Scientific Computing, TU Berlin, WS 2019/2020, Lecture 25

Jürgen Fuhrmann, WIAS Berlin

Nonlinear systems

In [1]:
using Printf
using VoronoiFVM
using  Plots
using Printf
injupyter()=isdefined(Main, :IJulia) && Main.IJulia.inited
Out[1]:
injupyter (generic function with 1 method)

The method

Construction of control volumes

  • Start with a triangulation of a polygonal domain (intervals in 1D,triangles in 2D, tetrahedra in 3D).

  • Join triangle circumcenters by lines $\rightarrow$ create Voronoi cells which can serve as control volumes, akin to representative elementary volumes (REV) used to derive conservation laws.

  • Black + green: triangle nodes
  • Gray: triangle edges
  • Blue: triangle circumcenters
  • Red: Boundaries of Voronoi cells

Condition on triangulation

  • There is a 1:1 incidence between triangulation nodes and Voronoi cells. Moreover, the angle between the interface between two Voronoi cells and the edge between their corresponding nodes is $\frac\pi2$.

  • Requires (in 2D) that sums of angles opposite to triangle edges are less than $\pi$ and that angles opposite to boudary edges are less than $\frac\pi2$.

  • "boundary conforming Delaunay property". It has different equivalent definitions and analogues in 3D.

  • Construction:

    • "by hand" (or script) from tensor product meshes
    • Mesh generators: Triangle, TetGen

The discretization approach

  • Use Voronoi cells as REVs aka control volumes aka finite volume cells.

  • Given a continuity equation $\nabla\cdot \vec j=0$ in a domain $\Omega$, integrate this over a contol volume $\omega_k$ with associated node $\vec x_k$ and apply Gauss theorem:

\begin{align*} 0&=\int_{\omega_k} \nabla\cdot \vec j \ d\omega =\int_{\partial\omega_k} \vec j\cdot \vec n ds\\ &=\sum_{l\in N_k} \int_{\omega_k\cap \omega_l} \vec j\cdot \vec n ds + \int_{\partial\omega_k\cap \partial\Omega} \vec j\cdot \vec n ds\\ &\approx \sum_{l\in N_k} \frac{\sigma_{kl}}{h_{kl}}g(u_k, u_l)+ \gamma_k b(u_k) \end{align*}

  • Here, $N_k$ is the set of neighbor control volumes, $\sigma_{kl}=|\omega_k\cap \omega_l|$, $h_{kl}=|\vec x_k -\vec x_l|$, $\gamma_k=|\partial\omega_k\cap \partial\Omega|$, where $|\cdot|$ denotes the measure (length resp. area) of a geometrical entity.

Flux functions

For instance, for the diffusion flux $\vec j=-D\vec\nabla u$, we use $g(u_k, u_l)=D(u_k -u_l)$.

For a convective diffusion flux $\vec j = -D\vec \nabla u + u \vec v$, one can chose the upwind flux

\begin{align*} g(u_k, u_l)=D(u_k -u_l) + v_{kl}\begin{cases} u_k,& v_{kl}>0\\ u_l,& v_{kl}\leq 0, \end{cases} \end{align*}

where $v_{kl}=\frac{h_{kl}}{\sigma_{kl}}\int_{\omega_k\cap \omega_l} \vec v \cdot \vec n_{kl} \ ds$ Fluxes also can depend nonlinearily on $u$.

For Diffusion flux $\vec j = -D\vec \nabla u^m\vec v$, we can choose the flux $g(u_k, u_l)=D(u_k^m -u_l^m)$.

Software API and implementation

The entities describing the discrete system can be subdivided into two categories:

  • geometrical data: $|\omega_k|, \gamma_k, \sigma_{kl}, h_{kl}$ together with the connectivity information of the triangles
  • physical data: the number $m$ and the functions $s,g,r,b,f$ describing the particular problem.

This structure allows to describe the problem to be solved by data derived from the discretization grid and by the functions describing the physics, giving rise to a software API.

The solution of the nonlinear systems of equations can be performed by Newton's method combined with various direct and iterative linear solvers.

The generic programming capabilities of Julia allow for an implementation of the method which results in an API which consists in the implementation of functions $s,g,r,b,f$ without the need to write code for their derivatives.

Examples

For the following examples, we always start with an initival value and watch the evolution of the isolated system. Isolation is achieved by setting homogeneous Neumann boundary conditions, which don't occur in the discrete formulation and therefore done have to be set explicitely.

Diffusion problem $\partial_t u - \nabla\cdot D\nabla u=0$

In [2]:
function diffusion(;n=1000, nt=100, tstep0=1.0e-3, D=1.0, dtgrowth=1.0)

    # Grid on interval 0.1
    X=collect(0:1.0/n:1)
    grid=VoronoiFVM.Grid(X)

    # Diffusion flux between neigboring control volumes
    function flux!(f,u,edge,data)
        uk=viewK(edge,u)
        ul=viewL(edge,u)
        f[1]=D*(uk[1]-ul[1])
    end

    # Storage term (under time derivative)
    function storage!(f,u,node,data)
        f[1]=u[1]
    end

    # Initial value
    fpeak(x)=exp(-100*(x-0.25)^2)

    # Create a physics structure
    physics=VoronoiFVM.Physics(
        flux=flux!,
        storage=storage!)

    sys=VoronoiFVM.DenseSystem(grid,physics)
    enable_species!(sys,1,[1])
    # Assume homogeneous Neumann boundary conditions, so do nothig

    # Create a solution array
    inival=unknowns(sys)
    solution=unknowns(sys)

    # Broadcast the initial value
    inival[1,:].=map(x->fpeak(x),X)

    # Run time evolution and plot
    time=0.0
    tstep=tstep0
    @gif for i=1:nt
        time=time+tstep
        solve!(solution,inival,sys,tstep=tstep)
        inival.=solution
        plot(X,solution[1,:],
             title="time=$(@sprintf("%.4f",time))",
             xlabel="x",ylabel="y",
             ylims=(0,1))
        tstep*=dtgrowth
    end
end
Out[2]:
diffusion (generic function with 1 method)
In [3]:
injupyter()&& diffusion()

┌ Info: Saved animation to 
│   fn = /home/fuhrmann/Wias/teach/scicomp/course/tmp.gif
â”” @ Plots /home/fuhrmann/.julia/packages/Plots/qZHsp/src/animation.jl:98
Out[3]:

We observe an re-distribution of species while keeping the integral over the solution constant until a final steady state is reached.

Reaction-diffusion problem $\partial_t u - \nabla\cdot D\nabla u + R u=0$

In [4]:
function reaction_diffusion(;n=1000,tstep0=1.0e-3, nt=100, D=1.0, R=1.0, dtgrowth=1.0)

    # Grid on interval 0.1
    X=collect(0:1.0/n:1)
    grid=VoronoiFVM.Grid(X)

    # Diffusion flux between neigboring control volumes
    function flux!(f,u,edge,data)
        uk=viewK(edge,u)
        ul=viewL(edge,u)
        f[1]=D*(uk[1]-ul[1])
    end

    # Storage term (under time derivative)
    function storage!(f,u,node,data)
        f[1]=u[1]
    end

    # Reaction term
    function reaction!(f,u,node,data)
        f[1]=R*u[1]
    end

    # Initial value
    fpeak(x)=exp(-100*(x-0.25)^2)

    # Create a physics structure
    physics=VoronoiFVM.Physics(
        flux=flux!,
        reaction=reaction!,
        storage=storage!)

    sys=VoronoiFVM.DenseSystem(grid,physics)
    enable_species!(sys,1,[1])
    # Assume homogeneous Neumann boundary conditions, so do nothig

    # Create a solution array
    inival=unknowns(sys)
    solution=unknowns(sys)

    # Broadcast the initial value
    inival[1,:].=map(x->fpeak(x),X)

    # Run time evolution and plot
    time=0.0
    tstep=tstep0
    @gif for i=1:nt
        time=time+tstep
        solve!(solution,inival,sys,tstep=tstep)
        plot(X,solution[1,:],
             title="time=$(@sprintf("%.4f",time))",
             xlabel="x",ylabel="y",
             ylims=(0,1))
        inival.=solution
        tstep*=dtgrowth
    end
end
Out[4]:
reaction_diffusion (generic function with 1 method)
In [5]:
injupyter()&& reaction_diffusion()

┌ Info: Saved animation to 
│   fn = /home/fuhrmann/Wias/teach/scicomp/course/tmp.gif
â”” @ Plots /home/fuhrmann/.julia/packages/Plots/qZHsp/src/animation.jl:98
Out[5]:

We observe an re-distribution of species. It is accompanied by "eating" stuff away due to the reaction term.

Convection-Diffusion problem $\partial_t u - \nabla\cdot( D\nabla u - u \vec v) =0$

In [6]:
function convection_diffusion(;n=50,tstep0=1.0e-3, nt=100, D=0.01, v=10.0, dtgrowth=1.0, centered=true, expfit=false)

    # Grid on interval 0.1
    X=collect(0:1.0/n:1)
    grid=VoronoiFVM.Grid(X)

    B(x)=x/(exp(x)-1)
    # Diffusion flux between neigboring control volumes
    function flux!(f,u,edge,data)
        uk=viewK(edge,u)
        ul=viewL(edge,u)
        h=edgelength(edge)
        if centered
            f[1]=D*(uk[1]-ul[1])+ v*h*0.5*(uk[1]+ul[1])
        elseif expfit
            f[1]=D*(B(-v*h/D)*uk[1]- B(v*h/D)*ul[1])
        else
            f[1]=D*(uk[1]-ul[1])+ ( v>0.0 ? v*h*uk[1] : v*h*ul[1] )
        end
    end

    # Storage term (under time derivative)
    function storage!(f,u,node,data)
        f[1]=u[1]
    end

    # Initial value
    fpeak(x)=exp(-100*(x-0.25)^2)

    # Create a physics structure
    physics=VoronoiFVM.Physics(
        flux=flux!,
        storage=storage!)

    sys=VoronoiFVM.DenseSystem(grid,physics)
    enable_species!(sys,1,[1])
    # Assume homogeneous Neumann boundary conditions, so do nothig

    # Create a solution array
    inival=unknowns(sys)
    solution=unknowns(sys)

    # Broadcast the initial value
    inival[1,:].=map(x->fpeak(x),X)

    # Run time evolution and plot
    time=0.0
    tstep=tstep0
    @gif for i=1:nt
        time=time+tstep
        solve!(solution,inival,sys,tstep=tstep)
        inival.=solution
        plot(X,solution[1,:],
             title="time=$(@sprintf("%.4f",time))",
             xlabel="x",ylabel="y",
             ylims=(0,1))
        tstep*=dtgrowth
    end
end
Out[6]:
convection_diffusion (generic function with 1 method)
In [7]:
injupyter()&& convection_diffusion(n=40,tstep0=1.0e-3,D=0.01, v=10.0,centered=true,expfit=false)

┌ Info: Saved animation to 
│   fn = /home/fuhrmann/Wias/teach/scicomp/course/tmp.gif
â”” @ Plots /home/fuhrmann/.julia/packages/Plots/qZHsp/src/animation.jl:98
Out[7]:
In [8]:
injupyter()&& convection_diffusion(n=40,tstep0=1.0e-3,D=0.01, v=10.0,centered=false,expfit=true)

┌ Info: Saved animation to 
│   fn = /home/fuhrmann/Wias/teach/scicomp/course/tmp.gif
â”” @ Plots /home/fuhrmann/.julia/packages/Plots/qZHsp/src/animation.jl:98
Out[8]:

For the small choosen diffusion coefficient, we observe a convection dominant transport to the right end of the domain, where species start to "pile up". For the centered flux, unphysical oscillations start to occur. For the exponential fitting flux, these are avoided at the price of additional diffusion "smearing out" the solution.

