Build transformation matrix for linear recurrence

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Автор aajjbb, 10 лет назад, По-английски

Hello, I'm currently learning how to use matrix exponentiation to solve linear recurrence relations, and I'm in doubt on how to build transformation matrices, for example.

Let f(n) = 2 * f(n - 1) + 2 * (f - 2) + 2 * f(n - 3) + f(n - 4)

With f(1) = a, f(2) = b, f(3) = c, f(4) = d

The vector v with the initial values should be $\text{[math]}$

How to build it's transformation matrix ?

matrix, function

aajjbb
10 лет назад
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misof

10 лет назад, # |

You are looking for a matrix A such that if you take the vector ( f(n-4), f(n-3), f(n-2), f(n-1) ) and multiply it by A you will get the vector ( f(n-3), f(n-2), f(n-1), f(n) ) for any n. Knowing this property, can you see how A must look like?

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aajjbb

10 лет назад, # ^ |

Hello, it's an honor to be answered by misof.

I've got an matrix with an alike property, but it only works for f(5). For a initial matrix of $\text{[math]}$ . f(6) = 46. But this current matrix gives me 68.

$\text{[math]}$

Could you point what is wrong in this transformation matrix ?

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Miras321

10 лет назад, # ^ |

68 is correct. With [3, 4, 2, 5] f(5) will be 25. Now we have [4, 2, 5, 25], f(6) = 25 * 2 + 5 * 2 + 2 * 2 + 4 = 68.

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aajjbb

10 лет назад, # ^ |

Thank you, actually, it's right, I had an mistake in my recurrence relation, thanks.

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JuanMata

10 лет назад, # |

Let $\text{[math]}$ and $\text{[math]}$ .
now u can see that $\text{[math]}$ .

just handle cases when $\text{[math]}$ separately, otherwise find $\text{[math]}$ (can be done in $\text{[math]}$ ) and multiply it with $\text{[math]}$ to get $\text{[math]}$ (whose first element will be $\text{[math]}$ , which is exactly what u want).