Solve equation for one unknown with one known parameter

clear all
clc
a=0.5
x=linspace(-2*a,2*a,100)
eqn=@(x,y,a)(1/2*a^2)*[x^2./(2-(1-(x./(2*a))^2-(y./(2*a))^2)+sqrt((1-(x./(2*a))^2-(y./(2*a))^2)+(y./a)^2))-y^2./((1-(x./(2*a))^2-(y./(2*a))^2)+sqrt((1-(x./(2*a))^2-(y./(2*a))^2)+(y./a)^2))]-1==0;
solve(eqn,y)

Answer is

Unrecognized function or variable 'y'.

Error in naca (line 6)

solve(eqn,y)

how can I solve y for x matrix value

[onur karakurt]

Also

ı dont want to use syms subprogram

value must be numerical

[Star Strider]

The actual equations are:

I = imread('https://www.mathworks.com/matlabcentral/answers/uploaded_files/621828/image.png');
figure
imshow(I)

.

Answer 1

[Star Strider]

Try something like this —

a=0.5;
% x=linspace(-2*a,2*a,100)
x=linspace(-2*a,2*a,20);
eqn=@(x,y,a)(1/2*a^2)*[x^2./(2-(1-(x./(2*a))^2-(y./(2*a))^2)+sqrt((1-(x./(2*a))^2-(y./(2*a))^2)+(y./a)^2))-y^2./((1-(x./(2*a))^2-(y./(2*a))^2)+sqrt((1-(x./(2*a))^2-(y./(2*a))^2)+(y./a)^2))]-1;

for k = 1:numel(x)
    yv(k,:) = fsolve(@(y)eqn(x(k),y,a), 10);
end

Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.

Results = table(x', yv)

Results = 20×2 table
      Var1         yv  
    _________    ______

           -1    1.9833
     -0.89474    2.1056
     -0.78947    2.2028
     -0.68421    2.2812
     -0.57895    2.3444
     -0.47368    2.3946
     -0.36842    2.4335
     -0.26316    2.4619
     -0.15789    2.4805
    -0.052632    2.4897
     0.052632    2.4897
      0.15789    2.4805
      0.26316    2.4619
      0.36842    2.4335
      0.47368    2.3946
      0.57895    2.3444

It might be necessary to use the uniquetol function to eliminate duplicate (or near-duplicate) values of ‘yv’ and thier associated ‘x’ values.

.

[onur karakurt]

I think it is not correct

because first and last value of y must be zero.

Actually ı want to solve these equation.

(x/(2*a*cos(theta)))^2-(y/(2*a*sin(theta)))=1
(2*(sin(theta))^2)=p+sqrt(p^2+(y/a)^2)
p=1-(x/2*a)^2-(y/2*a)^2

Here I simplified these function to f(x,y) shown below. May be I make a mistake while simplifying equations. Here first and last y value must be zero

(1/2*a^2)*[x^2./(2-(1-(x./(2*a))^2-(y./(2*a))^2)+sqrt((1-(x./(2*a))^2-(y./(2*a))^2)+(y./a)^2))-y^2./((1-(x./(2*a))^2-(y./(2*a))^2)+sqrt((1-(x./(2*a))^2-(y./(2*a))^2)+(y./a)^2))]=1

[onur karakurt]

[Star Strider]

I’m lost.

What variables are known, what variables are to be solved for?

The fsolve fucntion can solve systems such as these, however more needs to be known about them.

[onur karakurt]

x and a are known, y unknown.

Now ı trying to find the roots of y because it is nonlinear function.

when I fits x and a to any constant to find equation answers against the y values from 0 to 1 with 100 pieces. I saw infinite roots of y, I guess. But it must be any solution for this.

clear all
clc
a=0.5;
x=0.5;
y=linspace(0,1,100);
for i=1:size(y)
    y=y(i,:);
z=(1e15)*((x./(2*a*cos(asin(sqrt(((1-(x./(2*a)).^2-(y./(2*a)).^2)+sqrt((1-(x./(2*a)).^2-(y./(2*a)).^2).^2+(y./a).^2))./2))))).^2-(y./(2*a*sin(asin(sqrt(((1-(x./(2*a)).^2-(y./(2*a)).^2)+sqrt((1-(x./(2*a)).^2-(y./(2*a)).^2).^2+(y./a).^2))./2))))).^2-1);
end
plot(y,z)

[onur karakurt]

can you fid the roots of y for each x value while z=zero, a=0.5 and x=linspace(-2*a,2*a,100)

equation is this

z=(1e15)*((x./(2*a*cos(asin(sqrt(((1-(x./(2*a)).^2-(y./(2*a)).^2)+sqrt((1-(x./(2*a)).^2-(y./(2*a)).^2).^2+(y./a).^2))./2))))).^2-(y./(2*a*sin(asin(sqrt(((1-(x./(2*a)).^2-(y./(2*a)).^2)+sqrt((1-(x./(2*a)).^2-(y./(2*a)).^2).^2+(y./a).^2))./2))))).^2-1);

[Star Strider]

Finding all the roots is going to be a challenge.