Nonlinear diffusion problem $\partial_t u - \nabla\cdot D\nabla u^m=0$

In [9]:
function nonlinear_diffusion(;n=1000, nt=100, tstep0=1.0e-3, D=1.0, dtgrowth=1.0, m=3)

    # Grid on interval 0.1
    X=collect(0:1.0/n:1)
    grid=VoronoiFVM.Grid(X)

    function flux!(f,u,edge,data)
        uk=viewK(edge,u)
        ul=viewL(edge,u)
        f[1]=uk[1]^m-ul[1]^m
    end
    function storage!(f,u,node,data)
        f[1]=u[1]
    end

    # initial value with finite support
    fpeak(x)= 0.45≤ x ≤ 0.55 ? 1.0 : 0.0

    physics=VoronoiFVM.Physics(
        flux=flux!,
        storage=storage!)

    sys=VoronoiFVM.DenseSystem(grid,physics)
    enable_species!(sys,1,[1])
    inival=unknowns(sys)
    solution=unknowns(sys)
    inival[1,:].=map(x->fpeak(x),X)
    time=0.0
    tstep=tstep0
    @gif for i=1:nt
        time=time+tstep
        solve!(solution,inival,sys,tstep=tstep)
        inival.=solution
        plot(X,solution[1,:],
             title="time=$(@sprintf("%.4f",time))",
             xlabel="x",ylabel="y",
             ylims=(0,1))
        tstep*=dtgrowth
    end
end
Out[9]:
nonlinear_diffusion (generic function with 1 method)
In [10]:
injupyter()&& nonlinear_diffusion(m=3)

┌ Info: Saved animation to 
│   fn = /home/fuhrmann/Wias/teach/scicomp/course/tmp.gif
â”” @ Plots /home/fuhrmann/.julia/packages/Plots/qZHsp/src/animation.jl:98
Out[10]:

For this example, we deliberately start with a solution with finite support. In the case of $m>1$, we observe, that during the evolution, the finit support property of the solution is conserved.

In [11]:
function nonlinear_diffusion2D(;n=50, nt=100, tstep0=1.0e-3, D=1.0, dtgrowth=1.0, m=3)

    # Grid on interval 0.1
    X=collect(0:1.0/n:1)
    grid=VoronoiFVM.Grid(X,X)
    function flux!(f,u,edge,data)
        uk=viewK(edge,u)
        ul=viewL(edge,u)
        f[1]=uk[1]^m-ul[1]^m
    end
    function storage!(f,u,node,data)
        f[1]=u[1]
    end

    # initial value with finite support
    fpeak(x,y)= 0.45≤ x ≤ 0.55 &&  0.45≤ y ≤ 0.55 ? 1.0 : 0.0

    physics=VoronoiFVM.Physics(
        flux=flux!,
        storage=storage!)

    sys=VoronoiFVM.DenseSystem(grid,physics)
    enable_species!(sys,1,[1])
    inival=unknowns(sys)
    solution=unknowns(sys)
    for i=1:num_nodes(grid)
        inival[1,i]=fpeak(grid.coord[1,i],grid.coord[2,i])
    end
    time=0.0
    tstep=tstep0
    for i=1:nt
        time=time+tstep
        solve!(solution,inival,sys,tstep=tstep)
        inival.=solution
        p=contourf(X,X,reshape(solution[1,:],length(X),length(X)),levels=collect(0:0.1:1.0),
                 clim=(0,1.0),
                 colorbar=:right,color=:viridis,
                 title=@sprintf("time=%.4f\n",time))
        tstep*=dtgrowth
        gui(p)
    end
end
Out[11]:
nonlinear_diffusion2D (generic function with 1 method)
In [12]:
injupyter()&& nonlinear_diffusion(m=3)

┌ Info: Saved animation to 
│   fn = /home/fuhrmann/Wias/teach/scicomp/course/tmp.gif
â”” @ Plots /home/fuhrmann/.julia/packages/Plots/qZHsp/src/animation.jl:98
Out[12]:

Brusselator system

The Brusselator is a system of two species interacting via a reaction:

\begin{align*} \partial_t u_1 - \nabla \cdot D_1 \nabla u_1 + (B+1)u_1-A-u_1^2u_2 &=0\\ \partial_t u_2 - \nabla \cdot D_2 \nabla u_2 + u_1^2u_2 -B u_1 &=0\\ \end{align*}
In [13]:
function brusselator(;n=20,nt=150,tstep=0.05)
    A=4.0
    B=6.0
    D1=0.01
    D2=0.1
    perturbation=0.1;

    function diffusion!(f,u,edge,data)
        uk=viewK(edge,u)
        ul=viewL(edge,u)
        f[1]=D1*(uk[1]-ul[1])
        f[2]=D2*(uk[2]-ul[2])
    end

    function reaction!(f,u,node,data)
        x=u[1]
        y=u[2]
        f[1]= (B+1.0)*x-A-x*x*y
	f[2]= x*x*y-B*x
    end

    function storage!(f,u,node,data)
        f[1]=u[1]
        f[2]=u[2]
    end

    h=1.0/convert(Float64,n-1)
    X=collect(-1:h:1)
    Y=collect(-1:h:1)
    nx=length(X)
    ny=length(Y)
    grid=VoronoiFVM.Grid(X,Y)

    # Create a physics structure
    physics=VoronoiFVM.Physics(
        num_species=2,
        flux=diffusion!,
        storage=storage!,
        reaction=reaction!)

    sys=VoronoiFVM.DenseSystem(grid,physics)

    enable_species!(sys,1,[1])
    enable_species!(sys,2,[1])

    # Create a solution array
    inival=unknowns(sys)
    solution=unknowns(sys)
    t0=0

    for i=1:nx*ny
        inival[1,i]=1.0+perturbation*randn()
        inival[2,i]=1.0+perturbation*randn()
    end
    time=t0
    control=VoronoiFVM.NewtonControl()
    control.verbose=true
    control.max_lureuse=0
    control.tol_relative=1.0e-5
    control.tol_linear=1.0e-4
    @gif for i=1:nt
        time=time+tstep
        solve!(solution,inival,sys,control=control,tstep=tstep)
        inival.=solution
        @views begin
            p1=contourf(X,Y,reshape(solution[1,:],nx,ny),colorbar=:right,color=:viridis,aspect_ratio=1)
	    p2=contourf(X,Y,reshape(solution[2,:],nx,ny),colorbar=:right,color=:viridis,aspect_ratio=1)
            p=plot(p1,p2,layout=(1,2),title="time=$(time)" )
        end
        tstep*=1.05
    end
end
Out[13]:
brusselator (generic function with 1 method)
In [14]:
injupyter()&& brusselator()

