pde_nl13a.adb running 
solve  u^2 u'' + 2 u u' + 3u = f(x)
       f(x) = (41/3)-(2116/9)x+(4780/3)x^2-(48976/9)x^3+
              (48976.0/9.0)*x*x*x + 9984.0*x*x*x*x -
              (88064.0/9.0)*x*x*x*x*x + (14336.0/3.0)*x*x*x*x*x*x -
              9984x^4-(88064/9)x^5+(14336/3)x^6-(8192/9)x^7
 
   0<x<1  Boundary U(0) = 1, U(1) = 1
Analytic solution 
U(x) = 1.0 -(20.0/3.0)*x +12.0*x^2 -(16.0/3.0)*x^3
xmin= 0.00000000000000E+00, xmax= 1.00000000000000E+00
nx= 5 points for numeric derivative
x grid and analytic solution 
i= 1, Ua(  0.00000)=  1.00000
i= 2, Ua(  0.25000)= -0.00000
i= 3, Ua(  0.50000)= -0.00000
i= 4, Ua(  0.75000)=  0.50000
i= 5, Ua(  1.00000)=  1.00000

k( 1, 1)= 1.00000000000000E+00
f( 0.00000000000000E+00)= 1.00000000000000E+00, for f at i= 1
k( 5, 5)= 1.00000000000000E+00
f( 1.00000000000000E+00)= 1.00000000000000E+00, for f at i= 5
compute non linear stiffness matrix
k( 2, 6)= 1.46666666666667E+01
k( 2, 7)=-2.66666666666667E+01
k( 2, 8)= 8.00000000000000E+00
k( 2, 9)= 5.33333333333333E+00
k( 2, 6)= 1.33333333333333E+01
k( 2, 2)=-2.00000000000000E+00
k( 2, 6)= 6.66666666666667E+00
k( 2, 10)= 1.20000000000000E+01
k( 2, 11)=-4.00000000000000E+00
k( 2, 2)=-1.33333333333333E+00
k( 3, 12)=-1.33333333333333E+00
k( 3, 13)= 2.13333333333333E+01
k( 3, 14)=-4.00000000000000E+01
k( 3, 15)= 2.13333333333333E+01
k( 3, 12)=-2.66666666666667E+00
k( 3, 3)= 6.66666666666667E-01
k( 3, 10)=-5.33333333333333E+00
k( 3, 12)=-2.66666666666667E+00
k( 3, 16)= 5.33333333333333E+00
k( 3, 3)= 0.00000000000000E+00
k( 4, 17)=-1.33333333333333E+00
k( 4, 18)= 5.33333333333333E+00
k( 4, 19)= 8.00000000000000E+00
k( 4, 20)=-2.66666666666667E+01
k( 4, 17)= 1.33333333333333E+01
k( 4, 4)=-6.66666666666667E-01
k( 4, 11)= 4.00000000000000E+00
k( 4, 16)=-1.20000000000000E+01
k( 4, 17)= 2.00000000000000E+01
k( 4, 4)= 1.33333333333333E+00
k computed stiffness matrix
i= 1, j= 1, k(i,j)= 1.00000000000000E+00
i= 1, j= 2, k(i,j)= 0.00000000000000E+00
i= 1, j= 3, k(i,j)= 0.00000000000000E+00
i= 1, j= 4, k(i,j)= 0.00000000000000E+00
i= 1, j= 5, k(i,j)= 0.00000000000000E+00
i= 1, j= 6, k(i,j)= 0.00000000000000E+00
i= 1, j= 7, k(i,j)= 0.00000000000000E+00
i= 1, j= 8, k(i,j)= 0.00000000000000E+00
i= 1, j= 9, k(i,j)= 0.00000000000000E+00
i= 1, j= 10, k(i,j)= 0.00000000000000E+00
i= 1, j= 11, k(i,j)= 0.00000000000000E+00
i= 1, j= 12, k(i,j)= 0.00000000000000E+00
i= 1, j= 13, k(i,j)= 0.00000000000000E+00
i= 1, j= 14, k(i,j)= 0.00000000000000E+00
i= 1, j= 15, k(i,j)= 0.00000000000000E+00
i= 1, j= 16, k(i,j)= 0.00000000000000E+00
i= 1, j= 17, k(i,j)= 0.00000000000000E+00
i= 1, j= 18, k(i,j)= 0.00000000000000E+00
i= 1, j= 19, k(i,j)= 0.00000000000000E+00
i= 1, j= 20, k(i,j)= 0.00000000000000E+00
i= 2, j= 1, k(i,j)= 0.00000000000000E+00
i= 2, j= 2, k(i,j)= 1.66666666666667E+00
i= 2, j= 3, k(i,j)= 0.00000000000000E+00
i= 2, j= 4, k(i,j)= 0.00000000000000E+00
i= 2, j= 5, k(i,j)= 0.00000000000000E+00
i= 2, j= 6, k(i,j)= 6.66666666666667E+00
i= 2, j= 7, k(i,j)=-2.66666666666667E+01
i= 2, j= 8, k(i,j)= 8.00000000000000E+00
