# Inverting a lengthy matrix

I have a rather lengthy 10X10 matrix that I want to invert it. The matrix reads:

mat={{-I ρ0 ϕa ω HankelH1[m, k0 ra],
b1a kp1a^2 BesselJ[m, kp1a ra], b2a kp2a^2 BesselJ[m, kp2a ra], 0,
b1a kp1a^2 HankelH1[m, kp1a ra], b2a kp2a^2 HankelH1[m, kp2a ra], 0,
0, 0, 0}, {-I ρ0 (-1 + ϕa) ω HankelH1[m, k0 ra],
1/2 kp1a^2 (Na BesselJ[-2 + m, kp1a ra] +
2 (a1a - Na) BesselJ[m, kp1a ra] + Na BesselJ[2 + m, kp1a ra]),
1/2 kp2a^2 (Na BesselJ[-2 + m, kp2a ra] +
2 (a2a - Na) BesselJ[m, kp2a ra] + Na BesselJ[2 + m, kp2a ra]), (
2 I m Na (ksa ra BesselJ[-1 + m, ksa ra] - (1 + m) BesselJ[m,
ksa ra]))/ra^2,
1/2 kp1a^2 (Na HankelH1[-2 + m, kp1a ra] +
2 (a1a - Na) HankelH1[m, kp1a ra] + Na HankelH1[2 + m, kp1a ra]),
1/2 kp2a^2 (Na HankelH1[-2 + m, kp2a ra] +
2 (a2a - Na) HankelH1[m, kp2a ra] +
Na HankelH1[2 + m, kp2a ra]), (
2 I m Na (ksa ra HankelH1[-1 + m, ksa ra] - (1 + m) HankelH1[m,
ksa ra]))/ra^2, 0, 0, 0}, {0,
2 I m (kp1a ra BesselJ[-1 + m, kp1a ra] - (1 + m) BesselJ[m,
kp1a ra]),
2 I m (kp2a ra BesselJ[-1 + m, kp2a ra] - (1 + m) BesselJ[m,
kp2a ra]), (2 m (1 + m) - ksa^2 ra^2) BesselJ[-2 + m,
ksa ra] + (
2 m (2 - 2 m^2 + ksa^2 ra^2) BesselJ[-1 + m, ksa ra])/(ksa ra),
2 I F m (kp1a ra HankelH1[-1 + m, kp1a ra] - (1 + m) HankelH1[m,
kp1a ra]),
2 I m (kp2a ra HankelH1[-1 + m, kp2a ra] - (1 + m) HankelH1[m,
kp2a ra]), (2 m (1 + m) - ksa^2 ra^2) HankelH1[-2 + m,
ksa ra] + (
2 m (2 - 2 m^2 + ksa^2 ra^2) HankelH1[-1 + m, ksa ra])/(ksa ra), 0,
0, 0}, {k0 ra HankelH1[-1 + m, k0 ra] -
m HankelH1[m,
k0 ra], -I (-1 + ϕa + ξ1a ϕa) ω (kp1a ra \
BesselJ[-1 + m, kp1a ra] -
m BesselJ[m,
kp1a ra]), -I (-1 + ϕa + ξ2a ϕa) ω (kp2a \
ra BesselJ[-1 + m, kp2a ra] - m BesselJ[m, kp2a ra]),
m (-1 + ϕa + ξ3a ϕa) ω BesselJ[m,
ksa ra], -I (-1 + ϕa + ξ1a ϕa) ω (kp1a ra \
HankelH1[-1 + m, kp1a ra] -
m HankelH1[m,
kp1a ra]), -I (-1 + ϕa + ξ2a ϕa) ω (kp2a \
ra HankelH1[-1 + m, kp2a ra] - m HankelH1[m, kp2a ra]),
m (-1 + ϕa + ξ3a ϕa) ω HankelH1[m, ksa ra], 0,
0, 0}, {0,
kp1a BesselJ[-1 + m, kp1a rb] - (m BesselJ[m, kp1a rb])/rb,
kp2a BesselJ[-1 + m, kp2a rb] - (m BesselJ[m, kp2a rb])/rb, (
I m BesselJ[m, ksa rb])/rb,
kp1a HankelH1[-1 + m, kp1a rb] - (m HankelH1[m, kp1a rb])/rb,
