From 68eb32f6182acd7b71c1a20e4b03809a4cb5bce1 Mon Sep 17 00:00:00 2001
From: lhye200 <lhye20507@outlook.com>
Date: Tue, 10 Jun 2025 14:29:12 +0800
Subject: [PATCH] 222

---
 120-3.py | 1 +
 120-4.py | 1 +
 120-5.py | 1 +
 159-1.py | 2 +-
 159-2.py | 1 +
 159-3.py | 1 +
 159-5.py | 3 ++-
 159-6.py | 3 ++-
 159-7.py | 1 +
 68-1.py  | 1 -
 69-5.py  | 6 +++++-
 89-1.py  | 6 +++++-
 89-2.py  | 2 +-
 89-3.py  | 2 +-
 all.py   | 0
 15 files changed, 23 insertions(+), 8 deletions(-)
 create mode 100644 all.py

diff --git a/120-3.py b/120-3.py
index 0de11a5..8258a8b 100644
--- a/120-3.py
+++ b/120-3.py
@@ -4,6 +4,7 @@ x1, x2 = 3, 6
 def fx(x):
     return x/(4+x**2)
 
+# 复合Newton-Cotes公式 复合牛顿-科特斯公式
 # n等分参数，x1到x2的区间，type=1表示梯形法，type=2表示辛普森法
 def CompositeNewtonCotes(n, type):
     if type == 1:
diff --git a/120-4.py b/120-4.py
index c5f9e2c..f14eb15 100644
--- a/120-4.py
+++ b/120-4.py
@@ -2,6 +2,7 @@ def fx(x):
     return 1/x
 
 
+# 逐次分半梯形递推公式
 def SplitTrapezoidal(a,b,err):
     t1 = (b-a)*(fx(a)+fx(b))/2
     k = 1
diff --git a/120-5.py b/120-5.py
index 03eafa2..b4753b1 100644
--- a/120-5.py
+++ b/120-5.py
@@ -5,6 +5,7 @@ def fx(x):
         x = 1e-10  # Avoid division by zero
     return math.sin(x)/x
 
+# 龙贝格方法 积分
 def Romberg(a, b, err):
     t00 = (b-a)*(fx(a)+fx(b))/2
     t01 = t10 = t11 = t20 = t21 = t30 = t31 = 0
diff --git a/159-1.py b/159-1.py
index 468eedc..be4a60f 100644
--- a/159-1.py
+++ b/159-1.py
@@ -1,7 +1,7 @@
 def fx(x):
     return x**4-3*x+1
 
-
+# 二分法求解方程的根
 def SolveByDivTwo(x1,x2,err):
     while abs(x2-x1) >= err:
         x = (x1+x2)/2
diff --git a/159-2.py b/159-2.py
index 17ff624..2adf555 100644
--- a/159-2.py
+++ b/159-2.py
@@ -4,6 +4,7 @@ import matplotlib.pyplot as plt
 def fx(x):
     return math.exp(x)-math.sin(x)
 
+# 绘制函数图像 并标记可能的零点
 def DrawGraph(a, b, stepper):
     x = [a + (b-a)*i*stepper for i in range(int(1/stepper+1))]
     y = [fx(i) for i in x]
diff --git a/159-3.py b/159-3.py
index ae065eb..8fb7c98 100644
--- a/159-3.py
+++ b/159-3.py
@@ -4,6 +4,7 @@ def fd1(x):
 def fd2(x):
     return 1/(x-1)**0.5
 
+# 迭代法求解方程
 def Renew(x,fd,err):
     count=0
     i = 0
diff --git a/159-5.py b/159-5.py
index 8db084b..fed4517 100644
--- a/159-5.py
+++ b/159-5.py
@@ -14,7 +14,7 @@ def f2(x):
 def df2(x):
     return 1/(1+x**2) - 1
 
-
+# 牛顿方法求解方程
 def NewtonSolve(fx, dfx, x0, err, N0):
     count = 0
     x1 = x0 + 1 + err
@@ -28,6 +28,7 @@ def NewtonSolve(fx, dfx, x0, err, N0):
         x0 = x1
     return x1, 1
 
+# 查找根区间
 def FindRootZone(fx,start,stop,step):
     x = start
     while x < stop:
diff --git a/159-6.py b/159-6.py
index 24e42f8..5f7c997 100644
--- a/159-6.py
+++ b/159-6.py
@@ -7,7 +7,7 @@ def f(x):
 def df(x):
     return 2*x
 
