Pending issues with solve

[Happypig375]

~``` solve((x-1)(x+1)x=3x,x) => [(-1/3)(-1)^(1/6)24^(1/3)sqrt(3)^(1/3)-4(-1)^(-1/6)24^(-1/3)sqrt(3)^(-1/3), (1/6)(-1)^(1/6)(1+isqrt(3))24^(1/3)sqrt(3)^(1/3)+2((-1)^(1/6)24^(1/3)sqrt(3)^(1/3))^(-1)(-isqrt(3)+1), (1/6)(-1)^(1/6)(-isqrt(3)+1)24^(1/3)sqrt(3)^(1/3)+2((-1)^(1/6)24^(1/3)sqrt(3)^(1/3))^(-1)(1+isqrt(3))] should=> [2,-2,0]~

Fixed. Adding test.

[Happypig375]

~solve(sqrt(x)+sqrt(2x+1)=5,x) => []~ should=> [4]

[jiggzson]

@Happypig375 Thanks. Can you start closing off some of the issues that are fixed?

[Happypig375]

~solve(x=2/(3-x),x)~ => [1,1.0000000000000002] (@0.7.13) => [2*(-x+3)^(-1)] (@0.7.14) should=> [1,2]

[Happypig375]

solveEquations([sqrt(2)a-sqrt(2)c+b+d=0,a+c+sqrt(2)b-sqrt(2)d=0,a+c=0,b+d=1]) => Error: System must contain all linear equations!

All four equations are linear, but multivariate.

[jiggzson]

@Happypig375 so which variable are you trying to solve for?

[Happypig375]

@jiggzson All of them. Currently solveEquations([a+b=1,a-b=2]) => [a,1.5,b,-0.5]. Therefore, solveEquations([sqrt(2)a-sqrt(2)c+b+d=0,a+c+sqrt(2)b-sqrt(2)d=0,a+c=0,b+d=1]) should=> [a,-1/(2 sqrt(2)), b, 1/2, c, 1/(2 sqrt(2)), d, 1/2]

[Happypig375]

Also, I think that solveEquations([a+b=1,a-b=2]) should=> [a=1.5,b=-0.5] This is more clear about the connection between the unknown and the value.

[jiggzson]

Oops. I confused it with solve for a moment. You're right. Something is causing it to incorrectly flag it as non-linear. I'll look into this.

[Happypig375]

~solve(sqrt(x)-1,x) => [] should=> [1]~ ~solve(sqrt(x)+1,x) => [0] should=> []~

[Happypig375]

~solve(sqrt(x^2+1),x) => [] should=> [i, -i] solve(sqrt(x^2-1),x) => [] should=> [1, -1]~

[Happypig375]

The second argument to solve (variable to solve for) should be inferred for univariate polynomials/equations, and solve for all variables in multivariate equations like in solveEquations.

[Happypig375]

Cross-ref: #345

[jiggzson]

~~*solve(((x+1)((x+1)+1))/2=n,x)** Expected: +/-(sqrt(8n+1)+3)/2 Gives: +/-(sqrt(4n+1)+3)/2~~ Fixed!

[Happypig375]

solve(x^x^x^2018=2018,x) => [] should=> [2018^(1/2018)]

See https://youtu.be/UNwhaN2s2ww

[Happypig375]

~solve(sqrt(10x+186)=x+9,x) => [7.000000000000002,7,6.999999999999999] should=> [7]~

[Happypig375]

~solve(x^3+8=x^2+6,x) eval=> [−0.9999999999999996,0.9999999999999999⋅i+1,−0.9999999999999999⋅i+1] should=> [-1,1+i,1-i]~

[Happypig375]

Also, the solve function itself looks like the variable x is assumed when the 2nd argument is not provided.

1. The minimum number of arguments needed to pass to solve should be 1 instead of 2.
2. If the equation given is in one unknown, then that unknown should be assumed the variable to solve, and return to using x if there are multiple variables (or throw).

[Happypig375]

Missing active label here 😃

[jiggzson]

The minimum number of arguments needed to pass to solve should be 1 instead of 2.

I'll fix this. Additionally I've extended the symbolic range of Solve but solve is in dire need of unit tests.