    Start Newton iteration
    it=  1( 0) du=5.127e-01 cont=5.127e-01 dnorm=7.733e-02 0
    it=  2( 0) du=3.889e-03 cont=7.585e-03 dnorm=2.272e-05 0
    it=  3( 0) du=6.844e-08 cont=1.760e-05 dnorm=4.804e-10 0
    Newton iteration successful
  0.772778 seconds (3.06 M allocations: 134.992 MiB, 5.26% gc time)
    Start Newton iteration
    it=  1( 0) du=2.614e-01 cont=2.614e-01 dnorm=7.654e-02 0
    it=  2( 0) du=1.313e-03 cont=5.022e-03 dnorm=1.584e-05 0
    it=  3( 0) du=1.323e-08 cont=1.008e-05 dnorm=1.579e-10 0
    Newton iteration successful
  0.102768 seconds (1.10 M allocations: 33.942 MiB, 3.73% gc time)
    Start Newton iteration
    it=  1( 0) du=2.264e-01 cont=2.264e-01 dnorm=7.552e-02 0
    it=  2( 0) du=8.197e-04 cont=3.621e-03 dnorm=1.114e-05 0
    it=  3( 0) du=6.241e-09 cont=7.614e-06 dnorm=1.719e-11 1
    Newton iteration successful
  0.104019 seconds (1.10 M allocations: 33.935 MiB, 4.81% gc time)
    Start Newton iteration
    it=  1( 0) du=2.169e-01 cont=2.169e-01 dnorm=7.432e-02 0
    it=  2( 0) du=5.501e-04 cont=2.536e-03 dnorm=7.639e-06 0
    it=  3( 0) du=5.381e-09 cont=9.783e-06 dnorm=2.305e-11 1
    Newton iteration successful
  0.106529 seconds (1.10 M allocations: 33.945 MiB, 4.74% gc time)
    Start Newton iteration
    it=  1( 0) du=2.146e-01 cont=2.146e-01 dnorm=7.298e-02 0
    it=  2( 0) du=3.628e-04 cont=1.690e-03 dnorm=4.758e-06 0
    it=  3( 0) du=3.984e-09 cont=1.098e-05 dnorm=2.246e-11 1
    Newton iteration successful
  0.103050 seconds (1.10 M allocations: 33.933 MiB, 2.60% gc time)
    Start Newton iteration
    it=  1( 0) du=2.151e-01 cont=2.151e-01 dnorm=7.155e-02 0
    it=  2( 0) du=2.089e-04 cont=9.715e-04 dnorm=2.093e-06 0
    it=  3( 0) du=2.098e-09 cont=1.004e-05 dnorm=8.046e-12 1
    Newton iteration successful
  0.109273 seconds (1.10 M allocations: 33.943 MiB, 4.71% gc time)
    Start Newton iteration
    it=  1( 0) du=2.164e-01 cont=2.164e-01 dnorm=7.007e-02 0
    it=  2( 0) du=1.099e-04 cont=5.077e-04 dnorm=6.895e-07 0
    it=  3( 0) du=7.659e-10 cont=6.969e-06 dnorm=1.986e-12 1
    Newton iteration successful
  0.106071 seconds (1.10 M allocations: 33.934 MiB, 2.53% gc time)
    Start Newton iteration
    it=  1( 0) du=2.184e-01 cont=2.184e-01 dnorm=6.855e-02 0
    it=  2( 0) du=2.540e-04 cont=1.163e-03 dnorm=3.939e-06 0
    it=  3( 0) du=4.218e-09 cont=1.660e-05 dnorm=4.471e-11 1
    Newton iteration successful
  0.104983 seconds (1.10 M allocations: 33.945 MiB, 4.78% gc time)
    Start Newton iteration
    it=  1( 0) du=2.203e-01 cont=2.203e-01 dnorm=6.700e-02 0
    it=  2( 0) du=4.688e-04 cont=2.128e-03 dnorm=8.136e-06 0
    it=  3( 0) du=1.605e-08 cont=3.423e-05 dnorm=2.399e-10 0
    Newton iteration successful
  0.102837 seconds (1.10 M allocations: 33.934 MiB, 2.57% gc time)
    Start Newton iteration
    it=  1( 0) du=2.215e-01 cont=2.215e-01 dnorm=6.540e-02 0
    it=  2( 0) du=8.055e-04 cont=3.636e-03 dnorm=1.407e-05 0
    it=  3( 0) du=5.315e-08 cont=6.598e-05 dnorm=8.837e-10 0
    Newton iteration successful
  0.107217 seconds (1.10 M allocations: 33.934 MiB, 4.74% gc time)
    Start Newton iteration
    it=  1( 0) du=2.208e-01 cont=2.208e-01 dnorm=6.372e-02 0
    it=  2( 0) du=1.336e-03 cont=6.048e-03 dnorm=2.319e-05 0
    it=  3( 0) du=1.720e-07 cont=1.288e-04 dnorm=2.893e-09 0
    Newton iteration successful
  0.104367 seconds (1.10 M allocations: 33.945 MiB, 2.54% gc time)
    Start Newton iteration
    it=  1( 0) du=2.165e-01 cont=2.165e-01 dnorm=6.191e-02 0
    it=  2( 0) du=2.239e-03 cont=1.034e-02 dnorm=3.849e-05 0
    it=  3( 0) du=5.600e-07 cont=2.501e-04 dnorm=9.415e-09 0
    Newton iteration successful
  0.108000 seconds (1.10 M allocations: 33.936 MiB, 4.72% gc time)
    Start Newton iteration
    it=  1( 0) du=2.054e-01 cont=2.054e-01 dnorm=5.986e-02 0
    it=  2( 0) du=3.957e-03 cont=1.926e-02 dnorm=6.691e-05 0
    it=  3( 0) du=1.977e-06 cont=4.997e-04 dnorm=3.279e-08 0
    Newton iteration successful
  0.105985 seconds (1.10 M allocations: 33.934 MiB, 2.47% gc time)
    Start Newton iteration
    it=  1( 0) du=1.812e-01 cont=1.812e-01 dnorm=5.736e-02 0
    it=  2( 0) du=7.649e-03 cont=4.220e-02 dnorm=1.267e-04 0
    it=  3( 0) du=8.108e-06 cont=1.060e-03 dnorm=1.306e-07 0
    it=  4( 0) du=9.068e-12 cont=1.118e-06 dnorm=1.381e-13 1
    Newton iteration successful
  0.138501 seconds (1.47 M allocations: 45.250 MiB, 3.64% gc time)
    Start Newton iteration
    it=  1( 0) du=1.507e-01 cont=1.507e-01 dnorm=5.397e-02 0
    it=  2( 0) du=1.696e-02 cont=1.125e-01 dnorm=2.756e-04 0
    it=  3( 0) du=3.986e-05 cont=2.350e-03 dnorm=6.250e-07 0
    it=  4( 0) du=2.172e-10 cont=5.450e-06 dnorm=3.133e-12 1
    Newton iteration successful
  0.141394 seconds (1.47 M allocations: 45.251 MiB, 3.64% gc time)
    Start Newton iteration
    it=  1( 0) du=2.667e-01 cont=2.667e-01 dnorm=4.849e-02 0
    it=  2( 0) du=4.566e-02 cont=1.712e-01 dnorm=7.290e-04 0
    it=  3( 0) du=1.775e-04 cont=3.888e-03 dnorm=2.797e-06 0
    it=  4( 0) du=2.495e-09 cont=1.406e-05 dnorm=3.569e-11 1
    Newton iteration successful
  0.143886 seconds (1.47 M allocations: 45.241 MiB, 3.75% gc time)
    Start Newton iteration
    it=  1( 0) du=5.842e-01 cont=5.842e-01 dnorm=3.705e-02 0
    it=  2( 0) du=1.065e-01 cont=1.823e-01 dnorm=1.760e-03 0
    it=  3( 0) du=2.165e-03 cont=2.032e-02 dnorm=3.122e-05 0
    it=  4( 0) du=9.971e-07 cont=4.606e-04 dnorm=1.169e-08 0
    Newton iteration successful
  0.142284 seconds (1.47 M allocations: 45.250 MiB, 3.69% gc time)
    Start Newton iteration
    it=  1( 0) du=1.344e+00 cont=1.344e+00 dnorm=1.051e-02 0
    it=  2( 0) du=2.916e-01 cont=2.170e-01 dnorm=4.550e-03 0
    it=  3( 0) du=3.817e-02 cont=1.309e-01 dnorm=5.136e-04 0
    it=  4( 0) du=6.235e-04 cont=1.634e-02 dnorm=6.058e-06 0
    it=  5( 0) du=1.743e-07 cont=2.796e-04 dnorm=8.471e-10 0
    Newton iteration successful
  0.180001 seconds (1.84 M allocations: 56.558 MiB, 4.22% gc time)
    Start Newton iteration
    it=  1( 0) du=1.042e+00 cont=1.042e+00 dnorm=4.295e-03 0
    it=  2( 0) du=2.655e-01 cont=2.547e-01 dnorm=4.731e-03 0
    it=  3( 0) du=2.658e-02 cont=1.001e-01 dnorm=4.518e-04 0
    it=  4( 0) du=2.525e-04 cont=9.500e-03 dnorm=3.933e-06 0
    it=  5( 0) du=2.288e-08 cont=9.061e-05 dnorm=2.981e-10 0
    Newton iteration successful
  0.174131 seconds (1.84 M allocations: 56.556 MiB, 2.79% gc time)
    Start Newton iteration
    it=  1( 0) du=4.158e-01 cont=4.158e-01 dnorm=6.832e-03 0
    it=  2( 0) du=4.983e-02 cont=1.199e-01 dnorm=8.934e-04 0
    it=  3( 0) du=8.241e-04 cont=1.654e-02 dnorm=1.355e-05 0
    it=  4( 0) du=2.252e-07 cont=2.733e-04 dnorm=3.104e-09 0
    Newton iteration successful
  0.139566 seconds (1.47 M allocations: 45.251 MiB, 3.70% gc time)
    Start Newton iteration
    it=  1( 0) du=1.390e-01 cont=1.390e-01 dnorm=8.182e-03 0
    it=  2( 0) du=4.559e-03 cont=3.279e-02 dnorm=8.241e-05 0
    it=  3( 0) du=6.709e-06 cont=1.472e-03 dnorm=1.064e-07 0
    it=  4( 0) du=1.516e-11 cont=2.259e-06 dnorm=1.788e-13 1
    Newton iteration successful
  0.141118 seconds (1.47 M allocations: 45.241 MiB, 3.74% gc time)
    Start Newton iteration
    it=  1( 0) du=3.606e-02 cont=3.606e-02 dnorm=8.231e-03 0
    it=  2( 0) du=2.601e-04 cont=7.213e-03 dnorm=4.564e-06 0
    it=  3( 0) du=2.217e-08 cont=8.523e-05 dnorm=3.117e-10 0
    Newton iteration successful
  0.108278 seconds (1.10 M allocations: 33.945 MiB, 4.83% gc time)
    Start Newton iteration
    it=  1( 0) du=4.755e-02 cont=4.755e-02 dnorm=7.579e-03 0
    it=  2( 0) du=1.300e-04 cont=2.733e-03 dnorm=2.573e-06 0
    it=  3( 0) du=5.948e-09 cont=4.577e-05 dnorm=9.498e-11 1
    Newton iteration successful
  0.105390 seconds (1.10 M allocations: 33.934 MiB, 2.53% gc time)
    Start Newton iteration
    it=  1( 0) du=5.242e-02 cont=5.242e-02 dnorm=6.663e-03 0
    it=  2( 0) du=1.067e-04 cont=2.036e-03 dnorm=2.181e-06 0
    it=  3( 0) du=3.785e-09 cont=3.546e-05 dnorm=6.587e-11 1
    Newton iteration successful
  0.106082 seconds (1.10 M allocations: 33.934 MiB, 5.00% gc time)
    Start Newton iteration
    it=  1( 0) du=4.906e-02 cont=4.906e-02 dnorm=5.696e-03 0
    it=  2( 0) du=1.605e-04 cont=3.271e-03 dnorm=3.732e-06 0
    it=  3( 0) du=8.606e-09 cont=5.363e-05 dnorm=1.864e-10 0
    Newton iteration successful
  0.105148 seconds (1.10 M allocations: 33.945 MiB, 2.54% gc time)
    Start Newton iteration
    it=  1( 0) du=4.315e-02 cont=4.315e-02 dnorm=4.769e-03 0
    it=  2( 0) du=1.483e-04 cont=3.437e-03 dnorm=3.642e-06 0
    it=  3( 0) du=7.513e-09 cont=5.066e-05 dnorm=1.719e-10 0
    Newton iteration successful
  0.106141 seconds (1.10 M allocations: 33.936 MiB, 4.91% gc time)
    Start Newton iteration
