问题
I'm trying to adapt some equations (implicit f(x,y)) in order to be able list the Y for corresponding X-Value.
The equations could be e.g. as follows:
y^2 = x^3 + 2x - 3xy
(X^2+y^2-1)^3-x^2y^3=0
X^3+y^3=3xy^2-x-1
X^3+y^2=6xy/sqrt(y/x)
cos(PI*Y) = cos(PI.X)
Below you can see the plotted equations:
Hint, I don't know, but maybe this can be helpful, the following applies:
Y^2 + X^2 =1 ==> Y= sqrt(1-X^2)
The equations are to be adapt (substitution), so that they are expressed by X (not Y).
For y^2=x^3+2x-3xy by means of substitution results:
y1 = (-(-3x) - sqr((-3x)^2 - 4(-1)(x^3+2x)))/2*(-1)
y2 = (-(-3x) + sqr((-3x)^2 - 4(-1)(x^3+2x)))/2*(-1)
By means of adapted equations I will be able to vary X and get the corresponding Y.
See here the solution of Arkadiusz Raszeja-Solution for the equation y^2=x^3+2x-3xy
The solution of "Arkadiusz Raszeja" is for Quadratic equation, but I need an algorithm, so that e.g. all above equations can be solved.
var x,y;
for(var n=0; n<=10; n++) {
x=n;
y = (-(-3*x)-Math.sqrt(((-3*x)*(-3*x)) - 4*(-1)*((x*x*x)+2*x)))/(2*(-1));
alert(y);
}
The above alert(y); will show for Y something like below list:
X= 1 ; Y=0.79
X=2 ; Y=1.58
X=3 ; Y=2.79
X=4 ; Y=4.39
X=5 ; Y=6.33
X=6 ; Y=8.57
X=7 ; Y=11.12
X=8 ; Y=13.92
X=9 ; Y=16.98
X=10 ; Y= 20.29
My question is how can I program an algorithm, which will adapt (solve) the equations like in above example?
(You can also use a JS library like math.js, but not a plot or graph library. The solution should be in javascript)
Thanks in advance.
回答1:
Hopefully I'm understanding your question correctly. Would nerdamer help? It can help solve algebraically up to a 3rd degree polynomial. The buildFunction method can be called to get a JS function which can be used for graphing. I use it in a somewhat similar manner on the project website in combination with function-plot.js
var solutions = nerdamer('y^2=x^3+2x-3x*y').solveFor('y');
//You'll get back two solutions since it's quadratic wrt to y
console.log(solutions.toString());
//You can then parse the solutions to native javascript function
var f = nerdamer(solutions[0]).buildFunction();
console.log(f.toString());
/* Evaluate */
var solutions = nerdamer('y^3*x^2=(x^2+y^2-1)').solveFor('y');
console.log(solutions.toString());
//You can then parse the solutions again to native javascript function
var f = nerdamer(solutions[0]);
var points = {};
for(var i=1; i<10; i++)
points[i] = f.evaluate({x: i}).text();
console.log(points)<script src="http://nerdamer.com/js/nerdamer.core.js"></script>
<script src="http://nerdamer.com/js/Algebra.js"></script>
<script src="http://nerdamer.com/js/Calculus.js"></script>
<script src="http://nerdamer.com/js/Solve.js"></script>You could always just evaluate. This is slower than a pure JS function but it might be what you need. You'll have to probably use a try catch block for division by zero.
回答2:
I'd like to point out that this problem cannot be solved exactly in general. The cited solution for the quadratic case (y^2) can be extended to the cubic case and quartic case (there are a general complicated solutions). But there is a math theorem (from Galois theory) that states that there is no general solution for the quintic equation (and so on). In your case, maximum degree is 3, so you can use the cubic equation from wikipedia. For the heart graphic write: x^2*y^3 - y^2 -(x^2-1) = 0 and treat x as constant. For the sqrt case, get rid of it. Square both sides of equation, isolate y and you end up with a quartic equation on y, that you can solve using wikipedia's quartic equation knowledge.
Anyway, if you don't have a very strong reason to do this, don't do it, as the computer can solve this numerically for you. Standard approach is to calculate this implicitly, as in the plots you made.
I hope this helps.
回答3:
There ia a possible solution for the general quintic equation, when you addapt the solutionmethod from Cardano for the general cubic equation and the solutionmethod from Ferrari for the general quartic equation.
来源：https://stackoverflow.com/questions/43080688/algorithm-which-adapt-solve-the-complex-equations-implicit-function-fx-y