[jiggzson]

@Happypig375, automatically assuming the variable won't work at the moment since solve is used for various cases including systems of equations. I'm planning to refactor Solve in the near future so I'll go ahead and take care of it then.

[Happypig375]

Absolutely wrong answer by solve: ~solve(x^3-10x^2+31x-30,x)~ should return [2,3,5]

[jiggzson]

Absolutely wrong answer by solve:

Have you verified the answer? The result you get is: A. Ugly B. Needlessly complex C. 100% correct The problem is that nerdamer does not simplify complex numbers efficiently and that is something I'm currently working on. If I solve it numerically I get the right answer right away. Would defaulting to that solve your problem? The downside is that you don't get nice algebraic answers anymore whenever possible.

[Happypig375]

Have you verified the answer? The result you get is: A. Ugly B. Needlessly complex C. 100% correct

A. Evaluate can't simplify it B. The length exceeds Wolfram Alpha's input limit

How can I verify the answer when I don't even know what to do with it?

If I solve it numerically I get the right answer right away.

Ok

Would defaulting to that solve your problem? The downside is that you don't get nice algebraic answers anymore whenever possible.

Just fix this but don't change the defaults.

[jiggzson]

@Happypig375, fixing this is a little more involved than it might appear. One might be tempted to solve this numerically but then you end up with a tiny imaginary part for all the answers which is not ideal. Not to mention that the real part will now have rounding errors. Until I can successfully simplify [(-1/3)*(-20+18*i*sqrt(3))^(1/3)*2^(-1/3)+(-7/3)*(-20+18*i*sqrt(3))^(-1/3)*2^(1/3)+10/3,(1/6)*(-20+18*i*sqrt(3))^(1/3)*(1+i*sqrt(3))*2^(-1/3)+(7/6)*((-20+18*i*sqrt(3))^(1/3)*2^(-1/3))^(-1)*(-i*sqrt(3)+1)+10/3,(1/6)*(-20+18*i*sqrt(3))^(1/3)*(-i*sqrt(3)+1)*2^(-1/3)+(7/6)*((-20+18*i*sqrt(3))^(1/3)*2^(-1/3))^(-1)*(1+i*sqrt(3))+10/3] algebraically, I've decided to take the following approach.

• Check if all coefficients are numeric.
• Solve the polynomial
• Check if all the roots are integers. If they are then return that answer. If they aren't then solve it algebraically.

The unfortunate part is that I now solve twice but that'll have to do until I find the time to simplify the algebraic solution above.

[Happypig375]

I think that this can easily solved by implementing cube root for complex numbers. Ideas?

[jiggzson]

I think that this can easily solved by implementing cube root for complex numbers. Ideas?

That's what I initially thought myself. I implemented nth root for complex numbers which might still need some final touch-ups but when doing this one of the roots becomes 0.000000000000000221377340243793938154443⋅i+1.999999999999999555910790149937482438160. This has rounding errors in both the real and imaginary part. I can solve this algebraically at which point this error goes away but that requires a bit more work. I eventually hope to get to that at some point but I won't have the time now.

[Happypig375]

I think I saw where the rounding error comes from - using Math.sin and Math.cos inside Parser.pow. The conversion between Frac and Javascript Number is what causes the error.

[jiggzson]

The conversion between Frac and Javascript Number is what causes the error.

I think that exacerbates the problem but doesn't eliminate it from what I remember. I'll give it a go later so let's see. Additionally, I still need to work on eliminating rounding errors in pow.

[Happypig375]

The workaround earlier is not useful when fractional answers are there... ~solve(8x^3-26x^2+3x+9,x) should=> [3,3/4,-1/2] It currently produces [(-1/24)*(-13984+15120*i*sqrt(3))^(1/3)*2^(-1/3)+(-151/6)*(-13984+15120*i*sqrt(3))^(-1/3)*2^(1/3)+13/12,(1/48)*(-13984+15120*i*sqrt(3))^(1/3)*(1+i*sqrt(3))*2^(-1/3)+(151/12)*((-13984+15120*i*sqrt(3))^(1/3)*2^(-1/3))^(-1)*(-i*sqrt(3)+1)+13/12,(1/48)*(-13984+15120*i*sqrt(3))^(1/3)*(-i*sqrt(3)+1)*2^(-1/3)+(151/12)*((-13984+15120*i*sqrt(3))^(1/3)*2^(-1/3))^(-1)*(1+i*sqrt(3))+13/12].~

