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Equation Solving Wolfram Language

Great equation to solve

Aleksey Ovinnikov

Posted 8 years ago

I try to solve equation in Mathematica 11: Solve[1 - Integrate[exp[0.5 (x - b)^2] log2[exp[-2 bx] + 1], {x, -5, 20}]/(sqrt[2pi] log[2]) == 0, b] by have a error Solve::inex: Solve was unable to solve the system with inexact coefficients or the system obtained by direct rationalization of inexact numbers present in the system. Since many of the methods used by Solve require exact input, providing Solve with an exact version of the system may help. What should I do to correct it?

POSTED BY: Aleksey Ovinnikov

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Mariusz Iwaniuk

Mariusz Iwaniuk, engineer, math hobbyist.

Posted 8 years ago

You did a mistakes in $e^{\frac{1}{2} (y-b)^2}$,it should be:$e^{-\frac{1}{2} (y-b)^2}$ Int[b_?NumericQ] := NIntegrate[Exp[-1/2(y - b)^2]Log[2, Exp[-2by] + 1], {y, -5, 20}, Method -> "LobattoKronrodRule"]; ListPlot[Table[{b, (1 - Int[b]/(Sqrt[2Pi]Log[2]))}, {b, -3, 3, 1/10}], Joined -> True,PlotLegends -> {"eq(b)"}] // Quiet FindRoot[(1 - Int[b]/(Sqrt[2Pi]Log[2])) == 0, {b, 7/10}, WorkingPrecision -> 20] ({b -> 0.73171520950283916124}) FindRoot[(1 - Int[b]/(Sqrt[2Pi]Log[2])) == 0, {b, -7/10}, WorkingPrecision -> 20] ({b -> -0.73171520950283916124}) Next time give us a correct equation.

You did a mistakes in $e^{\frac{1}{2} (y-b)^2}$,it should be:$e^{-\frac{1}{2} (y-b)^2}$

Int[b_?NumericQ] := NIntegrate[Exp[-1/2*(y - b)^2]*Log[2, Exp[-2*b*y] + 1], {y, -5, 20}, Method -> "LobattoKronrodRule"];
ListPlot[Table[{b, (1 - Int[b]/(Sqrt[2*Pi]*Log[2]))}, {b, -3, 3, 1/10}], Joined -> True,PlotLegends -> {"eq(b)"}] // Quiet

enter image description here

 FindRoot[(1 - Int[b]/(Sqrt[2*Pi]*Log[2])) == 0, {b, 7/10}, WorkingPrecision -> 20]
 (*{b -> 0.73171520950283916124}*)
 FindRoot[(1 - Int[b]/(Sqrt[2*Pi]*Log[2])) == 0, {b, -7/10}, WorkingPrecision -> 20]
 (*{b -> -0.73171520950283916124}*)

Next time give us a correct equation.

POSTED BY: Mariusz Iwaniuk

Mariusz Iwaniuk

Mariusz Iwaniuk, engineer, math hobbyist.

Posted 8 years ago

You did a lot syntax mistakes, it should be: Solve[1 - Integrate[Exp[1/2(x - b)^2]Log[2, Exp[-2bx] + 1], {x, -5, 20}]/(Sqrt[2Pi]Log[2]) == 0, b] but MMA can't find symbolic solution to this integral: Integrate[Exp[1/2(x - b)^2]Log[2, Exp[-2bx] + 1], {x, -5, 20}] You may find it numerically: Int[b_?NumericQ] := NIntegrate[Exp[1/2(x - b)^2]Log[2, Exp[-2bx] + 1], {x, -5, 20}, Method -> "LobattoKronrodRule"]; and plot equation: ListLogLogPlot[Table[{b, -(1 - Int[b]/(Sqrt[2 Pi]*Log[2]))}, {b, -2, 10, 1/1000}], Joined -> True] // Quiet for $b\in \mathbb{R}$ is no solution.

You did a lot syntax mistakes, it should be:

Solve[1 - Integrate[Exp[1/2*(x - b)^2]*Log[2, Exp[-2*b*x] + 1], {x, -5, 20}]/(Sqrt[2*Pi]*Log[2]) == 0, b]

but MMA can't find symbolic solution to this integral:

Integrate[Exp[1/2*(x - b)^2]*Log[2, Exp[-2*b*x] + 1], {x, -5, 20}]

You may find it numerically:

 Int[b_?NumericQ] := NIntegrate[Exp[1/2*(x - b)^2]*Log[2, Exp[-2*b*x] + 1], {x, -5, 20}, Method -> "LobattoKronrodRule"];

and plot equation:

 ListLogLogPlot[Table[{b, -(1 - Int[b]/(Sqrt[2 Pi]*Log[2]))}, {b, -2, 10, 1/1000}], Joined -> True] // Quiet

enter image description here

for $b\in \mathbb{R}$ is no solution.

