pde_nl21a.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= 7 points for numeric derivative
x grid, analytic solution, f(x) 
i= 1, Ua(  0.00000)=  1.00000, f(x)=  3.00000
i= 2, Ua(  0.16667)=  1.02778, f(x)=  3.72299
i= 3, Ua(  0.33333)=  1.11111, f(x)=  4.56790
i= 4, Ua(  0.50000)=  1.25000, f(x)=  5.56250
i= 5, Ua(  0.66667)=  1.44444, f(x)=  6.75309
i= 6, Ua(  0.83333)=  1.69444, f(x)=  8.20448
i= 7, Ua(  1.00000)=  2.00000, f(x)= 10.00000

compute non linear A matrix
nonlinear matrix A
i= 1, j= 1, A(i,j)=-4.48000000000000E+01
i= 1, j= 2, A(i,j)=-2.10000000000000E+01
i= 1, j= 3, A(i,j)= 7.40000000000000E+01
i= 1, j= 4, A(i,j)=-4.70000000000000E+01
i= 1, j= 5, A(i,j)= 1.56000000000000E+01
i= 1, j= 6, A(i,j)= 1.00000000000000E+00
i= 1, j= 7, A(i,j)= 0.00000000000000E+00
i= 1, j= 8, A(i,j)= 0.00000000000000E+00
i= 1, j= 9, A(i,j)= 0.00000000000000E+00
i= 1, j= 10, A(i,j)= 0.00000000000000E+00
i= 2, j= 1, A(i,j)= 4.08000000000000E+01
i= 2, j= 2, A(i,j)=-9.10000000000000E+01
i= 2, j= 3, A(i,j)= 5.60000000000000E+01
i= 2, j= 4, A(i,j)=-3.00000000000000E+00
i= 2, j= 5, A(i,j)=-8.00000000000000E-01
i= 2, j= 6, A(i,j)= 0.00000000000000E+00
i= 2, j= 7, A(i,j)= 1.00000000000000E+00
i= 2, j= 8, A(i,j)= 0.00000000000000E+00
i= 2, j= 9, A(i,j)= 0.00000000000000E+00
i= 2, j= 10, A(i,j)= 0.00000000000000E+00
i= 3, j= 1, A(i,j)=-3.60000000000000E+00
i= 3, j= 2, A(i,j)= 4.50000000000000E+01
i= 3, j= 3, A(i,j)=-9.80000000000000E+01
i= 3, j= 4, A(i,j)= 6.30000000000000E+01
i= 3, j= 5, A(i,j)=-7.20000000000000E+00
i= 3, j= 6, A(i,j)= 0.00000000000000E+00
i= 3, j= 7, A(i,j)= 0.00000000000000E+00
i= 3, j= 8, A(i,j)= 1.00000000000000E+00
i= 3, j= 9, A(i,j)= 0.00000000000000E+00
i= 3, j= 10, A(i,j)= 0.00000000000000E+00
i= 4, j= 1, A(i,j)=-4.00000000000000E+00
i= 4, j= 2, A(i,j)= 9.00000000000000E+00
i= 4, j= 3, A(i,j)= 2.40000000000000E+01
i= 4, j= 4, A(i,j)=-7.70000000000000E+01
i= 4, j= 5, A(i,j)= 5.04000000000000E+01
i= 4, j= 6, A(i,j)= 0.00000000000000E+00
i= 4, j= 7, A(i,j)= 0.00000000000000E+00
i= 4, j= 8, A(i,j)= 0.00000000000000E+00
i= 4, j= 9, A(i,j)= 1.00000000000000E+00
i= 4, j= 10, A(i,j)= 0.00000000000000E+00
i= 5, j= 1, A(i,j)= 2.16000000000000E+01
i= 5, j= 2, A(i,j)=-6.70000000000000E+01
i= 5, j= 3, A(i,j)= 1.14000000000000E+02
i= 5, j= 4, A(i,j)=-8.10000000000000E+01
i= 5, j= 5, A(i,j)=-1.40000000000000E+01
i= 5, j= 6, A(i,j)= 0.00000000000000E+00
i= 5, j= 7, A(i,j)= 0.00000000000000E+00
i= 5, j= 8, A(i,j)= 0.00000000000000E+00
i= 5, j= 9, A(i,j)= 0.00000000000000E+00
i= 5, j= 10, A(i,j)= 1.00000000000000E+00

Y computed RHS 
Y( 1)=-1.72770061728395E+01
Y( 2)= 6.36790123456790E+00
Y( 3)= 4.16250000000000E+00
Y( 4)= 1.21530864197531E+01
Y( 5)=-4.75955246913580E+01

system of equations to be solved, i=1.. 5
A(i, 1)*X1+
A(i, 2)*X2+
A(i, 3)*X3+
A(i, 4)*X4+
A(i, 5)*X5+
A(i, 6)*X1*X1+
A(i, 7)*X2*X2+
A(i, 8)*X3*X3+
A(i, 9)*X4*X4+
A(i, 10)*X5*X5
 = Y(i)

test, giving exact solution
X1 =  1.02777777777778E+00
X2 =  1.11111111111111E+00
X3 =  1.25000000000000E+00
X4 =  1.44444444444444E+00
X5 =  1.69444444444444E+00
Check_Equation residual =  0.00000000000000E+00

simeq_newton5.adb running
itr   0,                       initial residual=  4.52970994047064E-14
simeq_newton5 found solution

initial guess, all 1.0
simeq_newton5.adb running
itr   0,                       initial residual=  4.32020061728395E+01
itr   1, prev=  4.32020061728395E+01, residual=  6.96422932539432E-01

itr   2, prev=  6.96422932539432E-01, residual=  1.83902659684065E-03

itr   3, prev=  1.83902659684065E-03, residual=  1.21047625256665E-08

simeq_newton5 found solution
U computed from nonlinear equations, Ua analytic, error 
ug( 1)=  1.0000000, Ua=  1.0000000, err=  0.0000000
ug( 2)=  1.0277778, Ua=  1.0277778, err= -0.0000000
ug( 3)=  1.1111111, Ua=  1.1111111, err= -0.0000000
ug( 4)=  1.2500000, Ua=  1.2500000, err= -0.0000000
ug( 5)=  1.4444444, Ua=  1.4444444, err= -0.0000000
ug( 6)=  1.6944444, Ua=  1.6944444, err= -0.0000000
ug( 7)=  2.0000000, Ua=  2.0000000, err=  0.0000000
                   maxerr= 4.17917256356759E-10, avgerr= 3.11158743215856E-10

check_soln on computed solution against PDE
check_soln
i= 2, err= 1.40269684756333E-09
i= 3, err= 3.50242956859148E-09
i= 4, err= 4.00387223375986E-09
i= 5, err= 2.53607246314402E-09
i= 6, err= 6.59731824725895E-10
check_soln max error= 4.00387223375986E-09

just checking check_soln when given correct solution
check_soln
i= 2, err= 8.21565038222616E-15
i= 3, err=-1.89848137210902E-14
i= 4, err=-1.43218770176645E-14
i= 5, err=-5.55111512312578E-15
i= 6, err=-1.42108547152020E-14
check_soln max error= 1.89848137210902E-14

pde_nl21a.adb finished