# Fibonacci numbers (Eiffel)

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The Fibonacci numbers are the integer sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, ..., in which each item is formed by adding the previous two. The sequence can be defined recursively by

1 \\ \end{cases} ."/>

Fibonacci number programs that implement this definition directly are often used as introductory examples of recursion. However, many other algorithms for calculating (or making use of) Fibonacci numbers also exist.

## Implementation

There are several different ways to implement the Fibonacci numbers in Eiffel.

### Recursive

The most direct translation of the mathematical definition is as a recursive function, which can be implemented as an Eiffel feature as follows:

<<recursive fibonacci>>=
fib (n: INTEGER): INTEGER
require
n_non_negative: n >= 0
do
inspect n
when 0 then Result := 0
when 1 then Result := 1
else
Result := fib (n-1) + fib (n-2)
end
end


### Iteration

Although it is based directly on the definition of a Fibonacci number, the recursive Fibonacci algorithm is extremely expensive, requiring time O(2n). It also performs a huge amount of redundant work because it computes many Fibonnaci values from scratch many times. A simple linear-time iterative approach which calculates each value of fib successively can avoid these issues:

<<iterative fibonacci>>=
fib_iterative (n: INTEGER): INTEGER
require
n_non_negative: n >= 0
local
prev1, prev2: INTEGER
i: INTEGER
do
prev1 := 0
-- Ensure correct behavior for n = 0
if n = 0 then
Result := 0
else
Result := 1
end
from
i := 1
until
i >= n
loop
prev2 := prev1
prev1 := Result
Result := prev2 + Result
i := i + 1
end
end


## Testing the FIBONACCI class

We can pull both implementations together into a single class, in order to facilitate testing.

<<Fibonacci.e>>=
class FIBONACCI
create
make
feature
recursive fibonacci
iterative fibonacci
test
end


The test harness consists of the following feature:

<<test>>=
make
local
i: INTEGER
do
from
i := 0
until
i > 10
loop
io.put_integer (fib (i))
io.put_new_line
i := i + 1
end
from
i := 0
until
i > 10
loop
io.put_integer (fib_iterative (i))
io.put_new_line
i := i + 1
end
end