i= 2, j= 9, k(i,j)= 5.33333333333333E+00
i= 2, j= 10, k(i,j)= 1.20000000000000E+01
i= 2, j= 11, k(i,j)=-4.00000000000000E+00
i= 2, j= 12, k(i,j)= 0.00000000000000E+00
i= 2, j= 13, k(i,j)= 0.00000000000000E+00
i= 2, j= 14, k(i,j)= 0.00000000000000E+00
i= 2, j= 15, k(i,j)= 0.00000000000000E+00
i= 2, j= 16, k(i,j)= 0.00000000000000E+00
i= 2, j= 17, k(i,j)= 0.00000000000000E+00
i= 2, j= 18, k(i,j)= 0.00000000000000E+00
i= 2, j= 19, k(i,j)= 0.00000000000000E+00
i= 2, j= 20, k(i,j)= 0.00000000000000E+00
i= 3, j= 1, k(i,j)= 0.00000000000000E+00
i= 3, j= 2, k(i,j)= 0.00000000000000E+00
i= 3, j= 3, k(i,j)= 3.00000000000000E+00
i= 3, j= 4, k(i,j)= 0.00000000000000E+00
i= 3, j= 5, k(i,j)= 0.00000000000000E+00
i= 3, j= 6, k(i,j)= 0.00000000000000E+00
i= 3, j= 7, k(i,j)= 0.00000000000000E+00
i= 3, j= 8, k(i,j)= 0.00000000000000E+00
i= 3, j= 9, k(i,j)= 0.00000000000000E+00
i= 3, j= 10, k(i,j)=-5.33333333333333E+00
i= 3, j= 11, k(i,j)= 0.00000000000000E+00
i= 3, j= 12, k(i,j)=-2.66666666666667E+00
i= 3, j= 13, k(i,j)= 2.13333333333333E+01
i= 3, j= 14, k(i,j)=-4.00000000000000E+01
i= 3, j= 15, k(i,j)= 2.13333333333333E+01
i= 3, j= 16, k(i,j)= 5.33333333333333E+00
i= 3, j= 17, k(i,j)= 0.00000000000000E+00
i= 3, j= 18, k(i,j)= 0.00000000000000E+00
i= 3, j= 19, k(i,j)= 0.00000000000000E+00
i= 3, j= 20, k(i,j)= 0.00000000000000E+00
i= 4, j= 1, k(i,j)= 0.00000000000000E+00
i= 4, j= 2, k(i,j)= 0.00000000000000E+00
i= 4, j= 3, k(i,j)= 0.00000000000000E+00
i= 4, j= 4, k(i,j)= 4.33333333333333E+00
i= 4, j= 5, k(i,j)= 0.00000000000000E+00
i= 4, j= 6, k(i,j)= 0.00000000000000E+00
i= 4, j= 7, k(i,j)= 0.00000000000000E+00
i= 4, j= 8, k(i,j)= 0.00000000000000E+00
i= 4, j= 9, k(i,j)= 0.00000000000000E+00
i= 4, j= 10, k(i,j)= 0.00000000000000E+00
i= 4, j= 11, k(i,j)= 4.00000000000000E+00
i= 4, j= 12, k(i,j)= 0.00000000000000E+00
i= 4, j= 13, k(i,j)= 0.00000000000000E+00
i= 4, j= 14, k(i,j)= 0.00000000000000E+00
i= 4, j= 15, k(i,j)= 0.00000000000000E+00
i= 4, j= 16, k(i,j)=-1.20000000000000E+01
i= 4, j= 17, k(i,j)= 2.00000000000000E+01
i= 4, j= 18, k(i,j)= 5.33333333333333E+00
i= 4, j= 19, k(i,j)= 8.00000000000000E+00
i= 4, j= 20, k(i,j)=-2.66666666666667E+01
i= 5, j= 1, k(i,j)= 0.00000000000000E+00
i= 5, j= 2, k(i,j)= 0.00000000000000E+00
i= 5, j= 3, k(i,j)= 0.00000000000000E+00
i= 5, j= 4, k(i,j)= 0.00000000000000E+00
i= 5, j= 5, k(i,j)= 1.00000000000000E+00
i= 5, j= 6, k(i,j)= 0.00000000000000E+00
i= 5, j= 7, k(i,j)= 0.00000000000000E+00
i= 5, j= 8, k(i,j)= 0.00000000000000E+00
i= 5, j= 9, k(i,j)= 0.00000000000000E+00
i= 5, j= 10, k(i,j)= 0.00000000000000E+00
i= 5, j= 11, k(i,j)= 0.00000000000000E+00
i= 5, j= 12, k(i,j)= 0.00000000000000E+00
i= 5, j= 13, k(i,j)= 0.00000000000000E+00
i= 5, j= 14, k(i,j)= 0.00000000000000E+00
i= 5, j= 15, k(i,j)= 0.00000000000000E+00
i= 5, j= 16, k(i,j)= 0.00000000000000E+00
i= 5, j= 17, k(i,j)= 0.00000000000000E+00
i= 5, j= 18, k(i,j)= 0.00000000000000E+00
i= 5, j= 19, k(i,j)= 0.00000000000000E+00
i= 5, j= 20, k(i,j)= 0.00000000000000E+00

