Solve Y=1/(1+e^-(X(X-1)/2-Z))for X.

Hi waddington!
This equation can be solved in the following way. I'll assume here
that the -Z is not part of the denominator of the fraction in the
exponent, so your equation would look like this:


                1
Y = --------------------------
     1 + exp( -X(X-1)/2 ) - Z

We start by taking the reciprocal,

1/Y = 1 + exp( -X(X-1)/2 ) - Z

Substract (1-Z),

1/Y - 1 + Z = exp( -X(X-1)/2 )

Take natural logarithm,

ln( 1/Y - 1 + Z ) = -X(X-1)/2

Multiply by -2,

-2ln( 1/Y - 1 + Z ) = X(X-1) = X^2 - X

And finally, setting C = 2ln( 1/Y - 1 + Z ), and adding C,

X^2 - X + C = 0

This equation can be solved using the quadratic equation formula, so
we get the following two possible solutions for X:

X1 = ( 1 + sqrt(1 - 4C) )/2
X2 = ( 1 - sqrt(1 - 4C) )/2

where C=2ln(1/Y - 1 + Z). Bear in mind that, if Y and Z are constants,
they should be set such that

1/Y - 1 + Z > 0

otherwise, the equation has no solution.


I hope this helps!
Best wishes,
elmarto

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Request for Answer Clarification by waddington-ga on 11 Mar 2005 09:12 PST

I see I didn't write it clearly what I had as:
               1
Y = --------------------------
     1 + exp( -X(X-1)/2 ) - Z

should have been
               1
Y = --------------------------
     1 + exp((-X(X-1)/2)+Z)

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Clarification of Answer by elmarto-ga on 11 Mar 2005 09:19 PST

Hi waddington,
While the correct equation is a bit different, the procedure for
solving it is much the same. I'll copy my original answer here and
update the equations when appropiate:

We start by taking the reciprocal,

1/Y = 1 + exp( -X(X-1)/2 + Z )

Substract 1,

1/Y - 1 = exp( -X(X-1)/2 + Z )

Take natural logarithm,

ln( 1/Y - 1 ) = -X(X-1)/2 + Z

Subtract Z,

ln( 1/Y - 1 ) - Z = -X(X-1)/2

Multiply by -2,

-2(ln(1/Y - 1) - Z) = X(X-1) = X^2 - X

And finally, setting C = 2(ln(1/Y - 1) - Z) , and adding C,

X^2 - X + C = 0

This equation can be solved using the quadratic equation formula, so
we get the following two possible solutions for X:

X1 = ( 1 + sqrt(1 - 4C) )/2
X2 = ( 1 - sqrt(1 - 4C) )/2

where C=2(ln(1/Y - 1) - Z). Bear in mind that, if Y and Z are constants,
they should be set such that

1/Y - 1 > 0

otherwise, the equation has no solution.


Best regards,
elmarto

Thank you.

Reciprocal, subtract one, add natural log, and multiply by -1 puts us here:

-(ln(1/Y)+1)=X(X-1)/2-Z

Multiply out and Multiply by (2-Z) puts us here:

-(ln(1/Y)+1)(2-Z)=x^2-x

Anyone else want to finish?

Thank you for the tip!