Symbolic calculation with solve command

I want MATLAB to express k1 k2 and k3 in terms of r through the following code but I can't get any useful result. The equations are pretty nonlinear. Is that the reason?

syms k1 k2 k3 r
eqns = [2*r*k3 + r*k2+r*k1  == 6, 7*r*k1 + 6*r*k2+2*r^2*k2*k3-r*k3 == 48, 8*r*k1-r*k2-r*k3-2*r^2*k1*k2+5*r^2*k1*k3+2*r^3*k1*k2*k3==63];
S = solve(eqns,[k1 k2 k3])

S = struct with fields:
    k1: [4×1 sym]
    k2: [4×1 sym]
    k3: [4×1 sym]

S.k1

ans = 

S.k2

ans = 

S.k3

ans = 

[SL: formatted code as code and executed it. I also displayed each field of S.]

Answer 1

[Torsten]

Use

S = solve(eqns,[k1 k2 k3],'MaxDegree',4)

instead of

S = solve(eqns,[k1 k2 k3])

[Walter Roberson]

Be warned that the form without the root() expression is pretty long.

syms k1 k2 k3 r
eqns = [2*r*k3 + r*k2+r*k1  == 6, 7*r*k1 + 6*r*k2+2*r^2*k2*k3-r*k3 == 48, 8*r*k1-r*k2-r*k3-2*r^2*k1*k2+5*r^2*k1*k3+2*r^3*k1*k2*k3==63];
S = solve(eqns, [k1 k2 k3], 'maxdegree', 4)

S = struct with fields:
    k1: [4×1 sym]
    k2: [4×1 sym]
    k3: [4×1 sym]

S.k1

ans = 

char(S.k1(1))

ans = '-(96*r^2*(1/(40*r) + (9*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(2/3) - (1257*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(1/3))/(50*r^2) + 23213/(5120*r^4))^(1/2)/(6*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(1/6)) + (- (23213*(9*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(2/3) - (1257*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(1/3))/(50*r^2) + 23213/(5120*r^4))^(1/2))/(5120*r^4) - 9*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(2/3)*(9*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(2/3) - (1257*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(1/3))/(50*r^2) + 23213/(5120*r^4))^(1/2) - (1257*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(1/3)*(9*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(2/3) - (1257*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(1/3))/(50*r^2) + 23213/(5120*r^4))^(1/2))/(25*r^2) - (117459*6^(1/2)*(3*3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2) + 1302896879/(2048000*r^6))^(1/2))/(16000*r^3))^(1/2)/(6*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(1/6)*(9*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(2/3) - (1257*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(1/3))/(50*r^2) + 23213/(5120*r^4))^(1/4)))^2 - 160*r^3*(1/(40*r) + (9*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(2/3) - (1257*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(1/3))/(50*r^2) + 23213/(5120*r^4))^(1/2)/(6*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(1/6)) + (- (23213*(9*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(2/3) - (1257*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(1/3))/(50*r^2) + 23213/(5120*r^4))^(1/2))/(5120*r^4) - 9*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(2/3)*(9*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(2/3) - (1257*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(1/3))/(50*r^2) + 23213/(5120*r^4))^(1/2) - (1257*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(1/3)*(9*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(2/3) - (1257*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(1/3))/(50*r^2) + 23213/(5120*r^4))^(1/2))/(25*r^2) - (117459*6^(1/2)*(3*3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2) + 1302896879/(2048000*r^6))^(1/2))/(16000*r^3))^(1/2)/(6*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(1/6)*(9*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(2/3) - (1257*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(1/3))/(50*r^2) + 23213/(5120*r^4))^(1/4)))^3 - 773*r*(1/(40*r) + (9*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(2/3) - (1257*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(1/3))/(50*r^2) + 23213/(5120*r^4))^(1/2)/(6*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(1/6)) + (- (23213*(9*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(2/3) - (1257*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(1/3))/(50*r^2) + 23213/(5120*r^4))^(1/2))/(5120*r^4) - 9*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(2/3)*(9*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(2/3) - (1257*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(1/3))/(50*r^2) + 23213/(5120*r^4))^(1/2) - (1257*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(1/3)*(9*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(2/3) - (1257*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(1/3))/(50*r^2) + 23213/(5120*r^4))^(1/2))/(25*r^2) - (117459*6^(1/2)*(3*3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2) + 1302896879/(2048000*r^6))^(1/2))/(16000*r^3))^(1/2)/(6*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(1/6)*(9*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(2/3) - (1257*((3^(1/2)*(245367007511193/(16384000000*r^12))^(1/2))/18 + 1302896879/(110592000*r^6))^(1/3))/(50*r^2) + 23213/(5120*r^4))^(1/4))) + 42)/(27*r)'

Answer 2

[Steven Lord]

If you object to the presence of root() in the solutions, since you're using release R2023a or later you can use the rewrite function with the "expandroot" option. You could also try using vpa on that result to try to make it a little easier to read, but the expressions are long and complicated enough that it doesn't help that much IMO.

syms k1 k2 k3 r
eqns = [2*r*k3 + r*k2+r*k1  == 6, 7*r*k1 + 6*r*k2+2*r^2*k2*k3-r*k3 == 48, 8*r*k1-r*k2-r*k3-2*r^2*k1*k2+5*r^2*k1*k3+2*r^3*k1*k2*k3==63];
S = solve(eqns,[k1 k2 k3]);
rewrittenExpression = rewrite(S.k1, "expandroot")

rewrittenExpression = 

vpa(rewrittenExpression, 4)

ans = 

●

If you don't want to have it written in terms of those intermediate sub-expressions you can change one of the preferences, but again that makes the expression look even more complicated.

previousValue = sympref('AbbreviateOutput',false);
vpa(rewrittenExpression, 4)

ans = 

sympref('AbbreviateOutput',previousValue); % reset the preference

[Ali Kiral]

Thank you Steve! Equations are already horribly non-linear and I wasn't expecting a simple expression anyway