Yhyjyk

I've seen similar questions on this site but somehow the solutions there didn't manage to solve my specific problem.

I have a function mat1 that takes a square $n times n$ matrix G, and some final time tfinal, and solves the following ODE numerically:
$$u'(t) = G(t) u(t)$$
$$u(0) = mathrmid_ntimes n$$
The code is:

mat1[G_, tfinal_] := Block[t, NDSolveValue[u'[t] == G[t].u[t], u[0] == IdentityMatrix[Dimensions[G[0]][[1]]], u, t, 0, tfinal,
Method -> "ExplicitRungeKutta"]]

Let's take an example matrix-valued function $g(t)$:

g[t_?NumericQ] := Sin[t], 0, Cos[t], t

Mathematica has no problems solving the ODE with g as the input matrix:

mat1[g, 10][1.21]
(*Result: 1.90977, 0., 1.92296, 2.07912*)

But when I want to numerically integrate it, I get the following error:

NIntegrate[mat1[g, 10][t], t, 0, 10]
(*NIntegrate::inum: Integrand InterpolatingFunction[0.,10.,5,3,1,98,4,0,0,0,0,Automatic,,,False,0.,0.120666,0.60333,0.874901,<<43>>,6.97746,7.05172,7.12517,<<48>>,1.,0.,0.,1.,0.,0.,1.,0.,1.0073,0.,0.121253,1.00731,0.121252,0.,1.0146,0.121548,<<48>>,<<48>>,Automatic][t] is not numerical at t = 0.000960178.*)
(*NIntegrate::inum: Integrand InterpolatingFunction[0.,10.,5,3,1,98,4,0,0,0,0,Automatic,,,False,0.,0.120666,0.60333,0.874901,<<43>>,6.97746,7.05172,7.12517,<<48>>,1.,0.,0.,1.,0.,0.,1.,0.,1.0073,0.,0.121253,1.00731,0.121252,0.,1.0146,0.121548,<<48>>,<<48>>,Automatic][t] is not numerical at t = 0.000960178.*)

I've also tried defining a function in between:

mat2[t_?NumericQ] := mat1[g, 10][t]

But I get the same error:

NIntegrate[mat2[t], t, 0, 10]
(*NIntegrate::inum: Integrand mat2[t] is not numerical at t = 0.0795732.*)

It looks like even with the NumericQ, Mathematica is trying to manipulate the integrand with a symbolic $t$ before putting numbers in.

EDIT:

It looks like the above code works fine for a real-valued function, as opposed to matrices:

mat1[G_, tfinal_] := Block[t, NDSolveValue[u'[t] == G[t]*u[t], u[0] ==1,u, t, 0, tfinal, Method -> "ExplicitRungeKutta"]]

g[t_?NumericQ] := Sin[t]

NIntegrate[mat1[g, 10][t], t, 0, 10]

(*Result: 36.4662*)

So it looks like the problem has something to do with $g$ being a matrix. I'm not sure how though.

As the error message says, the problem is that mat2[0.0795732] is not numerical. It is instead a 2x2 matrix of numbers. You could do something like:

mat2[t_?NumericQ] := mat1[g, 10][t][[1,1]]
NIntegrate[mat2[t], t, 0, 10]

36.4662

On the other hand, it is much simpler to just have NDSolveValue do the integration for you:

mat1[G_,tfinal_] := NDSolveValue[
 
 int'[t] == u[t], int[0] == ConstantArray[0, Dimensions[G[0]]],
 u'[t]==G[t].u[t], u[0]==IdentityMatrix[Dimensions[G[0]][[1]]]
 ,
 u, int,
 t,0,tfinal
]

Then:

mat1[g, 10]

and:

mat1[g, 10][[2]][10]

36.4662, 0., 3.69638*10^20, 5.23821*10^20

agreeing with the above result.

You can use Integrate to directly antidifferentiate an interpolating function. If $f(t)$ is an interpolating function with domain $(a,b)$, Integrate[f[t], t] returns an interpolating function with the same domain equal to
$$int_a^t f(tau) ; dtau,.$$

To get the definite integral, plug the end point:

Integrate[mat1[g, 10][t], t] /. t -> 10
(* 36.4662, 0., 3.69611*10^20, 5.23781*10^20 *)