First, using fsolve (I also experimented with fzero) is not very revealing —

zfcn = @(x,y,a) (1e15)*((x./(2*a*cos(asin(sqrt(((1-(x./(2*a)).^2-(y./(2*a)).^2)+sqrt((1-(x./(2*a)).^2-(y./(2*a)).^2).^2+(y./a).^2))./2))))).^2-(y./(2*a*sin(asin(sqrt(((1-(x./(2*a)).^2-(y./(2*a)).^2)+sqrt((1-(x./(2*a)).^2-(y./(2*a)).^2).^2+(y./a).^2))./2))))).^2-1);
a=0.5;
x=linspace(-2*a,2*a,100);
S = zeros(numel(x),3);
for k = 1:numel(x)
    [y,fv] = fsolve(@(y)zfcn(x(k),y,a), x(k));
    S(k,:) = [x(k), y, fv];
end

Equation solved at initial point.

fsolve completed because the vector of function values at the initial point
is near zero as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


No solution found.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance squared, but the vector of function values
is not near zero as measured by the value of the function tolerance.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


No solution found.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance squared, but the vector of function values
is not near zero as measured by the value of the function tolerance.


No solution found.

fsolve stopped because the problem appears regular as measured by the gradient,
but the vector of function values is not near zero as measured by the
value of the function tolerance.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved at initial point.

fsolve completed because the vector of function values at the initial point
is near zero as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved at initial point.

fsolve completed because the vector of function values at the initial point
is near zero as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved at initial point.

fsolve completed because the vector of function values at the initial point
is near zero as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved at initial point.

fsolve completed because the vector of function values at the initial point
is near zero as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


No solution found.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance squared, but the vector of function values
is not near zero as measured by the value of the function tolerance.


Equation solved at initial point.

fsolve completed because the vector of function values at the initial point
is near zero as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


No solution found.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance squared, but the vector of function values
is not near zero as measured by the value of the function tolerance.


No solution found.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance squared, but the vector of function values
is not near zero as measured by the value of the function tolerance.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


No solution found.

fsolve stopped because the problem appears regular as measured by the gradient,
but the vector of function values is not near zero as measured by the
value of the function tolerance.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


No solution found.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance squared, but the vector of function values
is not near zero as measured by the value of the function tolerance.


No solution found.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance squared, but the vector of function values
is not near zero as measured by the value of the function tolerance.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


No solution found.

fsolve stopped because the problem appears regular as measured by the gradient,
but the vector of function values is not near zero as measured by the
value of the function tolerance.


No solution found.

fsolve stopped because the problem appears regular as measured by the gradient,
but the vector of function values is not near zero as measured by the
value of the function tolerance.


No solution found.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance squared, but the vector of function values
is not near zero as measured by the value of the function tolerance.


No solution found.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance squared, but the vector of function values
is not near zero as measured by the value of the function tolerance.


No solution found.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance squared, but the vector of function values
is not near zero as measured by the value of the function tolerance.


No solution found.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance squared, but the vector of function values
is not near zero as measured by the value of the function tolerance.


No solution found.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance squared, but the vector of function values
is not near zero as measured by the value of the function tolerance.


No solution found.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance squared, but the vector of function values
is not near zero as measured by the value of the function tolerance.


No solution found.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance squared, but the vector of function values
is not near zero as measured by the value of the function tolerance.


No solution found.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance squared, but the vector of function values
is not near zero as measured by the value of the function tolerance.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


No solution found.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance squared, but the vector of function values
is not near zero as measured by the value of the function tolerance.


No solution found.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance squared, but the vector of function values
is not near zero as measured by the value of the function tolerance.


No solution found.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance squared, but the vector of function values
is not near zero as measured by the value of the function tolerance.


No solution found.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance squared, but the vector of function values
is not near zero as measured by the value of the function tolerance.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


No solution found.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance squared, but the vector of function values
is not near zero as measured by the value of the function tolerance.


No solution found.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance squared, but the vector of function values
is not near zero as measured by the value of the function tolerance.


No solution found.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance squared, but the vector of function values
is not near zero as measured by the value of the function tolerance.


No solution found.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance squared, but the vector of function values
is not near zero as measured by the value of the function tolerance.


No solution found.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance squared, but the vector of function values
is not near zero as measured by the value of the function tolerance.


No solution found.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance squared, but the vector of function values
is not near zero as measured by the value of the function tolerance.


No solution found.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance squared, but the vector of function values
is not near zero as measured by the value of the function tolerance.


No solution found.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance squared, but the vector of function values
is not near zero as measured by the value of the function tolerance.