    it=  1( 0) du=3.684e-02 cont=3.684e-02 dnorm=3.924e-03 0
    it=  2( 0) du=1.178e-04 cont=3.198e-03 dnorm=3.000e-06 0
    it=  3( 0) du=4.953e-09 cont=4.204e-05 dnorm=1.130e-10 0
    Newton iteration successful
  0.103393 seconds (1.10 M allocations: 33.934 MiB, 2.47% gc time)
    Start Newton iteration
    it=  1( 0) du=3.091e-02 cont=3.091e-02 dnorm=3.176e-03 0
    it=  2( 0) du=8.667e-05 cont=2.804e-03 dnorm=2.270e-06 0
    it=  3( 0) du=3.020e-09 cont=3.485e-05 dnorm=6.268e-11 1
    Newton iteration successful
  0.107310 seconds (1.10 M allocations: 33.943 MiB, 4.90% gc time)
    Start Newton iteration
    it=  1( 0) du=2.571e-02 cont=2.571e-02 dnorm=2.530e-03 0
    it=  2( 0) du=6.086e-05 cont=2.367e-03 dnorm=1.623e-06 0
    it=  3( 0) du=1.759e-09 cont=2.890e-05 dnorm=3.105e-11 1
    Newton iteration successful
  0.111426 seconds (1.10 M allocations: 33.936 MiB, 4.83% gc time)
    Start Newton iteration
    it=  1( 0) du=2.154e-02 cont=2.154e-02 dnorm=1.984e-03 0
    it=  2( 0) du=4.476e-05 cont=2.077e-03 dnorm=1.108e-06 0
    it=  3( 0) du=1.001e-09 cont=2.236e-05 dnorm=1.406e-11 1
    Newton iteration successful
  0.103380 seconds (1.10 M allocations: 33.935 MiB, 2.65% gc time)
    Start Newton iteration
    it=  1( 0) du=1.822e-02 cont=1.822e-02 dnorm=1.532e-03 0
    it=  2( 0) du=3.347e-05 cont=1.837e-03 dnorm=7.256e-07 0
    it=  3( 0) du=5.660e-10 cont=1.691e-05 dnorm=5.887e-12 1
    Newton iteration successful
  0.106592 seconds (1.10 M allocations: 33.943 MiB, 4.98% gc time)
    Start Newton iteration
    it=  1( 0) du=1.573e-02 cont=1.573e-02 dnorm=1.163e-03 0
    it=  2( 0) du=2.563e-05 cont=1.630e-03 dnorm=4.576e-07 0
    it=  3( 0) du=3.213e-10 cont=1.253e-05 dnorm=2.308e-12 1
    Newton iteration successful
  0.102957 seconds (1.10 M allocations: 33.934 MiB, 2.67% gc time)
    Start Newton iteration
    it=  1( 0) du=1.403e-02 cont=1.403e-02 dnorm=8.696e-04 0
    it=  2( 0) du=2.026e-05 cont=1.444e-03 dnorm=2.786e-07 0
    it=  3( 0) du=1.840e-10 cont=9.080e-06 dnorm=8.633e-13 1
    Newton iteration successful
  0.106847 seconds (1.10 M allocations: 33.936 MiB, 5.16% gc time)
    Start Newton iteration
    it=  1( 0) du=1.310e-02 cont=1.310e-02 dnorm=6.391e-04 0
    it=  2( 0) du=1.654e-05 cont=1.263e-03 dnorm=1.644e-07 0
    it=  3( 0) du=1.069e-10 cont=6.461e-06 dnorm=3.194e-13 1
    Newton iteration successful
  0.105093 seconds (1.10 M allocations: 33.945 MiB, 2.72% gc time)
    Start Newton iteration
    it=  1( 0) du=1.289e-02 cont=1.289e-02 dnorm=4.614e-04 0
    it=  2( 0) du=1.385e-05 cont=1.075e-03 dnorm=9.464e-08 0
    it=  3( 0) du=6.344e-11 cont=4.579e-06 dnorm=1.251e-13 1
    Newton iteration successful
  0.111472 seconds (1.10 M allocations: 33.933 MiB, 5.04% gc time)
    Start Newton iteration
    it=  1( 0) du=1.340e-02 cont=1.340e-02 dnorm=3.267e-04 0
    it=  2( 0) du=1.177e-05 cont=8.785e-04 dnorm=5.384e-08 0
    it=  3( 0) du=3.882e-11 cont=3.298e-06 dnorm=5.627e-14 1
    Newton iteration successful
  0.105122 seconds (1.10 M allocations: 33.943 MiB, 2.66% gc time)
    Start Newton iteration
    it=  1( 0) du=1.465e-02 cont=1.465e-02 dnorm=2.263e-04 0
    it=  2( 0) du=9.979e-06 cont=6.813e-04 dnorm=3.104e-08 0
    it=  3( 0) du=2.452e-11 cont=2.457e-06 dnorm=3.043e-14 1
    Newton iteration successful
  0.107384 seconds (1.10 M allocations: 33.936 MiB, 5.12% gc time)
    Start Newton iteration
    it=  1( 0) du=1.669e-02 cont=1.669e-02 dnorm=1.524e-04 0
    it=  2( 0) du=8.318e-06 cont=4.983e-04 dnorm=1.905e-08 0
    it=  3( 0) du=1.655e-11 cont=1.990e-06 dnorm=1.848e-14 1
    Newton iteration successful
  0.104524 seconds (1.10 M allocations: 33.935 MiB, 2.66% gc time)
    Start Newton iteration
    it=  1( 0) du=1.964e-02 cont=1.964e-02 dnorm=9.871e-05 0
    it=  2( 0) du=8.344e-06 cont=4.248e-04 dnorm=1.345e-08 0
    it=  3( 0) du=1.526e-11 cont=1.829e-06 dnorm=1.174e-14 1
    Newton iteration successful
  0.107608 seconds (1.10 M allocations: 33.943 MiB, 4.98% gc time)
    Start Newton iteration
    it=  1( 0) du=2.368e-02 cont=2.368e-02 dnorm=5.994e-05 0
    it=  2( 0) du=8.920e-06 cont=3.767e-04 dnorm=1.176e-08 0
    it=  3( 0) du=1.868e-11 cont=2.094e-06 dnorm=6.522e-15 1
    Newton iteration successful
  0.105255 seconds (1.10 M allocations: 33.934 MiB, 2.61% gc time)
    Start Newton iteration
    it=  1( 0) du=2.908e-02 cont=2.908e-02 dnorm=3.181e-05 0
    it=  2( 0) du=1.645e-05 cont=5.656e-04 dnorm=1.287e-08 0
    it=  3( 0) du=4.131e-11 cont=2.512e-06 dnorm=1.087e-15 1
    Newton iteration successful
  0.121643 seconds (1.10 M allocations: 33.936 MiB, 5.35% gc time)
    Start Newton iteration
    it=  1( 0) du=3.623e-02 cont=3.623e-02 dnorm=1.096e-05 0
    it=  2( 0) du=3.158e-05 cont=8.718e-04 dnorm=1.662e-08 0
    it=  3( 0) du=1.612e-10 cont=5.103e-06 dnorm=5.652e-15 1
    Newton iteration successful
  0.105146 seconds (1.10 M allocations: 33.945 MiB, 2.56% gc time)
    Start Newton iteration
    it=  1( 0) du=4.567e-02 cont=4.567e-02 dnorm=5.320e-06 0
    it=  2( 0) du=5.780e-05 cont=1.266e-03 dnorm=2.370e-08 0
    it=  3( 0) du=5.562e-10 cont=9.622e-06 dnorm=1.000e-14 1
    Newton iteration successful
  0.109124 seconds (1.10 M allocations: 33.934 MiB, 5.08% gc time)
    Start Newton iteration
    it=  1( 0) du=5.817e-02 cont=5.817e-02 dnorm=1.920e-05 0
    it=  2( 0) du=1.049e-04 cont=1.803e-03 dnorm=3.562e-08 0
    it=  3( 0) du=1.874e-09 cont=1.787e-05 dnorm=2.370e-14 1
    Newton iteration successful
  0.105592 seconds (1.10 M allocations: 33.942 MiB, 2.53% gc time)
    Start Newton iteration
    it=  1( 0) du=7.473e-02 cont=7.473e-02 dnorm=3.254e-05 0
    it=  2( 0) du=2.054e-04 cont=2.748e-03 dnorm=5.473e-08 0
    it=  3( 0) du=6.429e-09 cont=3.130e-05 dnorm=3.117e-13 1
    Newton iteration successful
  0.109703 seconds (1.10 M allocations: 33.934 MiB, 4.95% gc time)
    Start Newton iteration
    it=  1( 0) du=9.659e-02 cont=9.659e-02 dnorm=4.702e-05 0
    it=  2( 0) du=4.742e-04 cont=4.909e-03 dnorm=8.328e-08 0
    it=  3( 0) du=2.281e-08 cont=4.810e-05 dnorm=2.101e-12 1
    Newton iteration successful
  0.106475 seconds (1.10 M allocations: 33.945 MiB, 2.50% gc time)
    Start Newton iteration
    it=  1( 0) du=1.251e-01 cont=1.251e-01 dnorm=6.437e-05 0
    it=  2( 0) du=1.091e-03 cont=8.717e-03 dnorm=1.183e-07 0
    it=  3( 0) du=9.310e-08 cont=8.537e-05 dnorm=1.261e-11 1
    Newton iteration successful
  0.110895 seconds (1.10 M allocations: 33.933 MiB, 4.78% gc time)
    Start Newton iteration
    it=  1( 0) du=1.612e-01 cont=1.612e-01 dnorm=8.647e-05 0
    it=  2( 0) du=2.466e-03 cont=1.530e-02 dnorm=1.303e-07 0
    it=  3( 0) du=4.306e-07 cont=1.746e-04 dnorm=7.351e-11 1
    Newton iteration successful
  0.106870 seconds (1.10 M allocations: 33.944 MiB, 5.23% gc time)
    Start Newton iteration
    it=  1( 0) du=2.040e-01 cont=2.040e-01 dnorm=1.154e-04 0
    it=  2( 0) du=5.318e-03 cont=2.607e-02 dnorm=1.049e-08 0
    it=  3( 0) du=1.905e-06 cont=3.583e-04 dnorm=4.070e-10 0
    Newton iteration successful
  0.102799 seconds (1.10 M allocations: 33.935 MiB, 2.72% gc time)
    Start Newton iteration
    it=  1( 0) du=2.482e-01 cont=2.482e-01 dnorm=1.531e-04 0
    it=  2( 0) du=1.039e-02 cont=4.184e-02 dnorm=7.427e-07 0
    it=  3( 0) du=7.567e-06 cont=7.286e-04 dnorm=1.924e-09 0
    it=  4( 0) du=1.090e-11 cont=1.441e-06 dnorm=1.304e-15 1
    Newton iteration successful
  0.141264 seconds (1.47 M allocations: 45.251 MiB, 4.10% gc time)
    Start Newton iteration
    it=  1( 0) du=2.820e-01 cont=2.820e-01 dnorm=1.993e-04 0
    it=  2( 0) du=1.716e-02 cont=6.083e-02 dnorm=3.187e-06 0
    it=  3( 0) du=2.822e-05 cont=1.645e-03 dnorm=6.543e-09 0
    it=  4( 0) du=1.255e-10 cont=4.448e-06 dnorm=1.933e-14 1
    Newton iteration successful
  0.141780 seconds (1.47 M allocations: 45.241 MiB, 3.92% gc time)
    Start Newton iteration
    it=  1( 0) du=3.064e-01 cont=3.064e-01 dnorm=2.475e-04 0
    it=  2( 0) du=2.245e-02 cont=7.328e-02 dnorm=9.179e-06 0
    it=  3( 0) du=6.682e-05 cont=2.976e-03 dnorm=1.305e-08 0
    it=  4( 0) du=7.942e-10 cont=1.189e-05 dnorm=1.086e-13 1
    Newton iteration successful
  0.143941 seconds (1.47 M allocations: 45.250 MiB, 3.84% gc time)
    Start Newton iteration
    it=  1( 0) du=3.673e-01 cont=3.673e-01 dnorm=2.793e-04 0
    it=  2( 0) du=2.259e-02 cont=6.150e-02 dnorm=1.969e-05 0
    it=  3( 0) du=9.971e-05 cont=4.414e-03 dnorm=9.549e-09 0
    it=  4( 0) du=2.340e-09 cont=2.347e-05 dnorm=1.655e-13 1
    Newton iteration successful
  0.138794 seconds (1.47 M allocations: 45.251 MiB, 3.96% gc time)
    Start Newton iteration
    it=  1( 0) du=4.260e-01 cont=4.260e-01 dnorm=2.669e-04 0
    it=  2( 0) du=2.753e-02 cont=6.463e-02 dnorm=3.184e-05 0
