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Problems in getting Real number solutions in NSolve

hwoarang Polar

Posted 11 years ago

Hello, Wolfram, Simply speaking, the purpose of this small programming is to get the Real values of An and Ak. I think that this is a very simple equation to solve, but I have no idea why NSolve does not give any Real solution (even Imaginary solution). Could you help me ? I pasted the context of the programming below. Thank you in advance, n = 2.48; k = 4.38; (* dn/d\[Eta], dk/d\[Eta] ) \[Epsilon] = (n + I k)^2; \[Eta] = -5 10^-3; d\[Epsilon][An_, Ak_] := 2 (n + I k) (An + I Ak) \[Eta]; rp[\[Theta]_] := (\[Epsilon] Cos[\[Theta]] - (\[Epsilon] - Sin[\[Theta]]^2)^0.5)/(\[Epsilon] Cos[\[Theta]] + (\[Epsilon] - Sin[\[Theta]]^2)^0.5); drp[\[Theta]_, An_, Ak_] := ((d\[Epsilon][An, Ak] Cos[\[Theta]] - d\[Epsilon][An, Ak]/(2 (\[Epsilon] - Sin[\[Theta]]^2)^0.5)) (\[Epsilon] Cos[\[Theta]] + (\ \[Epsilon] - Sin[\[Theta]]^2)^0.5) - (\[Epsilon] Cos[\[Theta]] - (\ \[Epsilon] - Sin[\[Theta]]^2)^0.5) (d\[Epsilon][An, Ak] Cos[\[Theta]] + d\[Epsilon][An, Ak]/(2 (\[Epsilon] - Sin[\[Theta]]^2)^0.5)))/(\[Epsilon] Cos[\[Theta]] + (\ \[Epsilon] - Sin[\[Theta]]^2)^0.5)^2; NSolve[{2 Re[drp[10.0 \[Pi]/180, An, Ak]/rp[10.0 \[Pi]/180]] == 0.001509, 2 Re[drp[60.0 \[Pi]/180, An, Ak]/rp[60.0 \[Pi]/180]] == 0.00287}, {An, Ak}] Attachments:*

Hello, Wolfram,

Simply speaking, the purpose of this small programming is to get the Real values of An and Ak. I think that this is a very simple equation to solve, but I have no idea why NSolve does not give any Real solution (even Imaginary solution). Could you help me ? I pasted the context of the programming below.

Thank you in advance,

n = 2.48; k = 4.38; (* dn/d\[Eta], dk/d\[Eta] *) \[Epsilon] = (n + 
   I k)^2; \[Eta] = -5 10^-3; 
d\[Epsilon][An_, Ak_] := 2 (n + I k) (An + I Ak) \[Eta];
rp[\[Theta]_] := (\[Epsilon] Cos[\[Theta]] - (\[Epsilon] - 
       Sin[\[Theta]]^2)^0.5)/(\[Epsilon] Cos[\[Theta]] + (\[Epsilon] -
        Sin[\[Theta]]^2)^0.5);
drp[\[Theta]_, An_, 
   Ak_] := ((d\[Epsilon][An, Ak] Cos[\[Theta]] - 
        d\[Epsilon][An, 
          Ak]/(2 (\[Epsilon] - 
             Sin[\[Theta]]^2)^0.5)) (\[Epsilon] Cos[\[Theta]] + (\
\[Epsilon] - 
          Sin[\[Theta]]^2)^0.5) - (\[Epsilon] Cos[\[Theta]] - (\
\[Epsilon] - Sin[\[Theta]]^2)^0.5) (d\[Epsilon][An, 
          Ak] Cos[\[Theta]] + 
        d\[Epsilon][An, 
          Ak]/(2 (\[Epsilon] - 
             Sin[\[Theta]]^2)^0.5)))/(\[Epsilon] Cos[\[Theta]] + (\
\[Epsilon] - Sin[\[Theta]]^2)^0.5)^2;
NSolve[{2 Re[drp[10.0 \[Pi]/180, An, Ak]/rp[10.0 \[Pi]/180]] == 
   0.001509, 
  2 Re[drp[60.0 \[Pi]/180, An, Ak]/rp[60.0 \[Pi]/180]] == 
   0.00287}, {An, Ak}]

POSTED BY: hwoarang Polar

4 Replies

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Daniel Lichtblau

Daniel Lichtblau, Wolfram Research

Posted 11 years ago

POSTED BY: Daniel Lichtblau

hwoarang Polar

Posted 11 years ago

Thank you soooo much. It really works.

