# Essays/Linear Recurrences

## Linear Recurrences and Matrix Powers

I read with enjoyment and admiration accounts of Eugene McDonnell's excellent
adventure in Heron's Rule & Integer-Area Triangles in the January Vector.
The article discusses Sloane sequence A001075,` s=: 1 2 7 26 97 362 … `with
recurrence relation

A(n) = 4A(n-1) - A(n-2)

where` (2*s)+/_1 0 1 `are the sides of integer-area triangles. Computing the
sequence using the double recursion is found to be extremely slow, and is made
faster using generating functions, Taylor's series, and partial fractions.
There is another fast solution, based on array operations. The recurrence can be
written as the matrix equation:

┌ ┐ ┌ ┐ ┌ ┐ │A(n-1)│ │ 0 1 │ │A(n-2)│ │ │ = │ │ │ │ │A(n) │ │ _1 4 │ │A(n-1)│ └ ┘ └ ┘ └ ┘

This form makes clear why it's called a linear recurrence. In J,

mp=: +/ .* NB. Matrix multiplication M=: 0 1,:_1 4 A0=: 1 2 M mp A0 2 7 M mp M mp A0 (M mp M) mp A0 7 26 7 26 M mp M mp M mp A0 (M mp M mp M) mp A0 26 97 26 97 M&mp^:(i.5) A0 (M&mp^:(i.5) =i.2) mp A0 1 2 1 2 2 7 2 7 7 26 7 26 26 97 26 97 97 362 97 362

Thus the` n`-th element of the sequence obtains by n repeated multiplications
by` M `or, equivalently, by multiplying by the` n`-th power of` M` . The` n`-th power of a
matrix can be computed by repeated squaring, with a consequent reduction
from` O(n) `operations to` O(log n) `operations.
We illustrate the process for` M `to the 30-th power:

] b=: #:30 NB. Binary representation of 30 1 1 1 1 0 I.|.b NB. Powers corresponding to 1s in b 1 2 3 4 M mp M NB. Squaring a matrix _1 4 _4 15 mp~ M _1 4 _4 15 mp~^:1 2 3 4 M NB. Powers with squaring as the function _1 4 _4 15 _15 56 _56 209 _2911 10864 _10864 40545 _109552575 408855776 _408855776 1525870529 (M&mp^:(2^1 2 3 4) =i.2) -: mp~^:1 2 3 4 M 1 mp/ mp~^:1 2 3 4 M NB. Product of selected powers _1.11401e16 4.15753e16 _4.15753e16 1.55161e17 pow=: 4 : 'mp/ mp~^:(I.|.#:y) x' M pow 30 _1.11401e16 4.15753e16 _4.15753e16 1.55161e17 M pow 0 1 0 0 1 M&mp^:30 =i.2 NB. Power by repeated multiplication _1.11401e16 4.15753e16 _4.15753e16 1.55161e17 mp / 30# ,:M NB. Insert mp between 30 copies of M _1.11401e16 4.15753e16 _4.15753e16 1.55161e17

The three techniques for computing matrix powers are applied for various n, with the following timings (microseconds; J6.01 on an AMD 64 Athlon 3200+ 2.2 GHz):

30 60 90 120 `M&mp^:n =i.2`61.9 93.2 124.9 155.3 `mp/n#,:M`55.6 89.1 129.5 163.2 `M pow n`48.6 67.2 68.3 67.7

It remains to define a function to compute the n-th element of the sequence:

seq=: [: {. M&pow mp A0"_ seq"0 i.10 1 2 7 26 97 362 1351 5042 18817 70226 seq 30 7.20106e16

The preceding techniques are applicable to any linear recurrence (on any number
of terms). For example, the Fibonacci sequence obtains
from` M=:0 1,:1 1 `and` A0=:0 1` .

### See also

Contributed by Roger Hui. Substantially the same text previously appeared in Vector, Volume 12, Number 4, April 1996.