    it=  3( 0) du=1.597e-04 cont=5.800e-03 dnorm=1.647e-08 0
    it=  4( 0) du=5.167e-09 cont=3.235e-05 dnorm=4.785e-13 1
    Newton iteration successful
  0.141611 seconds (1.47 M allocations: 45.254 MiB, 3.98% gc time)
    Start Newton iteration
    it=  1( 0) du=4.331e-01 cont=4.331e-01 dnorm=1.912e-04 0
    it=  2( 0) du=3.248e-02 cont=7.500e-02 dnorm=3.898e-05 0
    it=  3( 0) du=2.465e-04 cont=7.589e-03 dnorm=4.368e-08 0
    it=  4( 0) du=1.317e-08 cont=5.342e-05 dnorm=1.589e-12 1
    Newton iteration successful
  0.142496 seconds (1.47 M allocations: 45.249 MiB, 3.95% gc time)
    Start Newton iteration
    it=  1( 0) du=3.920e-01 cont=3.920e-01 dnorm=6.672e-05 0
    it=  2( 0) du=2.764e-02 cont=7.052e-02 dnorm=3.702e-05 0
    it=  3( 0) du=1.918e-04 cont=6.938e-03 dnorm=3.849e-08 0
    it=  4( 0) du=8.430e-09 cont=4.396e-05 dnorm=1.039e-12 1
    Newton iteration successful
  0.140980 seconds (1.47 M allocations: 45.250 MiB, 3.89% gc time)
    Start Newton iteration
    it=  1( 0) du=3.836e-01 cont=3.836e-01 dnorm=6.204e-05 0
    it=  2( 0) du=2.564e-02 cont=6.684e-02 dnorm=2.854e-05 0
    it=  3( 0) du=1.416e-04 cont=5.523e-03 dnorm=1.566e-08 0
    it=  4( 0) du=4.415e-09 cont=3.118e-05 dnorm=2.659e-13 1
    Newton iteration successful
  0.145388 seconds (1.47 M allocations: 45.249 MiB, 4.03% gc time)
    Start Newton iteration
    it=  1( 0) du=3.253e-01 cont=3.253e-01 dnorm=1.525e-04 0
    it=  2( 0) du=2.022e-02 cont=6.216e-02 dnorm=1.887e-05 0
    it=  3( 0) du=9.485e-05 cont=4.690e-03 dnorm=1.157e-09 0
    it=  4( 0) du=2.014e-09 cont=2.123e-05 dnorm=5.449e-14 1
    Newton iteration successful
  0.139105 seconds (1.47 M allocations: 45.249 MiB, 4.00% gc time)
    Start Newton iteration
    it=  1( 0) du=2.880e-01 cont=2.880e-01 dnorm=1.889e-04 0
    it=  2( 0) du=1.595e-02 cont=5.536e-02 dnorm=1.146e-05 0
    it=  3( 0) du=4.347e-05 cont=2.726e-03 dnorm=6.022e-09 0
    it=  4( 0) du=4.086e-10 cont=9.399e-06 dnorm=1.005e-13 1
    Newton iteration successful
  0.140392 seconds (1.47 M allocations: 45.249 MiB, 4.07% gc time)
    Start Newton iteration
    it=  1( 0) du=2.699e-01 cont=2.699e-01 dnorm=1.816e-04 0
    it=  2( 0) du=1.643e-02 cont=6.089e-02 dnorm=6.885e-06 0
    it=  3( 0) du=4.275e-05 cont=2.601e-03 dnorm=3.887e-09 0
    it=  4( 0) du=3.319e-10 cont=7.765e-06 dnorm=4.213e-14 1
    Newton iteration successful
  0.140722 seconds (1.47 M allocations: 45.249 MiB, 4.07% gc time)
    Start Newton iteration
    it=  1( 0) du=2.364e-01 cont=2.364e-01 dnorm=1.509e-04 0
    it=  2( 0) du=1.343e-02 cont=5.680e-02 dnorm=4.231e-06 0
    it=  3( 0) du=3.257e-05 cont=2.426e-03 dnorm=1.614e-09 0
    it=  4( 0) du=1.988e-10 cont=6.102e-06 dnorm=9.123e-15 1
    Newton iteration successful
  0.141721 seconds (1.47 M allocations: 45.249 MiB, 4.01% gc time)
    Start Newton iteration
    it=  1( 0) du=1.956e-01 cont=1.956e-01 dnorm=1.139e-04 0
    it=  2( 0) du=9.042e-03 cont=4.623e-02 dnorm=2.656e-06 0
    it=  3( 0) du=1.647e-05 cont=1.822e-03 dnorm=8.190e-10 0
    it=  4( 0) du=5.459e-11 cont=3.314e-06 dnorm=3.476e-15 1
    Newton iteration successful
  0.143508 seconds (1.47 M allocations: 45.249 MiB, 4.02% gc time)
    Start Newton iteration
    it=  1( 0) du=1.688e-01 cont=1.688e-01 dnorm=8.013e-05 0
    it=  2( 0) du=5.716e-03 cont=3.387e-02 dnorm=1.739e-06 0
    it=  3( 0) du=7.429e-06 cont=1.300e-03 dnorm=5.147e-10 0
    it=  4( 0) du=1.260e-11 cont=1.696e-06 dnorm=1.955e-15 1
    Newton iteration successful
  0.143355 seconds (1.47 M allocations: 45.260 MiB, 3.91% gc time)
    Start Newton iteration
    it=  1( 0) du=1.438e-01 cont=1.438e-01 dnorm=5.308e-05 0
    it=  2( 0) du=4.698e-03 cont=3.267e-02 dnorm=1.268e-06 0
    it=  3( 0) du=3.571e-06 cont=7.601e-04 dnorm=2.569e-10 0
    it=  4( 0) du=3.166e-12 cont=8.865e-07 dnorm=4.345e-16 1
    Newton iteration successful
  0.141702 seconds (1.47 M allocations: 45.249 MiB, 4.05% gc time)
    Start Newton iteration
    it=  1( 0) du=1.488e-01 cont=1.488e-01 dnorm=3.302e-05 0
    it=  2( 0) du=4.560e-03 cont=3.064e-02 dnorm=1.098e-06 0
    it=  3( 0) du=3.611e-06 cont=7.919e-04 dnorm=5.413e-11 1
    it=  4( 0) du=2.184e-12 cont=6.048e-07 dnorm=4.346e-16 2
    Newton iteration successful
  0.141517 seconds (1.47 M allocations: 45.250 MiB, 3.99% gc time)
    Start Newton iteration
    it=  1( 0) du=1.665e-01 cont=1.665e-01 dnorm=1.896e-05 0
    it=  2( 0) du=4.347e-03 cont=2.611e-02 dnorm=1.125e-06 0
    it=  3( 0) du=3.781e-06 cont=8.698e-04 dnorm=7.273e-11 1
    it=  4( 0) du=2.721e-12 cont=7.197e-07 dnorm=4.346e-16 2
    Newton iteration successful
  0.142728 seconds (1.47 M allocations: 45.249 MiB, 3.98% gc time)
    Start Newton iteration
    it=  1( 0) du=1.861e-01 cont=1.861e-01 dnorm=9.754e-06 0
    it=  2( 0) du=4.881e-03 cont=2.622e-02 dnorm=1.272e-06 0
    it=  3( 0) du=4.581e-06 cont=9.386e-04 dnorm=1.624e-10 0
    it=  4( 0) du=5.013e-12 cont=1.094e-06 dnorm=0.000e+00 1
    Newton iteration successful
  0.141806 seconds (1.47 M allocations: 45.249 MiB, 4.12% gc time)
    Start Newton iteration
    it=  1( 0) du=1.979e-01 cont=1.979e-01 dnorm=4.584e-06 0
    it=  2( 0) du=6.218e-03 cont=3.143e-02 dnorm=1.454e-06 0
    it=  3( 0) du=8.226e-06 cont=1.323e-03 dnorm=2.805e-10 0
    it=  4( 0) du=1.581e-11 cont=1.922e-06 dnorm=4.346e-16 1
    Newton iteration successful
  0.142063 seconds (1.47 M allocations: 45.249 MiB, 4.03% gc time)
    Start Newton iteration
    it=  1( 0) du=1.988e-01 cont=1.988e-01 dnorm=2.883e-06 0
    it=  2( 0) du=7.092e-03 cont=3.567e-02 dnorm=1.563e-06 0
    it=  3( 0) du=1.178e-05 cont=1.661e-03 dnorm=4.128e-10 0
    it=  4( 0) du=3.319e-11 cont=2.817e-06 dnorm=8.692e-16 1
    Newton iteration successful
  0.142656 seconds (1.47 M allocations: 45.249 MiB, 4.05% gc time)
    Start Newton iteration
    it=  1( 0) du=2.217e-01 cont=2.217e-01 dnorm=3.979e-06 0
    it=  2( 0) du=7.244e-03 cont=3.268e-02 dnorm=1.497e-06 0
    it=  3( 0) du=1.288e-05 cont=1.778e-03 dnorm=3.520e-10 0
    it=  4( 0) du=4.108e-11 cont=3.189e-06 dnorm=8.692e-16 1
    Newton iteration successful
  0.141843 seconds (1.47 M allocations: 45.265 MiB, 4.01% gc time)
    Start Newton iteration
    it=  1( 0) du=2.456e-01 cont=2.456e-01 dnorm=6.741e-06 0
    it=  2( 0) du=6.505e-03 cont=2.648e-02 dnorm=1.224e-06 0
    it=  3( 0) du=1.083e-05 cont=1.665e-03 dnorm=1.840e-10 0
    it=  4( 0) du=3.005e-11 cont=2.775e-06 dnorm=8.692e-16 1
    Newton iteration successful
  0.146380 seconds (1.47 M allocations: 45.254 MiB, 4.01% gc time)
    Start Newton iteration
    it=  1( 0) du=2.718e-01 cont=2.718e-01 dnorm=9.549e-06 0
    it=  2( 0) du=9.540e-03 cont=3.510e-02 dnorm=8.170e-07 0
    it=  3( 0) du=1.588e-05 cont=1.665e-03 dnorm=1.311e-09 0
    it=  4( 0) du=3.387e-11 cont=2.133e-06 dnorm=3.911e-15 1
    Newton iteration successful
  0.140540 seconds (1.47 M allocations: 45.252 MiB, 4.10% gc time)
    Start Newton iteration
    it=  1( 0) du=2.871e-01 cont=2.871e-01 dnorm=1.064e-05 0
    it=  2( 0) du=1.239e-02 cont=4.316e-02 dnorm=4.158e-07 0
    it=  3( 0) du=2.944e-05 cont=2.376e-03 dnorm=2.807e-09 0
    it=  4( 0) du=1.354e-10 cont=4.599e-06 dnorm=1.239e-14 1
    Newton iteration successful
  0.142423 seconds (1.47 M allocations: 45.921 MiB, 4.15% gc time)
    Start Newton iteration
    it=  1( 0) du=2.864e-01 cont=2.864e-01 dnorm=8.608e-06 0
    it=  2( 0) du=1.331e-02 cont=4.645e-02 dnorm=1.569e-07 0
    it=  3( 0) du=3.876e-05 cont=2.913e-03 dnorm=4.110e-09 0
    it=  4( 0) du=2.753e-10 cont=7.101e-06 dnorm=2.369e-14 1
    Newton iteration successful
  0.144742 seconds (1.47 M allocations: 45.611 MiB, 4.09% gc time)
    Start Newton iteration
    it=  1( 0) du=2.699e-01 cont=2.699e-01 dnorm=2.810e-06 0
    it=  2( 0) du=1.220e-02 cont=4.519e-02 dnorm=1.117e-07 0
    it=  3( 0) du=3.746e-05 cont=3.071e-03 dnorm=4.865e-09 0
    it=  4( 0) du=3.004e-10 cont=8.019e-06 dnorm=3.151e-14 1
    Newton iteration successful
  0.143204 seconds (1.47 M allocations: 45.249 MiB, 4.11% gc time)
    Start Newton iteration
    it=  1( 0) du=2.538e-01 cont=2.538e-01 dnorm=6.531e-06 0
    it=  2( 0) du=1.168e-02 cont=4.601e-02 dnorm=2.922e-07 0
    it=  3( 0) du=3.125e-05 cont=2.676e-03 dnorm=5.315e-09 0
    it=  4( 0) du=2.103e-10 cont=6.730e-06 dnorm=3.520e-14 1
    Newton iteration successful
  0.148339 seconds (1.47 M allocations: 45.249 MiB, 4.05% gc time)
    Start Newton iteration
    it=  1( 0) du=2.486e-01 cont=2.486e-01 dnorm=1.846e-05 0
    it=  2( 0) du=1.435e-02 cont=5.773e-02 dnorm=6.871e-07 0
    it=  3( 0) du=4.068e-05 cont=2.834e-03 dnorm=5.663e-09 0
    it=  4( 0) du=2.680e-10 cont=6.589e-06 dnorm=4.129e-14 1
    Newton iteration successful
