What is the fastest method to compute the $nth$ number in Lucas sequences?

Question

Lucas sequences $U_n(P,Q)$ and $V_n(P,Q)$ are defined by the following relations:

$U_0(P,Q)=0,$

$U_1(P,Q)=1,$

$U_n(P,Q)=P\cdot U_{n-1}(P,Q)-Q\cdot U_{n-2}(P,Q)$

and

$V_0(P,Q)=2,$

$V_1(P,Q)=P,$

$V_n(P,Q)=P\cdot V_{n-1}(P,Q)-Q\cdot V_{n-2}(P,Q).$

I knew that we can use the following method to compute whether $U_n(P,Q)$ or $V_n(P,Q)$:

$\begin{pmatrix} P & -Q \\ 1 & 0 \end{pmatrix}^{n-1} \begin{pmatrix} U_1 & V_1 \\ U_0 & V_0 \end{pmatrix} = \begin{pmatrix} U_n & V_n \\ U_{n-1} & V_{n-1} \end{pmatrix}$

Now, my question is: is this method is the fastest method to compute the $nth$ number in Lucas sequences ?

can we use a method similar to (the fast doubling method) to compute the $nth$ number in Lucas sequences instead of the above method (lucas sequences matrix method) ?

Note that the fast doubling method here is the fastest method to compute the $n$-th Fibonacci number and it is an alternative of the following method (matrix method of Fibonacci numbers):

$\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^{n} = \begin{pmatrix} F_{n+1} & F_{n} \\ F_{n} & F_{n-1} \end{pmatrix}$

Support your answer with an example please.

Answer 1

Disclaimer $\newcommand{\two}[2]{\begin{pmatrix} #1 \\ #2 \end{pmatrix}} \newcommand{\four}[4]{\begin{pmatrix} #1 & #2 \\ #3 & #4 \end{pmatrix}}$

I am not well-versed with complexity theory. From the link you quote, I don't know why the ``fast doubling method'' is actually faster than fast exponentiation. It looks as if constructing a recurrence which relates $a_{2n}$ and $a_{2n - 1}$ with $a_{n}$ and $a_{n - 1}$ seems to make computation of $a_{n}$ fast. This is what I will try to do.

One more thing: I shall assume $Q \neq 0$. If $Q = 0$, then both the sequences are simply geometric progressions, and we have the general term quite easily. (You can also easily put it in a recursive format, but that is essentially fast exponentiation for a $1 \times 1$ matrix.)

---

Instead of $(U_{n})_{n}$ and $(V_{n})_{n}$, let us first focus on some other sequences.

Define $(P_{n})_{n \ge 0}$ as \begin{align} P_{0} &:= 1,\, P_{1} := P; \\ P_{n + 1} &:= P P_{n} - Q P_{n - 1} \quad \text{for } n \ge 1. \end{align}

and $(Q_{n})_{n \ge 0}$ as \begin{align} Q_{0} &:= 0,\, Q_{1} := -Q; \\ Q_{n + 1} &:= P Q_{n} - Q Q_{n - 1} \quad \text{for } n \ge 1. \end{align}

Note that the recurrence relation that they follow is the same as $(U_{n})_{n}$ and $(V_{n})_{n}$. We have simply changed the initial values. The upshot of these values is that we get the neat expression as

$$\four{P}{-Q}{1}{0}^n = \four{P_{n}}{Q_{n}}{P_{n - 1}}{Q_{n - 1}}.$$

(This is essentially the formula with matrices that you wrote, except that we have convenient initial values here, which clubs the LHS as a the power of a single matrix.)

The above formula now gives

$$\four{P_{n}}{Q_{n}}{P_{n - 1}}{Q_{n - 1}}^2 = \four{P_{2n}}{Q_{2n}}{P_{2n - 1}}{Q_{2n - 1}}.$$

Thus, we get the coupled recurrence relations as \begin{align} P_{2n} &= P_{n}^2 + Q_{n}P_{n - 1}, \\ P_{2n - 1} &= P_{n}P_{n - 1} + Q_{n - 1}P_{n - 1}, \\ Q_{2n} &= P_{n}Q_{n} + Q_{n}Q_{n - 1}, \\ Q_{2n - 1} &= P_{n - 1}Q_{n - 1} + Q_{n - 1}^2. \end{align}

This is the (apparently) fast recurrence relation which helps us solve for $P_{n}$ and $Q_{n}$ quickly. The question now is obviously how can we get $U_{n}$ and $V_{n}$ in terms of these two sequences. Fortunately, it is not that difficult. The key point to note that all four sequences satisfy the same recurrence relation. More this is a linear recurrence relation. Thus, if we can find $\alpha, \beta \in \mathbb{R}$ such that

$$\alpha\two{P_0}{P_1} + \beta\two{Q_0}{Q_1} = \two{U_0}{U_1},$$

then we actually have

$$U_{n} = \alpha P_{n} + \beta Q_{n}$$

for all $n \ge 0$. (Similarly for $V_{n}$.)

The above is simply a constant time operation. (Of course, it depends on how many $U_{n}$ you are computing, but it should still be asymptotically the same as fast doubling.)

Now, the question is whether we can actually find such solutions. Note that we have

$$\two{P_0}{P_1} = \two{1}{P} \quad\text{and}\quad \two{Q_0}{Q_1} = \two{0}{-Q}.$$

Since $Q \neq 0$, the two vectors above are linearly independent, and we can actually solve to get $\alpha$ and $\beta$. (If you're not familiar with this lingo, you can simply solve the system of two linear equations and see that you can get the roots.)

Answer 2

For fastest retrieving, use a lookup table: First, calculate a million of u's and a million of v's and store these numbers somewhere. The nth u is just u(n), and the same with the nth v. Nobody is interested in Lucas numbers with an index larger than a million (smile!).