NSolve solution table as input for FindRoot

Question

Goal is to use NSolve table of solutions for a set of linear equations as initial input values for FindRoot. Here is example of 3 Eqns:

Nsol = Table[NSolve[{
0 == x1 + 2*y1 + 3*z1,
0 == 2*x1 - 3*y1 + z1,
a == 1 + x1 + y1/z1 }, {x1, y1, z1}], {a, 1, 3, 1}]

This provides unique solutions for each value of a = 1,2,3. I'd like to now use these solutions as input to a FindRoot of 3 Eqns (one of which is nonlinear) for same values of a:

FRsol = Table[FindRoot[{
0 == x2 + 2*y2 + 3*z2,
0 == 2*x2 - 3*y2 + z2,
a == E^x2 + y2/z2}, {{x2,?},{y2,?},{z2,?}}], {a, 1, 3, 1}]

I've tried replacing "?" with several delimiters involving Nsol, but without success. While this example FRsol can be solved by merely plugging in some example initial values for x2,y2,z2, the real system I'm looking at has solutions that differ by >15 orders of magnitude and a much larger table of solutions which I will ultimately export. The approach is to use a linear set of equations to generate input values reasonably close to solutions involving system with nonlinear equations.

Answer 1

Could do it like so.

Setup:

vars = {x1, y1, z1};
solns1 = Table[{a, 
   First[NSolve[{0 == x1 + 2*y1 + 3*z1, 0 == 2*x1 - 3*y1 + z1, 
      a == 1 + x1 + y1/z1}, vars]]}, {a, 1, 3, 1}]

(* Out[306]= {{1, {x1 -> 0.714285714286, y1 -> 0.324675324675, 
   z1 -> -0.454545454545}}, {2, {x1 -> 1.71428571429, 
   y1 -> 0.779220779221, 
   z1 -> -1.09090909091}}, {3, {x1 -> 2.71428571429, 
   y1 -> 1.23376623377, z1 -> -1.72727272727}}} *)

Now plug in as initial values for the harder system.

solns2 = 
 Map[FindRoot[{0 == x1 + 2*y1 + 3*z1, 
     0 == 2*x1 - 3*y1 + z1, #[[1]] == E^x1 + y1/z1}, 
    Transpose[{vars, vars /. #[[2]]}]] &, solns1]

(* Out[304]= {{x1 -> 0.538996500733, y1 -> 0.244998409424, 
  z1 -> -0.342997773194}, {x1 -> 0.998528830111, y1 -> 0.45387674096, 
  z1 -> -0.635427437343}, {x1 -> 1.31218638897, y1 -> 0.596448358621, 
  z1 -> -0.835027702069}} *)

An alternative is to reinitialize each iteration in the second run with the results from the prior run.

Answer 2

Nsol = Table[{a, x1, y1, z1} /. 
   NSolve[{0 == x1 + 2*y1 + 3*z1, 0 == 2*x1 - 3*y1 + z1, 
      a == 1 + x1 + y1/z1}, {x1, y1, z1}][[1]], {a, 1, 3, 1}]

(* {{1, 0.714286, 0.324675, -0.454545}, {2, 1.71429, 
  0.779221, -1.09091}, {3, 2.71429, 1.23377, -1.72727}} *)

FindRoot[{0 == x2 + 2*y2 + 3*z2, 
    0 == 2*x2 - 3*y2 + z2, #[[1]] == 
     E^x2 + y2/
       z2}, {{x2, #[[2]]}, {y2, #[[3]]}, {z2, #[[4]]}}] & /@ Nsol

{* {{x2 -> 0.538997, y2 -> 0.244998, z2 -> -0.342998}, {x2 -> 0.998529, 
  y2 -> 0.453877, z2 -> -0.635427}, {x2 -> 1.31219, y2 -> 0.596448, 
  z2 -> -0.835028}} *)

EDIT: However, note that the exact solutions can be obtained with Solve

eqns = {0 == x2 + 2*y2 + 3*z2, 0 == 2*x2 - 3*y2 + z2, a == E^x2 + y2/z2};

Table[{sol[a] = Solve[eqns, {x2, y2, z2}][[1]], And @@ (eqns /. sol[a])}, 
    {a, 1, 3}] // Quiet // Column

% // N