POSTED BY: hwoarang Polar

hwoarang Polar

Posted 11 years ago

POSTED BY: hwoarang Polar

Bill Simpson

Posted 11 years ago

A number of small, but significant changes In[1]:= n = 2.48; k = 4.38;(dn/d[Eta],dk/d[Eta]) \[Epsilon] = (n + I k)^2; \[Eta] = -5 10^-3; d\[Epsilon][An_, Ak_] := 2 (n + I k) (An + I Ak) \[Eta]; rp[\[Theta]_] := (\[Epsilon] Cos[\[Theta]] - Sqrt[\[Epsilon] - Sin[\[Theta]]^2])/(\[Epsilon] Cos[\[Theta]] + Sqrt[\[Epsilon] - Sin[\[Theta]]^2]); drp[\[Theta]_, An_, Ak_] := ((d\[Epsilon][An, Ak] Cos[\[Theta]] - d\[Epsilon][An, Ak]/(2 Sqrt[\[Epsilon] - Sin[\[Theta]]^2])) (\[Epsilon] Cos[\[Theta]] + Sqrt[\[Epsilon] - Sin[\[Theta]]^2]) - (\[Epsilon] Cos[\[Theta]] - Sqrt[\[Epsilon] - Sin[\[Theta]]^2]) (d\[Epsilon][An, Ak] Cos[\[Theta]] + d\[Epsilon][An, Ak]/(2 Sqrt[\[Epsilon] - Sin[\[Theta]]^2])))/(\[Epsilon] Cos[\[Theta]] + Sqrt[\[Epsilon] - Sin[\[Theta]]^2])^2; sol = Simplify[NSolve[{ 2 Re[drp[10.0 \[Pi]/180, An, Ak]/rp[10.0 \[Pi]/180]] == 0.001509, 2 Re[drp[60.0 \[Pi]/180, An, Ak]/rp[60.0 \[Pi]/180]] == 0.00287}, {An, Ak}]] During evaluation of In[1]:= NSolve::svars: Equations may not give solutions for all "solve" variables. >> During evaluation of In[1]:= NSolve::ratnz: NSolve was unable to solve the system with inexact coefficients. The answer was obtained by solving a corresponding exact system and numericizing the result. >> Out[3]= {{Ak -> (-1.48616 - 1.24023 I) - (1. - 1.9618310^-16 I) Im[An] + (0. + 1. I) Re[An]}} (thus it appears*) Ak == -1.48616 - 1.24023 I + I An

A number of small, but significant changes

In[1]:= n = 2.48; k = 4.38;(*dn/d[Eta],dk/d[Eta]*)
\[Epsilon] = (n + I k)^2; \[Eta] = -5 10^-3;
d\[Epsilon][An_, Ak_] := 2 (n + I k) (An + I Ak) \[Eta]; 
rp[\[Theta]_] := (\[Epsilon] Cos[\[Theta]] - Sqrt[\[Epsilon] - Sin[\[Theta]]^2])/(\[Epsilon] Cos[\[Theta]] +
Sqrt[\[Epsilon] - Sin[\[Theta]]^2]); 
drp[\[Theta]_, An_, Ak_] := ((d\[Epsilon][An, Ak] Cos[\[Theta]] - d\[Epsilon][An, Ak]/(2 Sqrt[\[Epsilon] -
Sin[\[Theta]]^2])) (\[Epsilon] Cos[\[Theta]] + Sqrt[\[Epsilon] - Sin[\[Theta]]^2]) - (\[Epsilon] Cos[\[Theta]] -
Sqrt[\[Epsilon] - Sin[\[Theta]]^2]) (d\[Epsilon][An, Ak] Cos[\[Theta]] + d\[Epsilon][An, Ak]/(2 Sqrt[\[Epsilon] -
Sin[\[Theta]]^2])))/(\[Epsilon] Cos[\[Theta]] + Sqrt[\[Epsilon] - Sin[\[Theta]]^2])^2;
sol = Simplify[NSolve[{
    2 Re[drp[10.0 \[Pi]/180, An, Ak]/rp[10.0 \[Pi]/180]] == 0.001509, 
    2 Re[drp[60.0 \[Pi]/180, An, Ak]/rp[60.0 \[Pi]/180]] == 0.00287}, {An, Ak}]]

During evaluation of In[1]:= NSolve::svars: Equations may not give solutions for all "solve" variables. >>

During evaluation of In[1]:= NSolve::ratnz: NSolve was unable to solve the system with inexact coefficients. The answer was
obtained by solving a corresponding exact system and numericizing the result. >>

Out[3]= {{Ak -> (-1.48616 - 1.24023 I) - (1. - 1.96183*10^-16 I) Im[An] + (0. + 1. I) Re[An]}}

(*thus it appears*)
Ak == -1.48616 - 1.24023 I + I An

POSTED BY: Bill Simpson

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