  0.143977 seconds (1.47 M allocations: 45.268 MiB, 4.06% gc time)
    Start Newton iteration
    it=  1( 0) du=2.912e-01 cont=2.912e-01 dnorm=3.159e-05 0
    it=  2( 0) du=1.693e-02 cont=5.814e-02 dnorm=1.287e-06 0
    it=  3( 0) du=4.644e-05 cont=2.743e-03 dnorm=5.022e-09 0
    it=  4( 0) du=2.506e-10 cont=5.396e-06 dnorm=3.325e-14 1
    Newton iteration successful
  0.143539 seconds (1.47 M allocations: 45.270 MiB, 4.14% gc time)
    Start Newton iteration
    it=  1( 0) du=3.627e-01 cont=3.627e-01 dnorm=4.459e-05 0
    it=  2( 0) du=1.606e-02 cont=4.428e-02 dnorm=2.257e-06 0
    it=  3( 0) du=3.212e-05 cont=2.000e-03 dnorm=1.643e-09 0
    it=  4( 0) du=1.332e-10 cont=4.147e-06 dnorm=2.173e-16 1
    Newton iteration successful
  0.143456 seconds (1.47 M allocations: 45.249 MiB, 4.11% gc time)
    Start Newton iteration
    it=  1( 0) du=4.500e-01 cont=4.500e-01 dnorm=5.599e-05 0
    it=  2( 0) du=1.469e-02 cont=3.265e-02 dnorm=4.345e-06 0
    it=  3( 0) du=4.961e-05 cont=3.376e-03 dnorm=2.414e-09 0
    it=  4( 0) du=5.324e-10 cont=1.073e-05 dnorm=3.042e-15 1
    Newton iteration successful
  0.149032 seconds (1.47 M allocations: 46.385 MiB, 5.86% gc time)
    Start Newton iteration
    it=  1( 0) du=5.133e-01 cont=5.133e-01 dnorm=6.174e-05 0
    it=  2( 0) du=2.566e-02 cont=5.000e-02 dnorm=7.612e-06 0
    it=  3( 0) du=1.910e-04 cont=7.442e-03 dnorm=2.774e-08 0
    it=  4( 0) du=1.034e-08 cont=5.416e-05 dnorm=1.337e-12 1
    Newton iteration successful
  0.145118 seconds (1.47 M allocations: 45.250 MiB, 5.94% gc time)
    Start Newton iteration
    it=  1( 0) du=4.878e-01 cont=4.878e-01 dnorm=5.613e-05 0
    it=  2( 0) du=4.225e-02 cont=8.661e-02 dnorm=8.799e-06 0
    it=  3( 0) du=5.104e-04 cont=1.208e-02 dnorm=6.548e-08 0
    it=  4( 0) du=7.428e-08 cont=1.455e-04 dnorm=9.636e-12 1
    Newton iteration successful
  0.147446 seconds (1.47 M allocations: 45.250 MiB, 5.85% gc time)
    Start Newton iteration
    it=  1( 0) du=3.804e-01 cont=3.804e-01 dnorm=4.277e-05 0
    it=  2( 0) du=3.354e-02 cont=8.815e-02 dnorm=6.795e-06 0
    it=  3( 0) du=3.285e-04 cont=9.795e-03 dnorm=3.898e-08 0
    it=  4( 0) du=3.148e-08 cont=9.585e-05 dnorm=3.827e-12 1
    Newton iteration successful
  0.145977 seconds (1.47 M allocations: 45.370 MiB, 5.86% gc time)
    Start Newton iteration
    it=  1( 0) du=2.774e-01 cont=2.774e-01 dnorm=3.193e-05 0
    it=  2( 0) du=1.870e-02 cont=6.740e-02 dnorm=4.610e-06 0
    it=  3( 0) du=1.032e-04 cont=5.517e-03 dnorm=1.315e-08 0
    it=  4( 0) du=3.136e-09 cont=3.040e-05 dnorm=2.242e-13 1
    Newton iteration successful
  0.146072 seconds (1.47 M allocations: 45.281 MiB, 5.83% gc time)
    Start Newton iteration
    it=  1( 0) du=2.989e-01 cont=2.989e-01 dnorm=2.710e-05 0
    it=  2( 0) du=1.659e-02 cont=5.551e-02 dnorm=4.107e-06 0
    it=  3( 0) du=6.962e-05 cont=4.196e-03 dnorm=9.523e-09 0
    it=  4( 0) du=1.063e-09 cont=1.526e-05 dnorm=6.734e-14 1
    Newton iteration successful
  0.162546 seconds (1.47 M allocations: 45.250 MiB, 5.55% gc time)
    Start Newton iteration
    it=  1( 0) du=3.233e-01 cont=3.233e-01 dnorm=2.298e-05 0
    it=  2( 0) du=2.594e-02 cont=8.023e-02 dnorm=5.196e-06 0
    it=  3( 0) du=2.004e-04 cont=7.726e-03 dnorm=1.230e-08 0
    it=  4( 0) du=1.086e-08 cont=5.418e-05 dnorm=6.082e-15 1
    Newton iteration successful
  0.142590 seconds (1.47 M allocations: 45.231 MiB, 4.25% gc time)
    Start Newton iteration
    it=  1( 0) du=3.418e-01 cont=3.418e-01 dnorm=1.093e-05 0
    it=  2( 0) du=2.802e-02 cont=8.198e-02 dnorm=6.193e-06 0
    it=  3( 0) du=2.681e-04 cont=9.569e-03 dnorm=1.558e-08 0
    it=  4( 0) du=2.300e-08 cont=8.580e-05 dnorm=2.867e-14 1
    Newton iteration successful
  0.142335 seconds (1.47 M allocations: 45.201 MiB, 4.17% gc time)
    Start Newton iteration
    it=  1( 0) du=3.141e-01 cont=3.141e-01 dnorm=1.208e-05 0
    it=  2( 0) du=2.239e-02 cont=7.127e-02 dnorm=5.100e-06 0
    it=  3( 0) du=1.894e-04 cont=8.461e-03 dnorm=1.076e-08 0
    it=  4( 0) du=1.397e-08 cont=7.373e-05 dnorm=3.628e-13 1
    Newton iteration successful
  0.141940 seconds (1.47 M allocations: 45.249 MiB, 4.16% gc time)
    Start Newton iteration
    it=  1( 0) du=2.975e-01 cont=2.975e-01 dnorm=3.868e-05 0
    it=  2( 0) du=1.698e-02 cont=5.709e-02 dnorm=1.690e-06 0
    it=  3( 0) du=1.077e-04 cont=6.340e-03 dnorm=7.914e-09 0
    it=  4( 0) du=4.431e-09 cont=4.115e-05 dnorm=1.086e-14 1
    Newton iteration successful
  0.146450 seconds (1.47 M allocations: 45.249 MiB, 4.25% gc time)
    Start Newton iteration
    it=  1( 0) du=3.124e-01 cont=3.124e-01 dnorm=5.589e-05 0
    it=  2( 0) du=1.733e-02 cont=5.546e-02 dnorm=2.605e-06 0
    it=  3( 0) du=1.102e-04 cont=6.359e-03 dnorm=3.912e-08 0
    it=  4( 0) du=4.299e-09 cont=3.901e-05 dnorm=1.266e-12 1
    Newton iteration successful
  0.144249 seconds (1.47 M allocations: 45.249 MiB, 4.13% gc time)
    Start Newton iteration
    it=  1( 0) du=2.973e-01 cont=2.973e-01 dnorm=5.335e-05 0
    it=  2( 0) du=3.137e-02 cont=1.055e-01 dnorm=6.577e-06 0
    it=  3( 0) du=3.000e-04 cont=9.562e-03 dnorm=7.663e-08 0
    it=  4( 0) du=2.543e-08 cont=8.477e-05 dnorm=4.768e-12 1
    Newton iteration successful
  0.144308 seconds (1.47 M allocations: 45.249 MiB, 4.15% gc time)
    Start Newton iteration
    it=  1( 0) du=4.429e-01 cont=4.429e-01 dnorm=2.594e-05 0
    it=  2( 0) du=5.533e-02 cont=1.249e-01 dnorm=1.093e-05 0
    it=  3( 0) du=6.873e-04 cont=1.242e-02 dnorm=1.348e-07 0
    it=  4( 0) du=7.814e-08 cont=1.137e-04 dnorm=1.452e-11 1
    Newton iteration successful
  0.141646 seconds (1.47 M allocations: 45.249 MiB, 4.13% gc time)
    Start Newton iteration
    it=  1( 0) du=7.288e-01 cont=7.288e-01 dnorm=3.305e-05 0
    it=  2( 0) du=2.924e-02 cont=4.012e-02 dnorm=9.009e-06 0
    it=  3( 0) du=1.535e-04 cont=5.249e-03 dnorm=8.085e-08 0
    it=  4( 0) du=5.323e-09 cont=3.468e-05 dnorm=2.888e-12 1
    Newton iteration successful
  0.143547 seconds (1.47 M allocations: 45.271 MiB, 4.18% gc time)
    Start Newton iteration
    it=  1( 0) du=9.358e-01 cont=9.358e-01 dnorm=1.034e-04 0
    it=  2( 0) du=1.692e-01 cont=1.808e-01 dnorm=1.351e-05 0
    it=  3( 0) du=1.326e-02 cont=7.836e-02 dnorm=1.543e-06 0
    it=  4( 0) du=7.197e-05 cont=5.428e-03 dnorm=8.304e-09 0
    it=  5( 0) du=2.017e-09 cont=2.803e-05 dnorm=2.379e-13 1
    Newton iteration successful
  0.182887 seconds (1.84 M allocations: 57.690 MiB, 4.71% gc time)
    Start Newton iteration
    it=  1( 0) du=6.211e-01 cont=6.211e-01 dnorm=1.141e-04 0
    it=  2( 0) du=1.071e-01 cont=1.724e-01 dnorm=1.886e-05 0
    it=  3( 0) du=4.674e-03 cont=4.364e-02 dnorm=7.134e-07 0
    it=  4( 0) du=8.502e-06 cont=1.819e-03 dnorm=1.114e-09 0
    it=  5( 0) du=2.747e-11 cont=3.231e-06 dnorm=3.475e-15 1
    Newton iteration successful
  0.180620 seconds (1.84 M allocations: 56.678 MiB, 4.62% gc time)
    Start Newton iteration
    it=  1( 0) du=6.242e-01 cont=6.242e-01 dnorm=8.997e-05 0
    it=  2( 0) du=8.871e-02 cont=1.421e-01 dnorm=1.548e-05 0
    it=  3( 0) du=3.349e-03 cont=3.776e-02 dnorm=3.985e-07 0
    it=  4( 0) du=4.453e-06 cont=1.329e-03 dnorm=4.268e-10 0
    Newton iteration successful
  0.146112 seconds (1.47 M allocations: 45.629 MiB, 5.92% gc time)
    Start Newton iteration
    it=  1( 0) du=4.304e-01 cont=4.304e-01 dnorm=5.593e-05 0
    it=  2( 0) du=5.586e-02 cont=1.298e-01 dnorm=7.784e-06 0
    it=  3( 0) du=1.262e-03 cont=2.260e-02 dnorm=1.168e-07 0
    it=  4( 0) du=6.289e-07 cont=4.982e-04 dnorm=5.065e-11 1
    Newton iteration successful
  0.144694 seconds (1.47 M allocations: 45.593 MiB, 5.93% gc time)
    Start Newton iteration
    it=  1( 0) du=2.505e-01 cont=2.505e-01 dnorm=3.141e-05 0
    it=  2( 0) du=2.211e-02 cont=8.826e-02 dnorm=3.203e-06 0
    it=  3( 0) du=1.918e-04 cont=8.674e-03 dnorm=1.766e-08 0
    it=  4( 0) du=1.429e-08 cont=7.452e-05 dnorm=1.069e-12 1
    Newton iteration successful
  0.147510 seconds (1.47 M allocations: 45.416 MiB, 5.97% gc time)
    Start Newton iteration
    it=  1( 0) du=2.060e-01 cont=2.060e-01 dnorm=1.703e-05 0
    it=  2( 0) du=7.866e-03 cont=3.818e-02 dnorm=1.606e-06 0
    it=  3( 0) du=2.389e-05 cont=3.037e-03 dnorm=3.723e-09 0
    it=  4( 0) du=2.198e-10 cont=9.198e-06 dnorm=2.584e-14 1
    Newton iteration successful
  0.146113 seconds (1.47 M allocations: 45.250 MiB, 5.84% gc time)
    Start Newton iteration
    it=  1( 0) du=2.236e-01 cont=2.236e-01 dnorm=8.082e-06 0
    it=  2( 0) du=7.158e-03 cont=3.201e-02 dnorm=1.502e-06 0
    it=  3( 0) du=1.720e-05 cont=2.403e-03 dnorm=3.457e-09 0
    it=  4( 0) du=1.255e-10 cont=7.294e-06 dnorm=2.237e-14 1
    Newton iteration successful
  0.147073 seconds (1.47 M allocations: 45.417 MiB, 5.85% gc time)
