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Consider the system of linear homogeneous differential equations of first order

**dy/dx = A(x) y**

where

**x**denotes the independent variable,

**A(x)**is a square matrix, and

**y**is an unknown vector-function to be calculated.

Many-many years ago, when I was reading a good book on ordinary differential equations, I found a promising numerical method to approximately calculate the fundamental matrix (the matrizant or the Cauchy matrix)

**Ф(x)**for the above linear system. The method is based on the replacement of the variable matrix

**A(x)**by the constant matrix

**A(c)**where

**c**is the reasonably chosen numerical value for the variable

**x**. Then the approximate value of the matrizant

**Ф(x)**can be calculated by means of the formula

**Ф(x) = Exp ( (x – a) A(c) )**

where

**a**is the initial value of

**and**

*x***Exp**is so called

*matrix exponential*that can be calculated by means of the existing very fast and precise computer algorithms.

Regrettably, I forgot the title of the book in which the idea of the method was briefly explained. It’s a petty because the method seems to be promising for stiff systems of differential equations.

My questions:

**Have you ever heard about the described method to approximate the matrizant?****In what book or article was the method originally published?****Is there any development or improvement of the original method to make it more accurate?****Do you know any implementation of the method in any computing environment like Mathematica, Maple, or MATLAB?**

*I would appreciate it if you could provide any additional information about the described numerical method. I mean names, titles, references, formulae, or Web links.*

Thank you for your attention to this topic.

Respectfully,

Alexander