Solving equations

We have a critical equation

eqn=exp(-Ki*c)*(sin(c*k)*(sin(b*k)*(1/2*exp(Ki*b)*(exp(-Ki*a)*(1/2*cos(a*k) - 1/2/Ki*k*mb_star/mw_star*sin(a*k)) + Ki/mb_star*exp(-Ki*a)*(1/2/Ki*mb_star*cos(a*k) + 1/2/k*mw_star*sin(a*k))) + 1/2*exp(-Ki*b)*(exp(Ki*a)*(1/2*cos(a*k) - 1/2/Ki*k*mb_star/mw_star*sin(a*k)) - Ki/mb_star*exp(Ki*a)*(1/2/Ki*mb_star*cos(a*k) + 1/2/k*mw_star*sin(a*k)))) - k/mw_star*cos(b*k)*(1/2/Ki*mb_star*exp(Ki*b)*(exp(-Ki*a)*(1/2*cos(a*k) - 1/2/Ki*k*mb_star/mw_star*sin(a*k)) + Ki/mb_star*exp(-Ki*a)*(1/2/Ki*mb_star*cos(a*k) + 1/2/k*mw_star*sin(a*k))) - 1/2/Ki*mb_star*exp(-Ki*b)*(exp(Ki*a)*(1/2*cos(a*k) - 1/2/Ki*k*mb_star/mw_star*sin(a*k)) - Ki/mb_star*exp(Ki*a)*(1/2/Ki*mb_star*cos(a*k) + 1/2/k*mw_star*sin(a*k))))) + cos(c*k)*(cos(b*k)*(1/2*exp(Ki*b)*(exp(-Ki*a)*(1/2*cos(a*k) - 1/2/Ki*k*mb_star/mw_star*sin(a*k)) + Ki/mb_star*exp(-Ki*a)*(1/2/Ki*mb_star*cos(a*k) + 1/2/k*mw_star*sin(a*k))) + 1/2*exp(-Ki*b)*(exp(Ki*a)*(1/2*cos(a*k) - 1/2/Ki*k*mb_star/mw_star*sin(a*k)) - Ki/mb_star*exp(Ki*a)*(1/2/Ki*mb_star*cos(a*k) + 1/2/k*mw_star*sin(a*k)))) + k/mw_star*sin(b*k)*(1/2/Ki*mb_star*exp(Ki*b)*(exp(-Ki*a)*(1/2*cos(a*k) - 1/2/Ki*k*mb_star/mw_star*sin(a*k)) + Ki/mb_star*exp(-Ki*a)*(1/2/Ki*mb_star*cos(a*k) + 1/2/k*mw_star*sin(a*k))) - 1/2/Ki*mb_star*exp(-Ki*b)*(exp(Ki*a)*(1/2*cos(a*k) - 1/2/Ki*k*mb_star/mw_star*sin(a*k)) - Ki/mb_star*exp(Ki*a)*(1/2/Ki*mb_star*cos(a*k) + 1/2/k*mw_star*sin(a*k)))))) - Ki/mb_star*exp(-Ki*c)*(1/k*mw_star*cos(c*k)*(sin(b*k)*(1/2*exp(Ki*b)*(exp(-Ki*a)*(1/2*cos(a*k) - 1/2/Ki*k*mb_star/mw_star*sin(a*k)) + Ki/mb_star*exp(-Ki*a)*(1/2/Ki*mb_star*cos(a*k) + 1/2/k*mw_star*sin(a*k))) + 1/2*exp(-Ki*b)*(exp(Ki*a)*(1/2*cos(a*k) - 1/2/Ki*k*mb_star/mw_star*sin(a*k)) - Ki/mb_star*exp(Ki*a)*(1/2/Ki*mb_star*cos(a*k) + 1/2/k*mw_star*sin(a*k)))) - k/mw_star*cos(b*k)*(1/2/Ki*mb_star*exp(Ki*b)*(exp(-Ki*a)*(1/2*cos(a*k) - 1/2/Ki*k*mb_star/mw_star*sin(a*k)) + Ki/mb_star*exp(-Ki*a)*(1/2/Ki*mb_star*cos(a*k) + 1/2/k*mw_star*sin(a*k))) - 1/2/Ki*mb_star*exp(-Ki*b)*(exp(Ki*a)*(1/2*cos(a*k) - 1/2/Ki*k*mb_star/mw_star*sin(a*k)) - Ki/mb_star*exp(Ki*a)*(1/2/Ki*mb_star*cos(a*k) + 1/2/k*mw_star*sin(a*k))))) - 1/k*mw_star*sin(c*k)*(cos(b*k)*(1/2*exp(Ki*b)*(exp(-Ki*a)*(1/2*cos(a*k) - 1/2/Ki*k*mb_star/mw_star*sin(a*k)) + Ki/mb_star*exp(-Ki*a)*(1/2/Ki*mb_star*cos(a*k) + 1/2/k*mw_star*sin(a*k))) + 1/2*exp(-Ki*b)*(exp(Ki*a)*(1/2*cos(a*k) - 1/2/Ki*k*mb_star/mw_star*sin(a*k)) - Ki/mb_star*exp(Ki*a)*(1/2/Ki*mb_star*cos(a*k) + 1/2/k*mw_star*sin(a*k)))) + k/mw_star*sin(b*k)*(1/2/Ki*mb_star*exp(Ki*b)*(exp(-Ki*a)*(1/2*cos(a*k) - 1/2/Ki*k*mb_star/mw_star*sin(a*k)) + Ki/mb_star*exp(-Ki*a)*(1/2/Ki*mb_star*cos(a*k) + 1/2/k*mw_star*sin(a*k))) - 1/2/Ki*mb_star*exp(-Ki*b)*(exp(Ki*a)*(1/2*cos(a*k) - 1/2/Ki*k*mb_star/mw_star*sin(a*k)) - Ki/mb_star*exp(Ki*a)*(1/2/Ki*mb_star*cos(a*k) + 1/2/k*mw_star*sin(a*k))))))

