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the matrizant

Generally speaking, the solution of a system of linear differential equations is an exponential, subject to the complication that matrix multiplication is noncommutative. The ordinary exponential can usually be approached either as a power series, or as the product of numerous factors nearly equal to unity. In the matrix context, the first alternative is realized by Picard's method Picard's method of iterative refinement, while the second corresponds more closely to solving the equation by Euler's method: Euler's method distance = velocity x time, applied repetitively over small intervals.