Another bug is present here. When evaluate is checked, the vector does not evaluate automatically, but evaluates when vecget one of the solutions. Aka solve(8x^3-26x^2+3x+9,x) eval=> [(-1/24)*(-13984+15120*i*sqrt(3))^(1/3)*2^(-1/3)+(-151/6)*(-13984+15120*i*sqrt(3))^(-1/3)*2^(1/3)+13/12,(1/48)*(-13984+15120*i*sqrt(3))^(1/3)*(1+i*sqrt(3))*2^(-1/3)+(151/12)*((-13984+15120*i*sqrt(3))^(1/3)*2^(-1/3))^(-1)*(-i*sqrt(3)+1)+13/12,(1/48)*(-13984+15120*i*sqrt(3))^(1/3)*(-i*sqrt(3)+1)*2^(-1/3)+(151/12)*((-13984+15120*i*sqrt(3))^(1/3)*2^(-1/3))^(-1)*(1+i*sqrt(3))+13/12]

vecget(solve(8x^3-26x^2+3x+9,x),0) eval=> 1.0515041619631423e+33/754813980813027500i-300239975158033/150119987579016

Really. The rounding errors are important to fix.

[Happypig375]

Also this. ~solve(x^3-1/2x^2-13/2x-3,x) => [(-1/3)*((315/4)*i*sqrt(3)-221/2)^(1/3)*2^(-1/3)+(-79/12)*((315/4)*i*sqrt(3)-221/2)^(-1/3)*2^(1/3)+1/6,(1/6)*((315/4)*i*sqrt(3)-221/2)^(1/3)*(1+i*sqrt(3))*2^(-1/3)+(79/24)*(((315/4)*i*sqrt(3)-221/2)^(1/3)*2^(-1/3))^(-1)*(-i*sqrt(3)+1)+1/6,(1/6)*((315/4)*i*sqrt(3)-221/2)^(1/3)*(-i*sqrt(3)+1)*2^(-1/3)+(79/24)*(((315/4)*i*sqrt(3)-221/2)^(1/3)*2^(-1/3))^(-1)*(1+i*sqrt(3))+1/6] should=> [-1/2,3,-2]~

[Happypig375]

And the quartic equations solve(sqrt(sqrt(x))+sqrt(sqrt(881-x))=9,x) => [159.165753255588098206102369881636862489634] can=>[7442 - 2916 881^(1/4) + 486 sqrt(881) - 36 881^(3/4)]

[Happypig375]

Where did solving in terms of other variables go? solve(sqrt(x^2-p)+2sqrt(x^2-1),x) => [] should=> [sqrt(4 - p)/sqrt(3),-sqrt(4 - p)/sqrt(3)]

[jiggzson]

Hmmm.... I'll take a look it.

[Happypig375]

solve(x^(2i)=-x,x) => [] It should have two numerical solutions according to Wolfram Alpha.

[Happypig375]

solve(x*log(x),x) => [1,0] 0 is not valid since 0*log(0) is 0*-Infinity which is undefined.

[Happypig375]

~solve(x^2=x^-2,x) => [1,e^((1/2)*i*pi),e^(i*pi),e^((3/2)*i*pi)] => [1,i,-1,i]~

I have to re-enter the first output to simplify the answer. It can be simplified in one go.

[jiggzson]

@Happypig375, why is this a problem?

[Happypig375]

solve(x^2=x^-2,x) should directly give [1,i,-1,i].

[Happypig375]

Here is one where even Wolfram Alpha can't simplify: solve(-8x^3+6x-1=0,x) should=> [sin(pi/18),sin(5pi/18),sin(-7pi/18)]

See https://youtu.be/JqagcCrdpQQ

[jiggzson]

~solve((9x+x^2)^3+10800x+40x^4+4440x^2+720x^3+20(9*x+x^2)^2+8000,x) throws~

[jiggzson]

~solve((x^3-4)/(x^3+7x-11),x) does not give solutions~

[jiggzson]

~solve(x^3+2x^2+3x-4=0,x) gives wrong answer. Solve for cubics still broken.~

[jiggzson]

@Happypig375, I'm just going through the cases and marking off those that have been resolved.

[Happypig375]

Nice!