the value of k is ((2*mw_star*E)^0.5)/hcut;

the value of Ki is ((2*mb_star*(v-E))^0.5)/hcut;

this equation is generated by consecutive multiplication of 8 matrices. Here all the variables are known, except E and b. I need to solve this equation and plot E versus b. Please help..

[Grzegorz Knor]

Can you read your post? :)

http://www.mathworks.com/matlabcentral/answers/help/markup

[Mitch Martelli]

So you have one equation and two unknown variables?

[Walter Roberson]

The equation compacts substantially if you do trig combinations and simplify it. Unfortunately solving for E is taking a long time, and I will not be able to copy in the compacted version until that stops running.

[Walter Roberson]

Question: is eqn a known value, or should the right hand side be solved assuming that the expression value is 0 ?

[Rajat]

actually i got this equation by multiplying 8 matrices, and this is the 4th coefficient. i set this equation as eqn for simplicity, eqn is not a known value u know.. i am assuming that the expression value is 0. I want to solve this equation for b and try to get a value which contains the value E. and then i want to plot b vs E. that's what i was thinking. but i find this pretty hard.

[Rajat]

by the way the known values are mw_star=0.067*9.11*10^-31

mb_star=2*9.11*10^-31, hcut=6.58e-16,

a=1*10^-09, c=3*10^-09

May be my approach is wrong, but i am trying it. If u guys have any better idea then share it

[Walter Roberson]

Does v have a known value?

[Walter Roberson]

This is pretty messy to solve when I guess at random values of E. It gets in to roots of polynomials in E and tan() and sin() and cos() of other polynomials in E in combination with v, and easily gets in to complex expressions. It also needs a fair bit of accuracy to deal with: some of the expressions cancel each other out to within the default floating point accuracy, requiring one to push up the digits in order to not end up with the expression mostly evaluating to r + i*infinity or infinity + i*infinity. A known v value would help confine the search a fair bit, as would information about whether we should expect E to be non-negative or real-valued (including negative), or complex valued. Are we intended to be seeking for one complex plot versus another?

[Rajat]

yep, v have a known value.. assuming v has the value 1*10^-04

Answer 1

[Walter Roberson]

Not a solution, but a more compact version:

eqn = (1 / 8) * (( - cos(2^(1 / 2) * (mw_star * E)^(1 / 2) * (c - b + a) / hcut) + cos(2^(1 / 2) * (mw_star * E)^(1 / 2) * ( - c + b + a) / hcut)) * ((mb_star - mw_star) * E + mw_star * v)^2 * (mw_star * E)^(1 / 2) * ( - mb_star * ( - v + E))^(1 / 2) * exp(( - c - b + a) * ( - 2 * mb_star * ( - v + E))^(1 / 2) / hcut) - 4 * exp( - (c - b + a) * ( - 2 * mb_star * ( - v + E))^(1 / 2) / hcut) * ((1 / 4) * ((mb_star - mw_star) * E + mw_star * v)^2 * (mw_star * E)^(1 / 2) * ( - mb_star * ( - v + E))^(1 / 2) * cos(2^(1 / 2) * (mw_star * E)^(1 / 2) * ( - c + b + a) / hcut) - (1 / 4) * ((mb_star^2 + mw_star^2 + 6 * mb_star * mw_star) * E^2 - 6 * mw_star * ((1 / 3) * mw_star + mb_star) * v * E + mw_star^2 * v^2) * (mw_star * E)^(1 / 2) * ( - mb_star * ( - v + E))^(1 / 2) * cos(2^(1 / 2) * (mw_star * E)^(1 / 2) * (c - b + a) / hcut) + ((mb_star + mw_star) * E - mw_star * v) * mw_star * ( - v + E) * mb_star * E * sin(2^(1 / 2) * (mw_star * E)^(1 / 2) * (c - b + a) / hcut))) / ((mb_star * v - mb_star * E)^(1 / 2) * (mw_star * E)^(1 / 2) * mw_star * ( - v + E) * mb_star * E)

[Walter Roberson]

Maple says that solving this for b is too hard for it.

[Walter Roberson]

Odd, if I take the right hand side of this (i.e., solve assuming the expression is 0) and ask to solve it for E, then the answer Maple gives back is E, which is to say that Maple thinks that all values of E (except perhaps those that introduce singularities) lead to an expression value of 0. I have played a little and that doesn't seem plausible so far (but I haven't proven it to be incorrect.)

[Walter Roberson]

With the constants given above, if one substitutes in negative E such as E=-2, then one can do numeric solutions to find b. Finding analytic solutions is difficult, though, and is rather meaningless considering that most of the constants are expressed in floating point format rather than in analytic format.

As E approaches 0 from below, b approaches 2057.888774 -- which you would not be able to establish with as little as 50 digits of precision.

The behaviour as E gets more negative takes longer to determine.