How to solve an equation contains norminv?

Hi all,

I'd like to solve the following equation.

sol = [];
for L = -20:10:120
    syms x
    eqn = 10*log10(10.^(0.1*(norminv(x)*7.6 + 9.88)) + 10.^(0.1*(norminv(x)*6.5 + 12.1)) + 10.^(-0.3)) == L
    S = solve(eqn);
    sol = [sol S];
end

I modified the equation by replacing the norminv (x) with -sqrt(2)*erfcinv(2*x), i.e.,

sol = [];
for L = -20:10:120
    syms x
    eqn = 10*log10(10.^(0.1*(-sqrt(2)*erfcinv(2*x)*7.6 + 9.88)) + 10.^(0.1*(-sqrt(2)*erfcinv(2*x)*6.5 + 12.1)) + 10.^(-0.3)) == L
    S = solve(eqn);
    sol = [sol S];
end

However, I got a warning saying, "Warning: Unable to find explicit solution. For options, see help.". So, I add the following lines, hoping to solve the equation, but still unsuccessful.

S = solve(eqn);
string(lhs(eqn)-rhs(eqn))
sol = [sol vpasolve(eqn, [-143, -109.5937])]

Could someone please help me out?

Thanks in advance

Answer 1

[Walter Roberson]

syms x
eqn = 10*log10(10.^(0.1*(-sqrt(2)*erfcinv(2*x)*7.6 + 9.88)) + 10.^(0.1*(-sqrt(2)*erfcinv(2*x)*6.5 + 12.1)) + 10.^(-0.3))

eqn = 

double(limit(eqn, x, 1e-200))

ans = -3.0000

double(limit(eqn, x, 1-sym(1e-50)))

ans = 123.5352

The left hand side is the same for all of them, and you cannot get it down below -3 and cannot get it much above 120-ish. So there is no real-valued solution for L = -20 or L = -10

For the other values... use vpasolve() with around 0.01 as an initial guess, and after that use the previous solution as the initial guess for the next round.

[Susan]

@Walter Roberson Thank you so much for your response. I modified the code as follows

sol = [];
InitialGuess = 0.01;
for L = -2:2:123
    syms x
    eqn = 10*log10(10.^(0.1*(-sqrt(2)*erfcinv(2*x)*7.6 + 9.88)) + 10.^(0.1*(-sqrt(2)*erfcinv(2*x)*6.5 + 12.1)) + 10.^(-0.3)) == L
    sol = [sol vpasolve(eqn, InitialGuess)];
    InitialGuess = sol;
end
figure;
cdfplot(sol)
grid on

but I get the error, "Starting points or search ranges are incompatible with the variables." Could you please point out what I'm doing wrongly?

[Walter Roberson]

sol = [];
InitialGuess = 0.01;
syms x
expr = 10*log10(10.^(0.1*(-sqrt(2)*erfcinv(2*x)*7.6 + 9.88)) + 10.^(0.1*(-sqrt(2)*erfcinv(2*x)*6.5 + 12.1)) + 10.^(-0.3));

for L = -2:2:123
    thissol = vpasolve(expr == L, InitialGuess);
    if isempty(thissol)
        sol(end+1) = NaN;
    else
        sol(end+1) = double(thissol);
        InitialGuess = thissol;
    end
end
figure;
cdfplot(sol)
grid on

[Susan]

@Walter Roberson thank you so much for your help. Appreciate it!

Sorry if I'm asking about the obvious, but could you please tell me how I can plot the cdf of ''sol" values versus the L? (x-axis are L values)

[Torsten]

I think the sol-values themselves are a cdf, aren't they ?

syms x L
eqn = 10*log10(10.^(0.1*(-sqrt(2)*erfcinv(2*x)*7.6 + 9.88)) + 10.^(0.1*(-sqrt(2)*erfcinv(2*x)*6.5 + 12.1)) + 10.^(-0.3)) == L;
L_num = -1:50;
x0 = 0.0001;
for i = 1:numel(L_num)
   x_sol(i) =  double(vpasolve(subs(eqn,L,L_num(i)),x,x0));
   x0 = x_sol(i);
end
plot(L_num,x_sol)

[Susan]

@Walter Roberson Yes! This is what I expected. Thank you so much again for your help!

[Walter Roberson]

I think it was @Torsten that gave that reply ;-)

[Susan]

@Torsten Thank you so much for your response! It is what I was looking for. :)

If you don't mind, I've got another question for you. I'm trying to solve the above equation in a for loop. The constant values in the equation change a bit. But I get the following error.

Unable to perform assignment because the left and right sides have a different number of elements.

Error in test (line 104)
   x_sol(i) =  double(vpasolve(subs(eqn,L,L_num(i)),x1,x0));

It seems double(vpasolve(subs(eqn,L,L_num(ii)),x1,x0)) is a "0×1 empty double column vector". Any thoughts?

[Walter Roberson]

@Susan

That is why I coded test for empty result in https://www.mathworks.com/matlabcentral/answers/1920765-how-to-solve-an-equation-contains-norminv#comment_2640465

[Susan]

@Walter Roberson Got you! Thans again