pde_nl21.adb running 
The PDE to be solved for u(x) is:
 Uxx(x) + 2*Ux(x) + U(x)^2  = f(x)
 
 for testing code:
       f(x) = x^4 + 2x^2 + 4x + 3
       u(xmin=0)=1  u(xmax=1)=2
 
analytic solution and boundary  ub(x) = x^2+1
       thus  Ux(x) = 2*x    Uxx(x) = 2
xmin= 0.00000000000000E+00, xmax= 1.00000000000000E+00
nx= 5 points for numeric derivative
x grid, analytic solution, f(x) 
i= 1, Ua(  0.00000)=  1.00000, f(x)=  3.00000
i= 2, Ua(  0.25000)=  1.06250, f(x)=  4.12891
i= 3, Ua(  0.50000)=  1.25000, f(x)=  5.56250
i= 4, Ua(  0.75000)=  1.56250, f(x)=  7.44141
i= 5, Ua(  1.00000)=  2.00000, f(x)= 10.00000

compute non linear A matrix
nonlinear matrix A
i= 1, j= 1, A(i,j)=-3.33333333333333E+01
i= 1, j= 2, A(i,j)= 2.00000000000000E+01
i= 1, j= 3, A(i,j)= 1.33333333333333E+00
i= 1, j= 4, A(i,j)= 1.00000000000000E+00
i= 1, j= 5, A(i,j)= 0.00000000000000E+00
i= 1, j= 6, A(i,j)= 0.00000000000000E+00
i= 2, j= 1, A(i,j)= 1.60000000000000E+01
i= 2, j= 2, A(i,j)=-4.00000000000000E+01
i= 2, j= 3, A(i,j)= 2.66666666666667E+01
i= 2, j= 4, A(i,j)= 0.00000000000000E+00
i= 2, j= 5, A(i,j)= 1.00000000000000E+00
i= 2, j= 6, A(i,j)= 0.00000000000000E+00
i= 3, j= 1, A(i,j)= 9.33333333333333E+00
i= 3, j= 2, A(i,j)=-4.00000000000000E+00
i= 3, j= 3, A(i,j)=-2.00000000000000E+01
i= 3, j= 4, A(i,j)= 0.00000000000000E+00
i= 3, j= 5, A(i,j)= 0.00000000000000E+00
i= 3, j= 6, A(i,j)= 1.00000000000000E+00

Y computed RHS 
Y( 1)=-7.20442708333333E+00
Y( 2)= 1.02291666666667E+01
Y( 3)=-2.38919270833333E+01

system of equations to be solved, i=1.. 3
A(i, 1)*X1+
A(i, 2)*X2+
A(i, 3)*X3+
A(i, 4)*X1*X1+
A(i, 5)*X2*X2+
A(i, 6)*X3*X3
 = Y(i)

test, giving exact solution
X1 =  1.06250000000000E+00
X2 =  1.25000000000000E+00
X3 =  1.56250000000000E+00
Check_Equation residual =  3.55271367880050E-15

simeq_newton5.adb running
itr   0,                       initial residual=  1.33226762955019E-14
simeq_newton5 found solution

initial guess, all 1.0
simeq_newton5.adb running
itr   0,                       initial residual=  2.05833333333333E+01
itr   1, prev=  2.05833333333333E+01, residual=  3.47668232556459E-01

itr   2, prev=  3.47668232556459E-01, residual=  1.16153163625210E-03

itr   3, prev=  1.16153163625210E-03, residual=  7.26445481546989E-09

simeq_newton5 found solution
U computed from nonlinear equations, Ua analytic, error 
ug( 1)=  1.0000000, Ua=  1.0000000, err=  0.0000000
ug( 2)=  1.0625000, Ua=  1.0625000, err= -0.0000000
ug( 3)=  1.2500000, Ua=  1.2500000, err= -0.0000000
ug( 4)=  1.5625000, Ua=  1.5625000, err= -0.0000000
ug( 5)=  2.0000000, Ua=  2.0000000, err=  0.0000000
                   maxerr= 3.41883410470700E-10, avgerr= 2.68845020319948E-10

check_soln on computed solution against PDE
check_soln
i= 2, err= 2.27318408541066E-09
i= 3, err= 3.61938767845515E-09
i= 4, err= 1.37188393978249E-09
check_soln max error= 3.61938767845515E-09

just checking check_soln when given correct solution
check_soln
i= 2, err=-8.21565038222616E-15
i= 3, err=-3.33066907387547E-15
i= 4, err=-1.06581410364015E-14
check_soln max error= 1.06581410364015E-14

pde_nl21.adb finished