No solution found.

fsolve stopped because the problem appears regular as measured by the gradient,
but the vector of function values is not near zero as measured by the
value of the function tolerance.


No solution found.

fsolve stopped because the problem appears regular as measured by the gradient,
but the vector of function values is not near zero as measured by the
value of the function tolerance.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


No solution found.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance squared, but the vector of function values
is not near zero as measured by the value of the function tolerance.


No solution found.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance squared, but the vector of function values
is not near zero as measured by the value of the function tolerance.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


No solution found.

fsolve stopped because the problem appears regular as measured by the gradient,
but the vector of function values is not near zero as measured by the
value of the function tolerance.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


No solution found.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance squared, but the vector of function values
is not near zero as measured by the value of the function tolerance.


No solution found.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance squared, but the vector of function values
is not near zero as measured by the value of the function tolerance.


Equation solved at initial point.

fsolve completed because the vector of function values at the initial point
is near zero as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


No solution found.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance squared, but the vector of function values
is not near zero as measured by the value of the function tolerance.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved at initial point.

fsolve completed because the vector of function values at the initial point
is near zero as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved at initial point.

fsolve completed because the vector of function values at the initial point
is near zero as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved at initial point.

fsolve completed because the vector of function values at the initial point
is near zero as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved at initial point.

fsolve completed because the vector of function values at the initial point
is near zero as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


No solution found.

fsolve stopped because the problem appears regular as measured by the gradient,
but the vector of function values is not near zero as measured by the
value of the function tolerance.


No solution found.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance squared, but the vector of function values
is not near zero as measured by the value of the function tolerance.


Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


No solution found.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance squared, but the vector of function values
is not near zero as measured by the value of the function tolerance.


Equation solved at initial point.

fsolve completed because the vector of function values at the initial point
is near zero as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.

Out = array2table(S,'VariableNames',{'x','y','FunctionValue'})

Out = 100×3 table
       x           y        FunctionValue
    ________    ________    _____________

          -1          -1             0   
     -0.9798     -0.9798      -0.22204   
     -0.9596     -0.9596             0   
    -0.93939    -0.93939       0.22204   
    -0.91919    -0.91919       0.22204   
    -0.89899    -0.89899             0   
    -0.87879    -0.87879             0   
    -0.85859    -0.85859             0   
    -0.83838    -0.83838             0   
    -0.81818    -0.81818             0   
    -0.79798    -0.79798             0   
    -0.77778    -0.77778             0   
    -0.75758    -0.75758             0   
    -0.73737    -0.73737             0   
    -0.71717    -0.71717             0   
    -0.69697    -0.69697             0   

Second, a relatively easy way to determine the roots is to plot them using the contour function, drawing contours at the roots of ‘z’ —

y = linspace(-2*pi, 2*pi);
[Xm,Ym] = ndgrid(x,y);
Zm = zfcn(Xm,Ym,a);

figure
[M,h] = contour(Xm,Ym,Zm, [0 0]);                           % Draw Contours Only At z=0
grid
title('Contour Map of (x,y) Coordinates Where z=0')
xlabel('x')
ylabel('y')

% axis('equal')

Choose whatever limits you want for ‘y’. Here, I chose the interval .

So there are myriad symmetrical roots —

format shortE
M

M = 2×10722

0 -1.0000e+00 -9.7980e-01 -9.5960e-01 -9.5296e-01 -9.5960e-01 -9.6839e-01 -9.7980e-01 -9.9197e-01 -1.0000e+00 0 -1.0000e+00 -9.7980e-01 -9.6466e-01 -9.5960e-01 -9.5339e-01 -9.3939e-01 -9.2945e-01 -9.1919e-01 -9.1044e-01 -9.1528e-01 -8.9899e-01 -8.8995e-01 -8.8131e-01 -8.7879e-01 -8.7605e-01 -8.5859e-01 -8.4236e-01 -8.3838e-01 -8.1818e-01 9.0000e+00 -6.2233e+00 -6.2400e+00 -6.1954e+00 -6.1563e+00 -6.0802e+00 -6.0293e+00 -5.9283e+00 -6.0293e+00 -6.0905e+00 6.0000e+01 -5.8454e+00 -5.8875e+00 -5.7755e+00 -5.7244e+00 -5.6485e+00 -5.5255e+00 -5.6485e+00 -5.7244e+00 -5.6485e+00 -5.5216e+00 -5.4117e+00 -5.5216e+00 -5.6485e+00 -5.7041e+00 -5.6485e+00 -5.5861e+00 -5.5216e+00 -5.4043e+00 -5.4184e+00

format short

See the documentation for contour and specifically for M to understand how to retrieve the values that contour creates.

This is the only way I can think of to solve this.

.

[onur karakurt]

thanks so much

[onur karakurt]

million thanks

[Star Strider]

As always, my pleasure!