How to calculate only four elements per column in a block matrix?

Question

I have a matrix generated like this :

Table[Sequence @@ {f[t[[i]], j, x], g[t[[i]], j, x]}, {i, 16}, {j, 3}]

where $f$ and $g$ are two different functions, $t$ is a vector or List containing values of time, and $x$ is a variable number.

This provides with the following matrix :

$\small \begin{pmatrix} f[t[[1]],1,x] & g[t[[1]],1,x] & f[t[[1]],2,x] &g[t[[1]],2,x] & f[t[[1]],3,x] & g[t[[1]],3,x]\\ f[t[[2]],1,x]&g[t[[2]],1,x]&f[t[[2]],2,x]&g[t[[2]],2,x]&f[t[[2]],3,x]&g[t[[2]],3,x]\\ f[t[[3]],1,x]&g[t[[3]],1,x]&f[t[[3]],2,x]&g[t[[3]],2,x]&f[t[[3]],3,x]&g[t[[3]],3,x]\\ f[t[[4]],1,x] & g[t[[4]],1,x] & f[t[[4]],2,x] &g[t[[4]],2,x] & f[t[[4]],3,x] & g[t[[4]],3,x]\\ f[t[[5]],1,x] & g[t[[5]],1,x] & f[t[[5]],2,x] &g[t[[5]],2,x] & f[t[[5]],3,x] & g[t[[5]],3,x]\\ f[t[[6]],1,x] & g[t[[6]],1,x] & f[t[[6]],2,x] &g[t[[6]],2,x] & f[t[[6]],3,x] & g[t[[6]],3,x]\\ f[t[[7]],1,x] & g[t[[7]],1,x] & f[t[[7]],2,x] &g[t[[7]],2,x] & f[t[[7]],3,x] & g[t[[7]],3,x]\\ f[t[[8]],1,x] & g[t[[8]],1,x] & f[t[[8]],2,x] &g[t[[8]],2,x] & f[t[[8]],3,x] & g[t[[8]],3,x]\\ f[t[[9]],1,x] & g[t[[9]],1,x] & f[t[[9]],2,x] &g[t[[9]],2,x] & f[t[[9]],3,x] & g[t[[9]],3,x]\\ f[t[[10]],1,x] & g[t[[10]],1,x] & f[t[[10]],2,x] &g[t[[10]],2,x] & f[t[[10]],3,x] & g[t[[10]],3,x]\\ f[t[[11]],1,x] & g[t[[11]],1,x] & f[t[[11]],2,x] &g[t[[11]],2,x] & f[t[[11]],3,x] & g[t[[11]],3,x]\\ f[t[[12]],1,x] & g[t[[12]],1,x] & f[t[[12]],2,x] &g[t[[12]],2,x] & f[t[[12]],3,x] & g[t[[12]],3,x] \end{pmatrix}$

Now, in my experiments, I've found out that I don't need to calculate every value of the matrix but only four elements for each column, because the other ones are equal to zero. So I need to obtain the following matrix :

$\small \begin{pmatrix} f[t[[1]],1,x] & g[t[[1]],1,x] & 0&0 & 0 & 0\\ f[t[[2]],1,x]&g[t[[2]],1,x]&0&0&0&0\\ f[t[[3]],1,x]&g[t[[3]],1,x]&f[t[[3]],2,x]&g[t[[3]],2,x]&0&0\\ f[t[[4]],1,x] & g[t[[4]],1,x] & f[t[[4]],2,x] &g[t[[4]],2,x] & 0 & 0\\ 0 & 0 & f[t[[5]],2,x] &g[t[[5]],2,x] & f[t[[5]],3,x] & g[t[[5]],3,x]\\ 0 & 0 & f[t[[6]],2,x] &g[t[[6]],2,x] & f[t[[6]],3,x] & g[t[[6]],3,x]\\ 0 & 0 & 0 &0 & f[t[[7]],3,x] & g[t[[7]],3,x]\\ 0 & 0 & 0 &0 & f[t[[8]],3,x] & g[t[[8]],3,x]\\ 0 & 0 & 0 &0 & 0 & 0\\ 0 & 0 & 0 &0 & 0 & 0\\ 0 & 0 & 0 &0 & 0 & 0\\ 0 & 0 & 0 &0 & 0 & 0 \end{pmatrix}$

Is there a way to construct the matrix avoiding the calculus of the elements which are always equal to zero?

I tried this

Table[Sequence @@ {f[t[[i]], j, x], g[t[[i]], j, x]}, {i, 2j-1,2j+2}, {j, 3}]

but obviously without success...

My concern is with the time consumption (when working with large matrices).

Answer 1

For example:

m[p_] := SparseArray[Flatten@Table[{j, i} -> p[t[[i]], j, x], {i, 2 j - 1, 2 j + 2}, {j, 3}]];
h = Quiet@Transpose@Riffle[m[f], m[g]];
MatrixForm@h