f computed forcing function 
f( 1)= 1.00000000000000E+00
f( 2)= 4.71844785465692E-16
f( 3)= 4.08562073062058E-14
f( 4)= 3.83333333333357E+00
f( 5)= 1.00000000000000E+00

system of equations to be solved, i=1.. 5
k(i, 1)*X1+
k(i, 2)*X2+
k(i, 3)*X3+
k(i, 4)*X4+
k(i, 5)*X5+
k(i, 6)*X2*X2+
k(i, 7)*X2*X2*X2+
k(i, 8)*X3*X2*X2+
k(i, 9)*X4*X2*X2+
k(i, 10)*X3*X2+
k(i, 11)*X4*X2+
k(i, 12)*X3*X3+
k(i, 13)*X3*X3*X2+
k(i, 14)*X3*X3*X3+
k(i, 15)*X4*X3*X3+
k(i, 16)*X4*X3+
k(i, 17)*X4*X4+
k(i, 18)*X4*X4*X2+
k(i, 19)*X4*X4*X3+
k(i, 20)*X4*X4*X4
 = f(i)

finds a solution, not the one expected
simeq_newton5.adb running
itr   0,                       initial residual=  6.83333333333352E+00
itr   1, prev=  6.83333333333352E+00, residual=  3.65931431215120E+00

itr   2, prev=  3.65931431215120E+00, residual=  3.37279896748199E-01

itr   3, prev=  3.37279896748199E-01, residual=  4.27381740945612E-03

itr   4, prev=  4.27381740945612E-03, residual=  7.62784081259626E-07

simeq_newton5 found solution
check against expected solution
U computed from nonlinear equations, Ua analytic, error 
U( 1)=  1.00000, Ua=  1.00000, err=  0.00000
U( 2)=  1.15706, Ua= -0.00000, err=  1.15706
U( 3)=  1.13546, Ua= -0.00000, err=  1.13546
U( 4)=  1.00628, Ua=  0.50000, err=  0.50628
U( 5)=  1.00000, Ua=  1.00000, err=  0.00000
                   maxerr= 1.15705894351907E+00, avgerr= 5.59760829458356E-01

check for any solution
check_soln on computed solution against PDE
check_soln
i= 2, err=-6.34236548835432E-07
i= 3, err=-1.26797782717958E-07
i= 4, err= 1.74975101074892E-09
check_soln max error= 6.34236548835432E-07
just checking check_soln when given correct solution
check_soln
i= 2, err=-4.48715139119334E-16
i= 3, err=-4.14853336868267E-14
i= 4, err=-2.33368879776208E-13
check_soln max error= 2.33368879776208E-13

just checking simeq_newton5 when given correct solution \n
simeq_newton5.adb running
itr   0,                       initial residual=  2.75080883997229E-13
simeq_newton5 found solution
pde_nl13a.adb finished