    Start Newton iteration
    it=  1( 0) du=2.568e-01 cont=2.568e-01 dnorm=4.455e-07 0
    it=  2( 0) du=1.053e-02 cont=4.103e-02 dnorm=2.334e-06 0
    it=  3( 0) du=4.331e-05 cont=4.112e-03 dnorm=5.677e-09 0
    it=  4( 0) du=7.498e-10 cont=1.731e-05 dnorm=2.063e-14 1
    Newton iteration successful
  0.146728 seconds (1.47 M allocations: 45.612 MiB, 6.03% gc time)
    Start Newton iteration
    it=  1( 0) du=2.874e-01 cont=2.874e-01 dnorm=9.931e-06 0
    it=  2( 0) du=1.831e-02 cont=6.373e-02 dnorm=3.650e-06 0
    it=  3( 0) du=1.476e-04 cont=8.058e-03 dnorm=6.968e-09 0
    it=  4( 0) du=9.014e-09 cont=6.108e-05 dnorm=8.101e-14 1
    Newton iteration successful
  0.146785 seconds (1.47 M allocations: 45.250 MiB, 5.89% gc time)
    Start Newton iteration
    it=  1( 0) du=3.402e-01 cont=3.402e-01 dnorm=2.671e-05 0
    it=  2( 0) du=2.476e-02 cont=7.276e-02 dnorm=3.612e-06 0
    it=  3( 0) du=3.132e-04 cont=1.265e-02 dnorm=7.372e-09 0
    it=  4( 0) du=4.385e-08 cont=1.400e-04 dnorm=9.539e-13 1
    Newton iteration successful
  0.146328 seconds (1.47 M allocations: 45.250 MiB, 4.08% gc time)
    Start Newton iteration
    it=  1( 0) du=4.076e-01 cont=4.076e-01 dnorm=4.739e-05 0
    it=  2( 0) du=2.831e-02 cont=6.946e-02 dnorm=7.699e-07 0
    it=  3( 0) du=4.966e-04 cont=1.754e-02 dnorm=9.672e-08 0
    it=  4( 0) du=1.336e-07 cont=2.691e-04 dnorm=2.216e-11 1
    Newton iteration successful
  0.143647 seconds (1.47 M allocations: 45.250 MiB, 4.21% gc time)
    Start Newton iteration
    it=  1( 0) du=4.196e-01 cont=4.196e-01 dnorm=5.655e-05 0
    it=  2( 0) du=4.512e-02 cont=1.075e-01 dnorm=5.703e-06 0
    it=  3( 0) du=1.184e-03 cont=2.624e-02 dnorm=2.652e-07 0
    it=  4( 0) du=6.317e-07 cont=5.335e-04 dnorm=1.343e-10 0
    Newton iteration successful
  0.145317 seconds (1.47 M allocations: 45.250 MiB, 4.10% gc time)
    Start Newton iteration
    it=  1( 0) du=3.792e-01 cont=3.792e-01 dnorm=4.594e-05 0
    it=  2( 0) du=3.840e-02 cont=1.013e-01 dnorm=5.061e-06 0
    it=  3( 0) du=8.241e-04 cont=2.146e-02 dnorm=1.870e-07 0
    it=  4( 0) du=3.161e-07 cont=3.836e-04 dnorm=7.100e-11 1
    Newton iteration successful
  0.144352 seconds (1.47 M allocations: 45.251 MiB, 4.07% gc time)
    Start Newton iteration
    it=  1( 0) du=3.365e-01 cont=3.365e-01 dnorm=2.751e-05 0
    it=  2( 0) du=2.237e-02 cont=6.647e-02 dnorm=2.537e-06 0
    it=  3( 0) du=2.855e-04 cont=1.276e-02 dnorm=6.007e-08 0
    it=  4( 0) du=4.092e-08 cont=1.433e-04 dnorm=9.272e-12 1
    Newton iteration successful
  0.144064 seconds (1.47 M allocations: 45.249 MiB, 4.15% gc time)
    Start Newton iteration
    it=  1( 0) du=3.082e-01 cont=3.082e-01 dnorm=1.050e-05 0
    it=  2( 0) du=1.400e-02 cont=4.543e-02 dnorm=9.838e-07 0
    it=  3( 0) du=1.241e-04 cont=8.865e-03 dnorm=8.826e-09 0
    it=  4( 0) du=8.255e-09 cont=6.652e-05 dnorm=1.303e-14 1
    Newton iteration successful
  0.146555 seconds (1.47 M allocations: 45.261 MiB, 4.10% gc time)
    Start Newton iteration
    it=  1( 0) du=3.148e-01 cont=3.148e-01 dnorm=3.080e-06 0
    it=  2( 0) du=1.388e-02 cont=4.409e-02 dnorm=3.694e-07 0
    it=  3( 0) du=1.225e-04 cont=8.824e-03 dnorm=6.070e-09 0
    it=  4( 0) du=8.029e-09 cont=6.556e-05 dnorm=1.004e-12 1
    Newton iteration successful
  0.143699 seconds (1.47 M allocations: 45.250 MiB, 4.23% gc time)
    Start Newton iteration
    it=  1( 0) du=3.059e-01 cont=3.059e-01 dnorm=1.372e-05 0
    it=  2( 0) du=1.308e-02 cont=4.276e-02 dnorm=2.640e-07 0
    it=  3( 0) du=1.015e-04 cont=7.758e-03 dnorm=8.346e-09 0
    it=  4( 0) du=5.740e-09 cont=5.656e-05 dnorm=8.723e-13 1
    Newton iteration successful
  0.142604 seconds (1.47 M allocations: 45.250 MiB, 4.21% gc time)
    Start Newton iteration
    it=  1( 0) du=2.866e-01 cont=2.866e-01 dnorm=2.237e-05 0
    it=  2( 0) du=1.136e-02 cont=3.964e-02 dnorm=3.857e-07 0
    it=  3( 0) du=7.240e-05 cont=6.373e-03 dnorm=6.393e-09 0
    it=  4( 0) du=3.278e-09 cont=4.527e-05 dnorm=5.424e-13 1
    Newton iteration successful
  0.160378 seconds (1.47 M allocations: 45.251 MiB, 4.25% gc time)
    Start Newton iteration
    it=  1( 0) du=2.694e-01 cont=2.694e-01 dnorm=2.991e-05 0
    it=  2( 0) du=1.309e-02 cont=4.858e-02 dnorm=5.736e-07 0
    it=  3( 0) du=8.966e-05 cont=6.850e-03 dnorm=3.060e-09 0
    it=  4( 0) du=3.317e-09 cont=3.700e-05 dnorm=2.600e-13 1
    Newton iteration successful
  0.163845 seconds (1.47 M allocations: 45.249 MiB, 3.75% gc time)
    Start Newton iteration
    it=  1( 0) du=3.112e-01 cont=3.112e-01 dnorm=3.716e-05 0
    it=  2( 0) du=1.271e-02 cont=4.083e-02 dnorm=7.841e-07 0
    it=  3( 0) du=6.780e-05 cont=5.336e-03 dnorm=2.817e-09 0
    it=  4( 0) du=2.103e-09 cont=3.102e-05 dnorm=4.170e-14 1
    Newton iteration successful
  0.145005 seconds (1.47 M allocations: 45.250 MiB, 4.25% gc time)
    Start Newton iteration
    it=  1( 0) du=3.890e-01 cont=3.890e-01 dnorm=4.495e-05 0
    it=  2( 0) du=3.093e-02 cont=7.953e-02 dnorm=1.243e-06 0
    it=  3( 0) du=7.696e-04 cont=2.488e-02 dnorm=1.098e-08 0
    it=  4( 0) du=3.556e-07 cont=4.620e-04 dnorm=3.416e-13 1
    Newton iteration successful
  0.146385 seconds (1.47 M allocations: 45.754 MiB, 5.91% gc time)
    Start Newton iteration
    it=  1( 0) du=3.996e-01 cont=3.996e-01 dnorm=5.459e-05 0
    it=  2( 0) du=4.488e-02 cont=1.123e-01 dnorm=2.110e-06 0
    it=  3( 0) du=1.429e-03 cont=3.185e-02 dnorm=2.440e-08 0
    it=  4( 0) du=1.291e-06 cont=9.035e-04 dnorm=6.708e-12 1
    Newton iteration successful
  0.154390 seconds (1.47 M allocations: 45.754 MiB, 6.19% gc time)
    Start Newton iteration
    it=  1( 0) du=3.165e-01 cont=3.165e-01 dnorm=6.693e-05 0
    it=  2( 0) du=2.813e-02 cont=8.888e-02 dnorm=3.108e-06 0
    it=  3( 0) du=5.353e-04 cont=1.903e-02 dnorm=4.120e-08 0
    it=  4( 0) du=1.741e-07 cont=3.253e-04 dnorm=3.770e-12 1
    Newton iteration successful
  0.147834 seconds (1.47 M allocations: 45.919 MiB, 6.01% gc time)
    Start Newton iteration
    it=  1( 0) du=3.132e-01 cont=3.132e-01 dnorm=8.210e-05 0
    it=  2( 0) du=2.095e-02 cont=6.689e-02 dnorm=4.398e-06 0
    it=  3( 0) du=2.563e-04 cont=1.224e-02 dnorm=8.810e-08 0
    it=  4( 0) du=4.861e-08 cont=1.897e-04 dnorm=1.451e-11 1
    Newton iteration successful
  0.146219 seconds (1.47 M allocations: 45.250 MiB, 6.05% gc time)
    Start Newton iteration
    it=  1( 0) du=3.693e-01 cont=3.693e-01 dnorm=1.024e-04 0
    it=  2( 0) du=2.985e-02 cont=8.083e-02 dnorm=6.559e-06 0
    it=  3( 0) du=6.711e-04 cont=2.248e-02 dnorm=2.216e-07 0
    it=  4( 0) du=2.868e-07 cont=4.273e-04 dnorm=9.045e-11 1
    Newton iteration successful
  0.140106 seconds (1.47 M allocations: 45.251 MiB, 4.28% gc time)
    Start Newton iteration
    it=  1( 0) du=4.579e-01 cont=4.579e-01 dnorm=1.350e-04 0
    it=  2( 0) du=4.629e-02 cont=1.011e-01 dnorm=1.092e-05 0
    it=  3( 0) du=2.061e-03 cont=4.451e-02 dnorm=6.897e-07 0
    it=  4( 0) du=2.669e-06 cont=1.295e-03 dnorm=8.353e-10 0
    Newton iteration successful
  0.150121 seconds (1.47 M allocations: 45.251 MiB, 4.09% gc time)
    Start Newton iteration
    it=  1( 0) du=6.132e-01 cont=6.132e-01 dnorm=1.991e-04 0
    it=  2( 0) du=7.753e-02 cont=1.264e-01 dnorm=1.861e-05 0
    it=  3( 0) du=6.627e-03 cont=8.548e-02 dnorm=2.267e-06 0
    it=  4( 0) du=2.442e-05 cont=3.684e-03 dnorm=8.354e-09 0
    it=  5( 0) du=5.887e-10 cont=2.411e-05 dnorm=1.992e-13 1
    Newton iteration successful
  0.181696 seconds (1.84 M allocations: 56.573 MiB, 4.82% gc time)
    Start Newton iteration
    it=  1( 0) du=9.971e-01 cont=9.971e-01 dnorm=3.350e-04 0
    it=  2( 0) du=1.292e-01 cont=1.295e-01 dnorm=2.360e-06 0
    it=  3( 0) du=2.339e-02 cont=1.811e-01 dnorm=8.151e-06 0
    it=  4( 0) du=3.933e-04 cont=1.682e-02 dnorm=1.275e-07 0
    it=  5( 0) du=1.298e-07 cont=3.300e-04 dnorm=4.291e-11 1
    Newton iteration successful
  0.185917 seconds (1.84 M allocations: 56.537 MiB, 4.66% gc time)
    Start Newton iteration
    it=  1( 0) du=1.044e+00 cont=1.044e+00 dnorm=3.944e-04 0
    it=  2( 0) du=2.577e-01 cont=2.468e-01 dnorm=6.901e-05 0
    it=  3( 0) du=6.728e-02 cont=2.610e-01 dnorm=2.592e-05 0
    it=  4( 0) du=3.187e-03 cont=4.737e-02 dnorm=1.125e-06 0
    it=  5( 0) du=7.480e-06 cont=2.347e-03 dnorm=2.719e-09 0
    Newton iteration successful
  0.181056 seconds (1.84 M allocations: 56.556 MiB, 4.86% gc time)
    Start Newton iteration
    it=  1( 0) du=5.100e-01 cont=5.100e-01 dnorm=2.108e-04 0
    it=  2( 0) du=8.618e-02 cont=1.690e-01 dnorm=2.707e-05 0
    it=  3( 0) du=6.084e-03 cont=7.059e-02 dnorm=2.291e-06 0
    it=  4( 0) du=2.249e-05 cont=3.697e-03 dnorm=8.363e-09 0
    it=  5( 0) du=3.445e-10 cont=1.531e-05 dnorm=1.293e-13 1