kp2a HankelH1[-1 + m, kp2a rb] - (m HankelH1[m, kp2a rb])/rb, (
I m HankelH1[m, ksa rb])/
rb, -kp1b BesselJ[-1 + m, kp1b rb] + (m BesselJ[m, kp1b rb])/
rb, -kp2b BesselJ[-1 + m, kp2b rb] + (m BesselJ[m, kp2b rb])/
rb, -((I m BesselJ[m, ksb rb])/rb)}, {0, I m BesselJ[m, kp1a rb],
I m BesselJ[m, kp2a rb], -ksa rb BesselJ[-1 + m, ksa rb] +
m BesselJ[m, ksa rb], I m HankelH1[m, kp1a rb],
I m HankelH1[m, kp2a rb], -ksa rb HankelH1[-1 + m, ksa rb] +
m HankelH1[m, ksa rb], -I m BesselJ[m, kp1b rb], -I m BesselJ[m,
kp2b rb],
ksb rb BesselJ[-1 + m, ksb rb] -
m BesselJ[m,
ksb rb]}, {0, ((-1 + ξ1a) ϕa + ϕb) (-kp1a rb \
BesselJ[-1 + m, kp1a rb] +
m BesselJ[m,
kp1a rb]), ((-1 + ξ2a) ϕa + ϕb) (-kp2a rb \
BesselJ[-1 + m, kp2a rb] +
m BesselJ[m,
kp2a rb]), -I m ((-1 + ξ3a) ϕa + ϕb) BesselJ[m,
ksa rb], ((-1 + ξ1a) ϕa + ϕb) (-kp1a rb HankelH1[-1 \
+ m, kp1a rb] +
m HankelH1[m,
kp1a rb]), ((-1 + ξ2a) ϕa + ϕb) (-kp2a rb \
HankelH1[-1 + m, kp2a rb] +
m HankelH1[m,
kp2a rb]), -I m ((-1 + ξ3a) ϕa + ϕb) HankelH1[m,
ksa rb], ξ1b ϕb (kp1b rb BesselJ[-1 + m, kp1b rb] -
m BesselJ[m, kp1b rb]), ξ2b ϕb (kp2b rb BesselJ[-1 + m,
kp2b rb] - m BesselJ[m, kp2b rb]),
I m ξ3b ϕb BesselJ[m, ksb rb]}, {0, (
b1a kp1a^2 BesselJ[m, kp1a rb])/ϕa, (
b2a kp2a^2 BesselJ[m, kp2a rb])/ϕa, 0, (
b1a kp1a^2 HankelH1[m, kp1a rb])/ϕa, (
b2a kp2a^2 HankelH1[m, kp2a rb])/ϕa,
0, -((b1b kp1b^2 BesselJ[m, kp1b rb] L[m])/ϕb), -((
b2b kp2b^2 BesselJ[m, kp2b rb])/ϕb), 0}, {0,
1/2 kp1a^2 rb^2 (Na ϕa BesselJ[-2 + m, kp1a rb] +
2 (a1a ϕa - (b1a + Na) ϕa + b1a ϕb) BesselJ[m,
kp1a rb] + Na ϕa BesselJ[2 + m, kp1a rb]),
1/2 kp2a^2 rb^2 (Na ϕa BesselJ[-2 + m, kp2a rb] +
2 (a2a ϕa - (b2a + Na) ϕa + b2a ϕb) BesselJ[m,
kp2a rb] + Na ϕa BesselJ[2 + m, kp2a rb]),
2 I m Na ϕa (ksa rb BesselJ[-1 + m, ksa rb] - (1 + m) BesselJ[
m, ksa rb]),
1/2 kp1a^2 rb^2 (Na ϕa HankelH1[-2 + m, kp1a rb] +
2 (a1a ϕa - (b1a + Na) ϕa + b1a ϕb) HankelH1[m,
kp1a rb] + Na ϕa HankelH1[2 + m, kp1a rb]),
1/2 kp2a^2 rb^2 (Na ϕa HankelH1[-2 + m, kp2a rb] +
2 (a2a ϕa - (b2a + Na) ϕa + b2a ϕb) HankelH1[m,
kp2a rb] + Na ϕa HankelH1[2 + m, kp2a rb]),
2 I m Na ϕa (ksa rb HankelH1[-1 + m,
ksa rb] - (1 + m) HankelH1[m, ksa rb]), -(1/2)
