Matrix Operator Method
#15
bo198214 Wrote:
Gottfried Wrote:
bo198214 Wrote:Did you anyway already realize that instead of
\( \exp(t\cdot\log(A)) \) you can directly use the binomial series for computation?
\( A^t = \sum_{n=0}^\infty \left(t\\n\right) (A-I)^n \)
No, but it looks very good. I'll give it a deeper look, thanks for the hint!
But it seems as if A must have a diagonal of 1.
So probably you have to convert A into a Jordan normal form first as you must do with the logarithm (or how do you compute the matrix logarithm?)
If in a triangular matrix A the diagonal are all ones, the eigenvalues of (A-I) are zero. So I have to compute the matrix-logarithm by the ordinary series expression
With A1 = A-I then log(A) = A1 - A1*A1/2 + A1*A1*A1/3 ....
which is a nice exercise... since it comes out, that A1 is nilpotent and we can compute an exact result using only as many terms as the dimension of A is. For the infinite dimensional case one can note, that the coefficients are constant when dim is increased step by step, only new coefficients are added below the previously last row. So even the infinite case is analytically accessible.

Gottfried
Gottfried Helms, Kassel


Messages In This Thread
Matrix Operator Method - by Gottfried - 08/12/2007, 08:08 PM
RE: Matrix Operator Method - by bo198214 - 08/13/2007, 04:15 AM
RE: Matrix Operator Method - by jaydfox - 08/13/2007, 05:40 AM
RE: Matrix Operator Method - by Gottfried - 08/13/2007, 09:22 AM
RE: Matrix Operator Method - by bo198214 - 08/14/2007, 03:43 PM
RE: Matrix Operator Method - by Gottfried - 08/14/2007, 04:15 PM
RE: Matrix Operator Method - by bo198214 - 08/26/2007, 12:18 AM
RE: Matrix Operator Method - by Gottfried - 08/26/2007, 11:24 AM
RE: Matrix Operator Method - by bo198214 - 08/26/2007, 11:39 AM
RE: Matrix Operator Method - by Gottfried - 08/26/2007, 04:22 PM
RE: Matrix Operator Method - by Gottfried - 08/26/2007, 10:54 PM
RE: Matrix Operator Method - by bo198214 - 08/27/2007, 08:29 AM
RE: Matrix Operator Method - by Gottfried - 08/27/2007, 11:04 AM
RE: Matrix Operator Method - by bo198214 - 08/27/2007, 11:35 AM
RE: Matrix Operator Method - by Gottfried - 08/27/2007, 11:58 AM
RE: Matrix Operator Method - by bo198214 - 08/27/2007, 12:13 PM
RE: Matrix Operator Method - by Gottfried - 08/27/2007, 01:19 PM
RE: Matrix Operator Method - by Gottfried - 08/27/2007, 02:29 PM
RE: Matrix Operator Method - by bo198214 - 08/27/2007, 02:36 PM
RE: Matrix Operator Method - by Gottfried - 08/27/2007, 03:09 PM
RE: Matrix Operator Method - by bo198214 - 08/27/2007, 07:15 PM
RE: Matrix Operator Method - by Gottfried - 08/27/2007, 08:15 PM
RE: Matrix Operator Method - by bo198214 - 08/29/2007, 05:28 PM
RE: Matrix Operator Method - by Gottfried - 08/27/2007, 12:43 PM
RE: Matrix Operator Method - by Gottfried - 10/08/2007, 12:11 PM
RE: Matrix Operator Method - by Gottfried - 10/14/2007, 09:32 PM
RE: Matrix Operator Method - by Gottfried - 04/04/2008, 09:41 AM
RE: Matrix Operator Method - by Gottfried - 04/17/2008, 09:21 PM
RE: Matrix Operator Method - by bo198214 - 04/25/2008, 03:39 PM
RE: Matrix Operator Method - by Gottfried - 04/26/2008, 06:09 PM
RE: Matrix Operator Method - by bo198214 - 04/26/2008, 06:47 PM
RE: Matrix Operator Method - by Gottfried - 04/18/2008, 01:55 PM
RE: Matrix Operator Method - by Gottfried - 07/08/2008, 06:46 AM

Possibly Related Threads…
Thread Author Replies Views Last Post
  The ultimate beta method JmsNxn 8 2,081 04/15/2023, 02:36 AM
Last Post: JmsNxn
  greedy method for tetration ? tommy1729 0 508 02/11/2023, 12:13 AM
Last Post: tommy1729
  tommy's "linear" summability method tommy1729 15 3,807 02/10/2023, 03:55 AM
Last Post: JmsNxn
  another infinite composition gaussian method clone tommy1729 2 938 01/24/2023, 12:53 AM
Last Post: tommy1729
  Semi-group iso , tommy's limit fix method and alternative limit for 2sinh method tommy1729 1 876 12/30/2022, 11:27 PM
Last Post: tommy1729
  Matrix question for Gottfried Daniel 6 2,221 12/10/2022, 09:33 PM
Last Post: MphLee
  [MSE] short review/implem. of Andy's method and a next step Gottfried 4 1,741 11/03/2022, 11:51 AM
Last Post: Gottfried
  Is this the beta method? bo198214 3 1,569 08/18/2022, 04:18 AM
Last Post: JmsNxn
  Describing the beta method using fractional linear transformations JmsNxn 5 2,329 08/07/2022, 12:15 PM
Last Post: JmsNxn
Question The Etas and Euler Numbers of the 2Sinh Method Catullus 2 1,279 07/18/2022, 10:01 AM
Last Post: Catullus



Users browsing this thread: 1 Guest(s)