Tridiagonal matrices multiplication

[nicolas1991]

I have a nxn tridiagonal matrix (let's name it A) and i want to find a way to solve A^p, p=1,2,3,...inf, most efficient* (using the structure of my matrix)
my first problem is how many calculations do i need for A²,
and then how many calculations for the hole A^p ? any help please!*by most efficient i mean with the least calculations possible

 

[AlephZero]

For A², work out which terms are nonzero. It is a banded matrix, but not tridiagonal.

Then work out how many calculations you have to do to find each nonzero entry. For most of the terms that will be 3 miltiples and 2 adds, but the first and last rows and columns of A don't have 3 nonzero entries.

Then go on to A³, A⁴, etc.

If you are want to calculate all the powers of A in order, there is more than one way to do the higher powers. For eaxmple A⁴ = A³.A or A²,A². One way might be cheaper than the other.

 

[nicolas1991]

For A², work out which terms are nonzero. It is a banded matrix, but not tridiagonal.

Then work out how many calculations you have to do to find each nonzero entry. For most of the terms that will be 3 miltiples and 2 adds, but the first and last rows and columns of A don't have 3 nonzero entries.

Then go on to A³, A⁴, etc.

If you are want to calculate all the powers of A in order, there is more than one way to do the higher powers. For eaxmple A⁴ = A³.A or A²,A². One way might be cheaper than the other.

first of all thank you ! but my real problem is how to calculation the number of calculations ( ) i need for a A^p, i can find out how many calculations i need for A², and for A³ etc. but every time the array change structure (tridiagonal->fivediagonal->sevendiagonal ...) any help please...