖ β cos sin ‖A‖ κ(A) test1 test2 timer
0 4.0e-01 0.0e+00 4.0e-01 -1.0e+00 0.0e+00 0.0e+00 0.0e+00 ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ 0.00s
10 5.7e-04 8.9e-04 9.3e-01 7.0e-01 7.1e-01 3.0e+01 2.4e+01 1.0e-03 3.7e-02 0.00s
20 4.6e-05 4.4e-05 6.2e-01 6.8e-01 7.4e-01 4.0e+01 3.5e+01 5.6e-05 1.7e-02 0.00s
30 7.6e-06 7.6e-06 9.9e-01 -4.9e-01 8.7e-01 5.0e+01 3.5e+01 7.3e-06 1.8e-02 0.00s
40 6.9e-07 1.2e-06 2.0e+00 -4.7e-01 8.8e-01 5.7e+01 3.7e+01 5.8e-07 2.8e-02 0.00s
50 7.5e-08 9.5e-08 9.6e-01 6.7e-01 7.4e-01 6.4e+01 3.7e+01 5.7e-08 1.5e-02 0.00s
2 3.51646316e-03 2.07810848e-02
┌ Warning: minres! doesn't support right preconditioning.
└ @ LinearSolve ~/.julia/packages/LinearSolve/B82H3/src/iterative_wrappers.jl:275
MINRES: system of size 202
k ‖r‖ ‖Aᴴr‖ β cos sin ‖A‖ κ(A) test1 test2 timer
0 4.7e-02 0.0e+00 4.7e-02 -1.0e+00 0.0e+00 0.0e+00 0.0e+00 ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ 0.00s
10 5.8e-05 5.6e-05 8.2e-01 5.9e-01 8.0e-01 2.5e+01 2.0e+01 8.5e-04 3.1e-02 0.00s
20 4.4e-06 4.3e-06 6.7e-01 6.6e-01 7.5e-01 3.3e+01 2.6e+01 4.7e-05 2.2e-02 0.00s
30 6.7e-07 6.4e-07 8.1e-01 -5.7e-01 8.2e-01 4.0e+01 2.6e+01 5.8e-06 2.0e-02 0.00s
40 6.7e-08 1.2e-07 2.2e+00 -4.4e-01 9.0e-01 4.2e+01 2.9e+01 5.6e-07 3.8e-02 0.00s
3 6.82363870e-05 2.88556208e-03
```
---
### Author:  [@Oscar\_Smith](https://discourse.julialang.org/u/Oscar_Smith)
#### Post date: [September 25, 2024, 5:08am UTC](https://discourse.julialang.org/t/implementing-inexact-newton-solves-in-nonlinearsolve-jl/119844/2 "2024-09-25T05:08:44Z")
what do you mean by adaptively set the accuracy?
---
### Author:  [@mpf01](https://discourse.julialang.org/u/mpf01)
#### Post date: [September 25, 2024, 5:55am UTC](https://discourse.julialang.org/t/implementing-inexact-newton-solves-in-nonlinearsolve-jl/119844/3 "2024-09-25T05:55:59Z")
In situations where the computation cost is primarily dominated by the Jacobian-vector products, further efficiency can be had by solving the Newton system inexactly. A common approach is to ask an iterative method (eg, MINRES, in my case) to obtain a search direction \Delta x\in\mathbb{R}^n that reduces the residual to first order:
\|J(x\_k) \Delta x + r(x\_k)\| \le \eta\_k\|r(x\_k)\|,
where r is the residual function, J is its Jacobian, and \eta\_k\in(0,1) is a dynamic accuracy parameter.
In each step of the loop, I’d like to pass \eta\_k to MINRES as an optimality tolerance. This iimplements the above criterion.