refactor(math): Organize math algorithms into dedicated directories

- Moved 'fibonacci', 'eratosthenes', and 'euclidean' implementations
  into their respective subdirectories within 'math/'.
- Updated 'math/README.md' to reflect the new structure.
- Added 'newtonRoot.hs' for Newton's method.
- Added 'tree.hs' under 'data_structure/tree'.

Author: RyosukeDTomita

data_structure/tree/tree.hs

@@ -0,0 +1,56 @@
+-- https://www.sampou.org/haskell/article/whyfp.html より
+data Tree a = Node a [Tree a]
+
+-- 木のリストに対する畳み込み
+redtree ::
+  (a -> b -> b) -> -- f : node を潰す
+  (b -> b -> b) -> -- g : cons を潰す
+  b -> -- a : nil を潰す
+  Tree a ->
+  b
+redtree f g a (Node label subtrees) =
+  f label (redtree' f g a subtrees)
+
+-- ツリーのリストを処理する関数
+redtree' ::
+  (a -> b -> b) ->
+  (b -> b -> b) ->
+  b ->
+  [Tree a] ->
+  b
+redtree' f g a (subtree : rest) =
+  g
+    (redtree f g a subtree) -- リストのサイズが1に分解して潰す --> f label (redtree' f g a [先頭の木]) ...という流れでredtree' _ _ a [] = aにたどりつく
+    (redtree' f g a rest) -- 残りで再帰
+redtree' _ _ a [] =
+  a
+
+tree :: Tree Int
+tree =
+  Node
+    1
+    ( (:)
+        (Node 2 [])
+        ( (:)
+            ( Node
+                3
+                ((:) (Node 4 []) [])
+            )
+            []
+        )
+    )
+
+sumtree :: (Num a) => Tree a -> a
+sumtree tree = redtree (+) (+) 0 tree
+
+labels :: (Num a) => Tree a -> [a]
+labels tree = redtree (:) (++) [] tree
+
+maptree :: (a -> b) -> Tree a -> Tree b
+maptree f tree = redtree (\label subtrees -> Node (f label) subtrees) (:) [] tree
+
+main :: IO ()
+main = do
+  print $ sumtree tree -- 10
+  print $ labels tree -- [1, 2, 3, 4]
+  print $ maptree (* 2) tree

math/README.md

@@ -1,5 +1,35 @@
 # 数学ぽいやつ
 
+## フィボナッチ数列
+
+[fibonacci](./fibonacci/)
+
+---
+
+## ニュートン法による平方根の近似
+
+[newton](./newton/)
+
+---
+
 ## ユークリッドの互助法
 
+[euclidean](./euclidean/)
+
+### 実装例
+
+- [] C
+- [] Python
+- [x] Haskell
+
+---
+
 ## エラトステネスのふるい
+
+[eratosthenes](./eratosthenes/)
+
+- [] C
+- [] Python
+- [x] Haskell
+
+---

math/eratosthenes.hs math/eratosthenes/eratosthenes.hsmath/eratosthenes.hs renamed to math/eratosthenes/eratosthenes.hs

@@ -0,0 +1,33 @@
+-- https://www.sampou.org/haskell/article/whyfp.html
+-- 近似を1つ進める関数
+next :: Double -> Double -> Double
+next n x = (x + n / x) / 2
+
+-- 許容誤差と近似値よりも差の小さい2つの連続する近似値を探す
+within :: Double -> [Double] -> Double
+within eps (a : b : rest)
+  | abs (a - b) <= eps = b -- 絶対誤差
+  | otherwise = within eps (b : rest)
+
+-- ２つの近似値の比が1に近づくようにして許容誤差よりも小さい２つの連続する近似値を探す
+relative :: Double -> [Double] -> Double
+relative eps (a : b : rest)
+  | abs (a - b) <= eps * abs b = b -- abs (a - b) / abs b <= epsを変形したもの。いわゆる相対誤差
+  | otherwise = relative eps (b : rest)
+
+-- withinを使って平方根をもとめる例
+sqrt' :: Double -> Double -> Double -> Double
+sqrt' a0 eps n = within eps $ iterate (next n) a0
+
+relativesqrt :: Double -> Double -> Double -> Double
+relativesqrt a0 eps n = relative eps $ iterate (next n) a0
+
+main :: IO ()
+main = do
+  let n = fromIntegral 2 -- 平方根を求めたい値
+  let a0 = fromInteger 1 -- 計算の初期値
+  let eps = 0.001
+  let epsRelative = 0.001
+  print $ take 10 $ iterate (next n) 1
+  print $ sqrt' a0 eps n
+  print $ relativesqrt a0 epsRelative n