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NSolve gives an empty set as a solution

S Sarkar

Posted 2 years ago

This gives a result NSolve[{7463^x2070^y15^w == 3, 7060^x2035^y15^w == 5, 8874^x1945^y15^w == 23}, {w, x, y}, Reals] This doesn't NSolve[{7463^x2070^y15^w10^z == 3, 7060^x2035^y15^w10^z == 5, 8874^x1945^y15^w10^z == 23, 9918^x1949^y15^w10^z == 17}, {w, x, y, z}, Reals]

This gives a result

NSolve[{7463^x*2070^y*15^w == 3, 7060^x*2035^y*15^w == 5, 
      8874^x*1945^y*15^w == 23}, {w, x, y}, Reals]

This doesn't

NSolve[{7463^x*2070^y*15^w*10^z == 3, 7060^x*2035^y*15^w*10^z == 5, 
  8874^x*1945^y*15^w*10^z == 23, 9918^x*1949^y*15^w*10^z == 17}, {w, 
  x, y, z}, Reals]

POSTED BY: S Sarkar

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Gianluca Gorni

Gianluca Gorni, University of Udine

Posted 2 years ago

Mathematica solves the equation and gives the empty set, which is correct, as you can check by turning the system into a linear one: eqs = {7463^x2070^y15^w10^z == 3, 7060^x2035^y15^w10^z == 5, 8874^x1945^y15^w10^z == 23, 9918^x1949^y15^w10^z == 17}; toLog = Map[Log, eqs, {2}] /. HoldPattern[Log[Times[x__]]] :> Plus @@ Map[Log, {x}] /. Log[a_^b_] :> b*Log[a] {b, mat} = CoefficientArrays[toLog, {w, x, y, z}] // Normal mat // Simplify // MatrixForm Det[mat]

Mathematica solves the equation and gives the empty set, which is correct, as you can check by turning the system into a linear one:

eqs = {7463^x*2070^y*15^w*10^z == 3,
   7060^x*2035^y*15^w*10^z == 5,
   8874^x*1945^y*15^w*10^z == 23,
   9918^x*1949^y*15^w*10^z == 17};
toLog = Map[Log, eqs, {2}] /. 
   HoldPattern[Log[Times[x__]]] :> Plus @@ Map[Log, {x}] /. 
  Log[a_^b_] :> b*Log[a]
{b, mat} = CoefficientArrays[toLog, {w, x, y, z}] // Normal
mat // Simplify // MatrixForm
Det[mat]

POSTED BY: Gianluca Gorni

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