Thanks Alexey, but I am looking for more of a loop approach. If I have a matrix of 5 columns and 100 rows with the 5 columns giving the interval values for the cubic spline fit and the 100 rows being the 100 cubic spline fits I would like stored for. I would like a loop that would take the 5x100 row matrix and output the parameters for the 100 fits. I don't know Mathematica that well but attempted to do this and failed. I tried the following:
For[i = 1, i < 101, i++,
F1[t_] = k1 + k2t + k3t^2 + k4*t^3;
F2[t_] = k5 + k6t + k7t^2 + k8*t^3;
F3[t_] = k9 + k10t + k11t^2 + k12*t^3;
F4[t_] = k13 + k14t + k15t^2 + k16*t^3;
F5[t_] = k17 + k18t + k19t^2 + k20*t^3
Data = Solve[{F1[0] == 0, F1'[5] == F2'[5], F1''[5] == F2''[5],
F2'[15] == F3'[15], F2''[15] == F3''[15],
F3'[30] == F4'[30], F3''[30] == F4''[30],
F4'[45] == F5'[45], F4''[45] == F5''[45],
F5''[0] == 0, F1''[0] == 0, F1[5] == matrix[i,1], F2[5] == matrix[i,1],
F2[15] == matrix[i,2], F3[15] == matrix[i,2], F3[30] == matrix[i,3],
F4[30] == matrix[i,3], F4[48] == matrix[i,4], F5[48] == matrix[i,4],
F5[88] == matrix[i,5]}]
{F1[t], F2[t], F3[t], F4[t], F5[t]} /. Data
]