kp1b^2 rb^2 ϕa (Nb BesselJ[-2 + m, kp1b rb] +
2 (a1b - Nb) BesselJ[m, kp1b rb] +
Nb BesselJ[2 + m, kp1b rb]), -(1/2)
kp2b^2 rb^2 ϕa (Nb BesselJ[-2 + m, kp2b rb] +
2 (a2b - Nb) BesselJ[m, kp2b rb] + Nb BesselJ[2 + m, kp2b rb]),
2 I m Nb ϕa (-ksb rb BesselJ[-1 + m, ksb rb] + (1 + m) BesselJ[
m, ksb rb])}, {0,
2 I m Na rb ϕa (kp1a rb BesselJ[-1 + m,
kp1a rb] - (1 + m) BesselJ[m, kp1a rb]),
2 I m Na rb ϕa (kp2a rb BesselJ[-1 + m,
kp2a rb] - (1 + m) BesselJ[m, kp2a rb]), (
Na ϕa (ksa rb (2 m (1 + m) - ksa^2 rb^2) BesselJ[-2 + m,
ksa rb] +
2 m (2 - 2 m^2 + ksa^2 rb^2) BesselJ[-1 + m, ksa rb]))/ksa,
2 I m Na rb ϕa (kp1a rb HankelH1[-1 + m,
kp1a rb] - (1 + m) HankelH1[m, kp1a rb]),
2 I m Na rb ϕa (kp2a rb HankelH1[-1 + m,
kp2a rb] - (1 + m) HankelH1[m, kp2a rb]),
Na ϕa (rb (2 m (1 + m) - ksa^2 rb^2) HankelH1[-2 + m,
ksa rb] + (
2 m (2 - 2 m^2 + ksa^2 rb^2) HankelH1[-1 + m, ksa rb])/ksa),
2 I m Nb rb ϕa (-kp1b rb BesselJ[-1 + m,
kp1b rb] + (1 + m) BesselJ[m, kp1b rb]),
2 I m Nb rb ϕa (-kp2b rb BesselJ[-1 + m,
kp2b rb] + (1 + m) BesselJ[m, kp2b rb]), (
Nb ϕa (ksb rb (-2 m (1 + m) + ksb^2 rb^2) BesselJ[-2 + m,
ksb rb] +
2 m (-2 + 2 m^2 - ksb^2 rb^2) BesselJ[-1 + m, ksb rb]))/ksb}}


All the involving parameters are positive. Is there any chance for an analytic inversion anyhow (Inverse[mat] seems to get stuck)? If the answer is no (most possible I guess), what is the best approach for a numerical inversion? Thanks in advance.

• Really, do it numerically. Assign machine precision numbers to all symbols and use Inverse. – Henrik Schumacher Jan 23 at 11:21
• @HenrikSchumacher Thank you very much. Generally speaking, above what order should one give up any attempt of analytically inverting a matrix with Inverse command? – Dimitris Jan 23 at 11:25
• That depends on the complexity and the structure of nonzero values of the matrix. In general, dense dymbolic matrices of size $6 \times 6$ or so might still work; everything above will probably cause trouble. – Henrik Schumacher Jan 23 at 11:54
• @HenrikSchumacher : I understand. Thanks for you comments once again. – Dimitris Jan 23 at 11:56
• You're welcome. – Henrik Schumacher Jan 23 at 11:57