- #1
Alexander122745
- 2
- 0
Hello,
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 x and 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:
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
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 x and 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