-
+# 牛顿方法求解方程
 def NewtonSolve(fx, dfx, x0, err, N0):
     count = 0
     x1 = x0 + 1 + err
@@ -21,6 +21,7 @@ def NewtonSolve(fx, dfx, x0, err, N0):
         x0 = x1
     return x1, 1
 
+# 查找根区间
 def FindRootZone(fx,start,stop,step):
     x = start
     while x < stop:
diff --git a/159-7.py b/159-7.py
index 219fe37..0555e94 100644
--- a/159-7.py
+++ b/159-7.py
@@ -1,3 +1,4 @@
+# 查找根区间
 def FindRootZone(fx,start,stop,step):
     x = start
     while x < stop:
diff --git a/68-1.py b/68-1.py
index 7935934..a451c33 100644
--- a/68-1.py
+++ b/68-1.py
@@ -1,4 +1,3 @@
-# 
 
 import math
 
diff --git a/69-5.py b/69-5.py
index e2589f1..36e56bf 100644
--- a/69-5.py
+++ b/69-5.py
@@ -1,6 +1,7 @@
 x = [0,1,2,3]
 y = [0,0,0,0]
 
+# 追赶法
 def ZGsolve(A,b):
     n = len(b)
     beta = [0]*n
@@ -21,19 +22,21 @@ def ZGsolve(A,b):
 
     return b
 
-
+# 获取相邻点的差分
 def GetDList(list_r):
     result = []
     for i in range(1,len(list_r)):
         result.append(list_r[i] - list_r[i-1])
     return result
 
+# 获取相邻点的差商
 def GetDQList(list_x, list_y):
     result = []
     for i in range(1,len(list_y)):
         result.append((list_y[i] - list_y[i-1]) / (list_x[i] - list_x[i-1]))
     return result
 
+# 三次样条插值
 def CubicSplineInterpolation(list_x,list_y,boundary_type,a1,a2):
     list_h = GetDList(list_x)
     list_dqxy = GetDQList(list_x, list_y)
@@ -68,6 +71,7 @@ def CubicSplineInterpolation(list_x,list_y,boundary_type,a1,a2):
 
     return M,list_h
 
+# 打印矩阵
 def PrintResult(M,list_h,list_x,list_y):
     for i in range(len(list_h)):
         k1 = (M[i+1]-M[i])/6/list_h[i]
diff --git a/89-1.py b/89-1.py
index 21c228f..bdd71c4 100644
--- a/89-1.py
+++ b/89-1.py
@@ -43,7 +43,7 @@ def SovleRowMain(A,b):
     return b
 
 
-
+# 最小二乘法拟合
 def LeastSquares(list_x,list_y,n):
     m = len(list_x)
     x_n = []
@@ -63,6 +63,7 @@ def LeastSquares(list_x,list_y,n):
         A.append(tmp)
     return SovleRowMain(A,b)
 
+# 计算多项式在给定x值上的值
 def CalculateY(list_x, coeff):
     re = []
     for i in range(len(list_x)):
@@ -71,6 +72,7 @@ def CalculateY(list_x, coeff):
             re[i] += coeff[j]*list_x[i]**j
     return re
 
+# 计算均方根误差
 def MeanSquareErr(list_y,list_y_approx):
     m = len(list_y)
     err = 0
@@ -78,6 +80,7 @@ def MeanSquareErr(list_y,list_y_approx):
         err += (list_y[i] - list_y_approx[i])**2
     return err**0.5
 
+# 计算最大误差
 def MaxErr(list_y,list_y_approx):
     m = len(list_y)
     err = 0
@@ -86,6 +89,7 @@ def MaxErr(list_y,list_y_approx):
             err = abs(list_y[i] - list_y_approx[i])
     return err
 
+# 打印拟合方程
 def PrintEquation(coeff):
     n = len(coeff)
     str_ = str(coeff[0]) + "+"
diff --git a/89-2.py b/89-2.py
index bbedc1a..f9de217 100644
--- a/89-2.py
+++ b/89-2.py
@@ -45,7 +45,7 @@ def SovleRowMain(A,b):
     return b
 
 
-
+# 最小二乘法拟合
 def LeastSquares(list_x,list_y,n):
     m = len(list_x)
     x_n = []
diff --git a/89-3.py b/89-3.py
index 1f5ceb5..564fa89 100644
--- a/89-3.py
+++ b/89-3.py
@@ -43,7 +43,7 @@ def SovleRowMain(A,b):
     return b
 
 
-
+# 最小二乘法拟合
 def LeastSquares(list_x,list_y,n):
     m = len(list_x)
     x_n = []
diff --git a/all.py b/all.py
new file mode 100644
index 0000000..e69de29