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Solving system of implicit functions: solution satisfies one equation

Guangye Cao

Posted 4 years ago

Attachments: supply_demand.nb

POSTED BY: Guangye Cao

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Rohit Namjoshi

Posted 4 years ago

Hi Guangye, By default plots in WL exclude singularities. In this case, they must occur where some denominator in `eq1 == 0`. Override that. ContourPlot[eq1 == 0, {T, -5.025(10^6), 0}, {p, -0.1, 0.4}, Exclusions -> None, PlotPoints -> 50] The behavior of the function in that region is quite erratic. Plot3D[eq1, {T, -5.02510^6, -4*10^6}, {p, -0.1, 0.15}, PlotRange -> All, Exclusions -> None, ColorFunction -> "TemperatureMap", ImageSize -> 600]

Hi Guangye,

By default plots in WL exclude singularities. In this case, they must occur where some denominator in eq1 == 0. Override that.

ContourPlot[eq1 == 0, {T, -5.025*(10^6), 0}, {p, -0.1, 0.4},
  Exclusions -> None, PlotPoints -> 50]

enter image description here

The behavior of the function in that region is quite erratic.

Plot3D[eq1, {T, -5.025*10^6, -4*10^6}, {p, -0.1, 0.15},
 PlotRange -> All,
 Exclusions -> None,
 ColorFunction -> "TemperatureMap",
 ImageSize -> 600]

enter image description here

POSTED BY: Rohit Namjoshi

Rohit Namjoshi

Posted 4 years ago

Hi Guangye, Define a symbol for the first equation eq1 = p - \[Phi]/ R + ((1 - \[Phi]) \[Phi] (1 - \[Lambda]) \[Beta])/(((W - p T) R + \[Phi] T) R + \[Phi] (1 - \[Phi]) T)/(R (\ \[Lambda]/((W - p T) R + \[Phi] T) + ((1 - \[Lambda]) \[Beta] R)/(((W - p T) R + \[Phi] T) R + \[Phi] (1 - \[Phi]) T))) sol = Solve[eq1 == 0 && p == A/T, {p, T}] (* {{p -> -2.3731310^-8, T -> -4.2138410^9}, {p -> 0.0607204, T -> 1646.89}} ) Verify. For `eq1` both results are zero to within machine precision. For `A / T` the result is the two `p` values from the solution. eq1 /. sol ( {5.5511210^-17, 0.} ) A / T /. sol (* {-2.3731310^-8, 0.0607204} ) For an exact solution eq1r = Rationalize@eq1 solr = Solve[eq1r == 0 && p == A/T, {p, T}] Verify as before.

Hi Guangye,

Define a symbol for the first equation

eq1 = p - \[Phi]/
   R + ((1 - \[Phi]) \[Phi] (1 - \[Lambda]) \[Beta])/(((W - 
            p T) R + \[Phi] T) R + \[Phi] (1 - \[Phi]) T)/(R (\
\[Lambda]/((W - 
             p T) R + \[Phi] T) + ((1 - \[Lambda]) \[Beta] R)/(((W - 
                p T) R + \[Phi] T) R + \[Phi] (1 - \[Phi]) T)))

sol = Solve[eq1 == 0 && p == A/T, {p, T}]
(* {{p -> -2.37313*10^-8, T -> -4.21384*10^9}, {p -> 0.0607204, T -> 1646.89}} *)

Verify. For eq1 both results are zero to within machine precision. For A / T the result is the two p values from the solution.

eq1 /. sol
(* {5.55112*10^-17, 0.} *)

A / T /. sol
(* {-2.37313*10^-8, 0.0607204} *)

For an exact solution

eq1r = Rationalize@eq1
solr = Solve[eq1r == 0 && p == A/T, {p, T}]

Verify as before.

POSTED BY: Rohit Namjoshi

Guangye Cao

Posted 4 years ago

That works, many thanks!

POSTED BY: Guangye Cao

Guangye Cao

Posted 4 years ago

POSTED BY: Guangye Cao

Rohit Namjoshi

Posted 4 years ago

Works fine for me. Try restarting the kernel. Maybe this is the problem `Clear[T]`, not `clear[T]`. WL is case-sensitive.

POSTED BY: Rohit Namjoshi

Guangye Cao

Posted 4 years ago

POSTED BY: Guangye Cao

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