How to solve an equation contains norminv?

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Susan 2023-2-28

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1920765-how-to-solve-an-equation-contains-norminv

评论： Susan 2023-3-1

采纳的回答： Walter Roberson

在 MATLAB Online 中打开

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

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采纳的回答

Walter Roberson 2023-2-28

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链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1920765-how-to-solve-an-equation-contains-norminv#answer_1182180

在 MATLAB Online 中打开

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.