# Getting error message from NSolve

I'm trying to solve for coefficients $k1$, $k2$, $k3$, $k4$ in the following equation:

$\det\left(\,\begin{bmatrix}k1+k2 & -k2 & 0\\-k2 & k2+k3 & -k3 \\ 0 & -k3 & k3+k4\end{bmatrix} - \begin{bmatrix}\omega^2 & 0 & 0\\0 & \omega^2 & 0 \\ 0 & 0 & \omega^2\end{bmatrix}\,\right)=0$

where $\omega>0$ is such that $\frac{\omega_1}{2\pi}=0.0028$, $\frac{\omega_2}{2\pi}=0.0036$, $\frac{\omega_3}{2\pi}=0.0042$.

I've tried to solve this by substituting $\omega^2$ with Replace, yielding

w ==
Root[-a b c - a b d - a c d - b c d +
(3909 a b + 7818 a c + 11727 b c + 3909 a d + 7818 b d +
3909 c d) #1 + (-15280281 a - 30560562 b - 30560562 c -
15280281 d) #1^2 + 59730618429 #1^3 &, 1] ||
w ==
Root[-a b c - a b d - a c d - b c d +
(3909 a b + 7818 a c + 11727 b c + 3909 a d + 7818 b d +
3909 c d) #1 + (-15280281 a - 30560562 b - 30560562 c -
15280281 d) #1^2 + 59730618429 #1^3 &, 2] ||
w ==
Root[-a b c - a b d - a c d - b c d +
(3909 a b + 7818 a c + 11727 b c + 3909 a d + 7818 b d +
3909 c d) #1 + (-15280281 a - 30560562 b - 30560562 c -
15280281 d) #1^2 + 59730618429 #1^3 &, 3]


upon which I use NSolve:

equations =
{Root[
-a b c - a b d - a c d - b c d +
(3909 a b + 7818 a c + 11727 b c + 3909 a d + 7818 b d + 3909 c d) #1 +
(-15280281 a - 30560562 b - 30560562 c - 15280281 d) #1^2 +
59730618429 #1^3 &,
1] == (0.0028*2*Pi)^2,
Root[
-a b c - a b d - a c d - b c d +
(3909 a b + 7818 a c + 11727 b c + 3909 a d + 7818 b d + 3909 c d) #1 +
(-15280281 a - 30560562 b - 30560562 c - 15280281 d) #1^2 +
59730618429 #1^3 &,
2] == (0.0036*2*Pi)^2,
Root[
-a b c - a b d - a c d - b c d +
(3909 a b + 7818 a c + 11727 b c + 3909 a d + 7818 b d + 3909 c d) #1 +
(-15280281 a - 30560562 b - 30560562 c - 15280281 d) #1^2 +
59730618429 #1^3 &,
3] == (0.0042*2*Pi)^2,
a > 0,
b > 0,
c > 0,
d > 0};

coeffs = NSolve[equations, {a, b, c, d}, Reals]


But I obtain the error message "Infinite solution set has dimension at least 1" and obtain {} as the output.

Does this mean that there are no solutions to the problem, or that I am solving this problem incorrectly? Is there a better way to solve this problem?

• You're solving essentially 2 equations for 4 unknowns. Use Reduce, but know that the solution space is likely (though not necessarily) a 2 dimensional object. The equation does have solutions, it has an infinite number of them. See FindInstance if you only need one possible solution chosen more or less at random. – eyorble Jun 27 '18 at 20:57
• I've tried using FindInstance, but the calculation seems to take a long time (I haven't actually tried letting it run all the way, I stopped it after around 5 minutes). – oswinso Jun 27 '18 at 21:06

For the first $\omega$, the following use of FindInstance finds a {k1, k2, k3, k4} tuple that satisfies the equation more or less instantly.

FindInstance[
Det[{{k1 + k2, -k2, 0}, {-k2, k2 + k3, -k3}, {0, -k3,
k3 + k4}} - {{w^2, 0, 0}, {0, w^2, 0}, {0, 0, w^2}}] == 0 /.
w -> (2 π 0.0028),
{k1, k2, k3, k4}]


The solution space for this value of $\omega$ can be found with Reduce, but it doesn't easily simplify to anything obviously and inherently meaningful:

Reduce[Det[{{k1 + k2, -k2, 0}, {-k2, k2 + k3, -k3}, {0, -k3,
k3 + k4}} - {{w^2, 0, 0}, {0, w^2, 0}, {0, 0, w^2}}] == 0 /.
w -> (2 π 0.0028)]

• This only results in finding one such {k1, k2, k3, k4} tuple which such that one root results in $\omega_1$ though. Simplifying the equation results in: k1 k2 k3 + k1 k2 k4 + k1 k3 k4 + k2 k3 k4 + (-3909 k1 k2 - 7818 k1 k3 - 11727 k2 k3 - 3909 k1 k4 - 7818 k2 k4 - 3909 k3 k4) w + (15280281 k1 + 30560562 k2 + 30560562 k3 + 15280281 k4) w^2 - 59730618429 w^3 == 0, which is a polynomial, and I want to find values for the coefficients {k1, k2, k3, k4} which results in those roots – oswinso Jun 27 '18 at 21:56
• FindInstance cannot find an infinite solution set, but you can have it find as many solutions as you wish by say, FindInstance[..., {k1, k2, k3, k4}, 10], to find 10 solutions. You could also do FindInstance[..., {k1, k2, k3, k4}, Reals, 10] to find 10 strictly real solutions. – eyorble Jun 27 '18 at 21:58