POSTED BY: Mariusz Iwaniuk

Frank Kampas

Frank Kampas, Physicist at Large Consulting

Posted 8 years ago

Changing the 0.5 to 1/2 will get rid of the inexact number message. However, when you do that you'll get a message that Solve can't handle the problem.

POSTED BY: Frank Kampas

Daniel Lichtblau

Daniel Lichtblau, Wolfram Research

Posted 8 years ago

The Wolfram Language is case sensitive.

POSTED BY: Daniel Lichtblau

Aleksey Ovinnikov

Posted 8 years ago

This equation evaluates the performance bound on error correcting codes from this paper: http://www.clemson.edu/sure/projects/jpursley_secon01.pdf (in the attachment) Equation (1), (6) and Figure 4 : Capacity of the Binary-Input Channel. Mariusz Iwaniuk, your result is far away from the Figure 4 plot. Are there any other way to calcucate equation? Attachments:

POSTED BY: Aleksey Ovinnikov

Aleksey Ovinnikov

Posted 8 years ago

Mariusz Iwaniuk, thanks for help. I apologize for the mistake committed by me. I'm doing my first steps in Mathematica. I try to plot figure from function: `1-NIntegrate[Exp[-1/2(y - Sqrt[2ebn0R])^2]Log[2, Exp[-2Sqrt[2ebn0R]y] + 1], {y, -5, 20}, Method -> "LobattoKronrodRule"]/(Sqrt[2Pi]Log[2])) == R;` I need to plot ebn0(R), but I don't know how can I do this. This plot should be equivalent to Figure 4 : Capacity of the Binary-Input Channel.`

POSTED BY: Aleksey Ovinnikov

EDITORIAL BOARD

EDITORIAL BOARD, WOLFRAM

Posted 8 years ago

Welcome to Wolfram Community! Please make sure you know the rules and how to format your code properly, which you can find here: https://wolfr.am/READ-1ST If you do not format code, it may become corrupted and useless to other members. Please EDIT your posts and make sure code blocks start on a new paragraph and look framed and colored like this. int = Integrate[1/(x^3 - 1), x]; Map[Framed, int, Infinity]

POSTED BY: EDITORIAL BOARD

Mariusz Iwaniuk

Mariusz Iwaniuk, engineer, math hobbyist.

Posted 8 years ago

I had to remove $log(2)$ from the denominator from equation and I must to replace the expression $Eb/N0$ with $10^{\text{Eb/N0}/10}$ in order for the plot to be correct . Int2[R_?NumericQ, ebNo_?NumericQ] := NIntegrate[Exp[-1/2(y - Sqrt[2ebNoR])^2] Log[2, Exp[-2Sqrt[2ebNoR]y] + 1], {y, -Infinity, Infinity}, Method -> "LobattoKronrodRule"]; ContourPlot[R == 1 - Int2[R, 10^(ebNo/10)]/Sqrt[2Pi], {R, 0.001, 1}, {ebNo, -2, 10}, Axes -> True, GridLines -> {Table[n, {n, 0, 1, 0.1}], Table[n, {n, -2, 10, 2}]}, ContourStyle -> {Black, Thick}, FrameTicks -> {{All, None}, {Table[n, {n, 0, 1, 0.1}], None}}, FrameLabel -> {"R", "Eb/N0"}] Have a nice day :)

I had to remove $log(2)$ from the denominator from equation and I must to replace the expression $Eb/N0$ with $10^{\text{Eb/N0}/10}$ in order for the plot to be correct .

     Int2[R_?NumericQ, ebNo_?NumericQ] := NIntegrate[Exp[-1/2*(y - Sqrt[2*ebNo*R])^2] 
     Log[2, Exp[-2*Sqrt[2*ebNo*R]*y] + 1], {y, -Infinity, Infinity}, Method -> "LobattoKronrodRule"];

     ContourPlot[R == 1 - Int2[R, 10^(ebNo/10)]/Sqrt[2*Pi], {R, 0.001, 1}, {ebNo, -2, 10}, Axes -> True, 
     GridLines -> {Table[n, {n, 0, 1, 0.1}], Table[n, {n, -2, 10, 2}]}, ContourStyle -> {Black, Thick}, 
     FrameTicks -> {{All, None}, {Table[n, {n, 0, 1, 0.1}], None}}, FrameLabel -> {"R", "Eb/N0"}]

enter image description here Have a nice day :)

POSTED BY: Mariusz Iwaniuk

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