    Newton iteration successful
  0.181813 seconds (1.84 M allocations: 56.658 MiB, 5.06% gc time)
    Start Newton iteration
    it=  1( 0) du=2.280e-01 cont=2.280e-01 dnorm=1.141e-04 0
    it=  2( 0) du=1.713e-02 cont=7.512e-02 dnorm=3.767e-06 0
    it=  3( 0) du=2.648e-04 cont=1.546e-02 dnorm=4.417e-08 0
    it=  4( 0) du=4.434e-08 cont=1.675e-04 dnorm=5.852e-12 1
    Newton iteration successful
  0.146385 seconds (1.47 M allocations: 45.251 MiB, 6.02% gc time)
    Start Newton iteration
    it=  1( 0) du=3.006e-01 cont=3.006e-01 dnorm=8.498e-05 0
    it=  2( 0) du=1.741e-02 cont=5.791e-02 dnorm=3.700e-06 0
    it=  3( 0) du=2.282e-04 cont=1.311e-02 dnorm=5.330e-09 0
    it=  4( 0) du=2.574e-08 cont=1.128e-04 dnorm=3.721e-13 1
    Newton iteration successful
  0.148869 seconds (1.47 M allocations: 45.251 MiB, 4.12% gc time)
    Start Newton iteration
    it=  1( 0) du=3.461e-01 cont=3.461e-01 dnorm=6.961e-05 0
    it=  2( 0) du=3.929e-02 cont=1.135e-01 dnorm=1.013e-05 0
    it=  3( 0) du=1.447e-03 cont=3.684e-02 dnorm=2.424e-07 0
    it=  4( 0) du=1.469e-06 cont=1.015e-03 dnorm=2.690e-10 0
    Newton iteration successful
  0.144804 seconds (1.47 M allocations: 45.434 MiB, 6.02% gc time)
    Start Newton iteration
    it=  1( 0) du=3.033e-01 cont=3.033e-01 dnorm=4.183e-05 0
    it=  2( 0) du=4.328e-02 cont=1.427e-01 dnorm=7.961e-06 0
    it=  3( 0) du=1.728e-03 cont=3.993e-02 dnorm=2.565e-07 0
    it=  4( 0) du=2.326e-06 cont=1.346e-03 dnorm=3.542e-10 0
    Newton iteration successful
  0.145022 seconds (1.47 M allocations: 45.305 MiB, 6.04% gc time)
    Start Newton iteration
    it=  1( 0) du=2.591e-01 cont=2.591e-01 dnorm=1.849e-05 0
    it=  2( 0) du=2.254e-02 cont=8.700e-02 dnorm=2.406e-06 0
    it=  3( 0) du=5.659e-04 cont=2.510e-02 dnorm=5.539e-08 0
    it=  4( 0) du=2.476e-07 cont=4.374e-04 dnorm=2.640e-11 1
    Newton iteration successful
  0.146263 seconds (1.47 M allocations: 45.257 MiB, 5.97% gc time)
    Start Newton iteration
    it=  1( 0) du=1.729e-01 cont=1.729e-01 dnorm=9.331e-06 0
    it=  2( 0) du=1.132e-02 cont=6.543e-02 dnorm=3.403e-07 0
    it=  3( 0) du=1.142e-04 cont=1.009e-02 dnorm=3.750e-09 0
    it=  4( 0) du=9.150e-09 cont=8.015e-05 dnorm=2.992e-13 1
    Newton iteration successful
  0.143387 seconds (1.47 M allocations: 45.274 MiB, 4.20% gc time)
    Start Newton iteration
    it=  1( 0) du=1.002e-01 cont=1.002e-01 dnorm=6.735e-06 0
    it=  2( 0) du=4.062e-03 cont=4.053e-02 dnorm=1.743e-08 0
    it=  3( 0) du=1.309e-05 cont=3.223e-03 dnorm=7.679e-11 1
    it=  4( 0) du=1.168e-10 cont=8.922e-06 dnorm=6.504e-16 2
    Newton iteration successful
  0.146014 seconds (1.47 M allocations: 45.281 MiB, 4.16% gc time)
    Start Newton iteration
    it=  1( 0) du=5.443e-02 cont=5.443e-02 dnorm=5.475e-06 0
    it=  2( 0) du=1.205e-03 cont=2.213e-02 dnorm=2.523e-09 0
    it=  3( 0) du=1.104e-06 cont=9.160e-04 dnorm=5.187e-12 1
    it=  4( 0) du=8.370e-13 cont=7.585e-07 dnorm=0.000e+00 2
    Newton iteration successful
  0.140753 seconds (1.47 M allocations: 45.250 MiB, 4.25% gc time)
    Start Newton iteration
    it=  1( 0) du=2.870e-02 cont=2.870e-02 dnorm=4.291e-06 0
    it=  2( 0) du=3.274e-04 cont=1.141e-02 dnorm=2.197e-10 0
    it=  3( 0) du=7.981e-08 cont=2.437e-04 dnorm=8.260e-13 1
    Newton iteration successful
  0.107271 seconds (1.10 M allocations: 33.944 MiB, 5.64% gc time)
    Start Newton iteration
    it=  1( 0) du=1.536e-02 cont=1.536e-02 dnorm=3.160e-06 0
    it=  2( 0) du=8.507e-05 cont=5.540e-03 dnorm=8.522e-10 0
    it=  3( 0) du=5.310e-09 cont=6.242e-05 dnorm=1.008e-13 1
    Newton iteration successful
  0.106166 seconds (1.10 M allocations: 33.934 MiB, 3.56% gc time)
    Start Newton iteration
    it=  1( 0) du=8.265e-03 cont=8.265e-03 dnorm=2.203e-06 0
    it=  2( 0) du=2.151e-05 cont=2.603e-03 dnorm=5.569e-10 0
    it=  3( 0) du=3.350e-10 cont=1.557e-05 dnorm=1.431e-14 1
    Newton iteration successful
  0.106697 seconds (1.10 M allocations: 33.943 MiB, 5.56% gc time)
    Start Newton iteration
    it=  1( 0) du=4.428e-03 cont=4.428e-03 dnorm=1.468e-06 0
    it=  2( 0) du=5.352e-06 cont=1.209e-03 dnorm=2.623e-10 0
    it=  3( 0) du=2.138e-11 cont=3.994e-06 dnorm=1.951e-15 1
    Newton iteration successful
  0.106295 seconds (1.10 M allocations: 33.932 MiB, 2.95% gc time)
    Start Newton iteration
    it=  1( 0) du=2.502e-03 cont=2.502e-03 dnorm=9.429e-07 0
    it=  2( 0) du=1.324e-06 cont=5.290e-04 dnorm=1.055e-10 0
    it=  3( 0) du=1.660e-12 cont=1.254e-06 dnorm=0.000e+00 1
    Newton iteration successful
  0.108290 seconds (1.10 M allocations: 33.945 MiB, 5.59% gc time)
    Start Newton iteration
    it=  1( 0) du=1.558e-03 cont=1.558e-03 dnorm=5.863e-07 0
    it=  2( 0) du=4.004e-07 cont=2.570e-04 dnorm=3.836e-11 1
    it=  3( 0) du=2.371e-13 cont=5.922e-07 dnorm=0.000e+00 2
    Newton iteration successful
  0.105471 seconds (1.10 M allocations: 33.934 MiB, 2.88% gc time)
    Start Newton iteration
    it=  1( 0) du=9.392e-04 cont=9.392e-04 dnorm=3.542e-07 0
    it=  2( 0) du=1.391e-07 cont=1.481e-04 dnorm=1.296e-11 1
    it=  3( 0) du=2.741e-14 cont=1.970e-07 dnorm=2.168e-16 2
    Newton iteration successful
  0.107793 seconds (1.10 M allocations: 33.943 MiB, 5.59% gc time)
    Start Newton iteration
    it=  1( 0) du=5.499e-04 cont=5.499e-04 dnorm=2.082e-07 0
    it=  2( 0) du=4.641e-08 cont=8.441e-05 dnorm=4.120e-12 1
    it=  3( 0) du=2.360e-14 cont=5.085e-07 dnorm=2.168e-16 2
    Newton iteration successful
  0.106046 seconds (1.10 M allocations: 33.933 MiB, 2.91% gc time)
    Start Newton iteration
    it=  1( 0) du=3.134e-04 cont=3.134e-04 dnorm=1.193e-07 0
    it=  2( 0) du=1.485e-08 cont=4.737e-05 dnorm=1.242e-12 1
    it=  3( 0) du=1.613e-14 cont=1.086e-06 dnorm=2.168e-16 2
    Newton iteration successful
  0.109799 seconds (1.10 M allocations: 33.945 MiB, 5.49% gc time)
    Start Newton iteration
    it=  1( 0) du=1.741e-04 cont=1.741e-04 dnorm=6.667e-08 0
    it=  2( 0) du=4.547e-09 cont=2.611e-05 dnorm=3.566e-13 1
    it=  3( 0) du=3.748e-14 cont=8.243e-06 dnorm=2.168e-16 2
    Newton iteration successful
  0.107113 seconds (1.10 M allocations: 33.934 MiB, 2.88% gc time)
    Start Newton iteration
    it=  1( 0) du=9.438e-05 cont=9.438e-05 dnorm=3.634e-08 0
    it=  2( 0) du=1.332e-09 cont=1.411e-05 dnorm=9.712e-14 1
    it=  3( 0) du=2.204e-14 cont=1.655e-05 dnorm=2.168e-16 2
    Newton iteration successful
  0.107756 seconds (1.10 M allocations: 33.943 MiB, 5.57% gc time)
    Start Newton iteration
    it=  1( 0) du=4.993e-05 cont=4.993e-05 dnorm=1.932e-08 0
    it=  2( 0) du=3.728e-10 cont=7.467e-06 dnorm=2.558e-14 1
    Newton iteration successful
  0.072189 seconds (736.55 k allocations: 22.627 MiB, 4.23% gc time)
    Start Newton iteration
    it=  1( 0) du=2.578e-05 cont=2.578e-05 dnorm=1.002e-08 0
    it=  2( 0) du=9.959e-11 cont=3.864e-06 dnorm=6.287e-15 1
    Newton iteration successful
  0.070257 seconds (736.55 k allocations: 22.628 MiB, 4.26% gc time)
    Start Newton iteration
    it=  1( 0) du=1.298e-05 cont=1.298e-05 dnorm=5.065e-09 0
    it=  2( 0) du=2.536e-11 cont=1.954e-06 dnorm=1.734e-15 1
    Newton iteration successful
  0.072516 seconds (736.55 k allocations: 22.628 MiB, 4.17% gc time)
    Start Newton iteration
    it=  1( 0) du=6.378e-06 cont=6.378e-06 dnorm=2.496e-09 0
    it=  2( 0) du=6.139e-12 cont=9.625e-07 dnorm=4.336e-16 1
    Newton iteration successful
  0.071008 seconds (736.55 k allocations: 22.628 MiB, 4.19% gc time)
    Start Newton iteration
    it=  1( 0) du=3.054e-06 cont=3.054e-06 dnorm=1.198e-09 0
    it=  2( 0) du=1.420e-12 cont=4.651e-07 dnorm=2.168e-16 1
    Newton iteration successful
  0.074016 seconds (736.56 k allocations: 22.629 MiB, 8.35% gc time)
    Start Newton iteration
    it=  1( 0) du=1.424e-06 cont=1.424e-06 dnorm=5.598e-10 0
    it=  2( 0) du=3.183e-13 cont=2.234e-07 dnorm=0.000e+00 1
    Newton iteration successful
  0.077436 seconds (736.55 k allocations: 22.628 MiB, 5.19% gc time)
    Start Newton iteration
    it=  1( 0) du=6.468e-07 cont=6.468e-07 dnorm=2.546e-10 0
    it=  2( 0) du=7.203e-14 cont=1.114e-07 dnorm=0.000e+00 1
    Newton iteration successful
  0.072186 seconds (736.55 k allocations: 22.628 MiB, 4.54% gc time)
    Start Newton iteration
    it=  1( 0) du=2.857e-07 cont=2.857e-07 dnorm=1.126e-10 0
    it=  2( 0) du=2.827e-14 cont=9.893e-08 dnorm=2.168e-16 1
    Newton iteration successful
  0.072392 seconds (736.55 k allocations: 22.628 MiB, 4.62% gc time)

┌ Info: Saved animation to 
│   fn = /home/fuhrmann/Wias/teach/scicomp/course/tmp.gif
â”” @ Plots /home/fuhrmann/.julia/packages/Plots/qZHsp/src/animation.jl:98
Out[14]:

This notebook was generated using Literate.jl.