20250513

159-5.py

@@ -0,0 +1,60 @@
+import math
+
+
+def f1(x):
+    return x**2 + 10*math.cos(x)
+
+def df1(x):
+    return 2*x - 10*math.sin(x)
+
+
+def f2(x):
+    return 1 + math.atan(x) - x
+
+def df2(x):
+    return 1/(1+x**2) - 1
+
+
+def NewtonSolve(fx, dfx, x0, err, N0):
+    count = 0
+    x1 = x0 + 1 + err
+    while abs(x1 - x0) > err or abs(fx(x1)) > err: # 添加条件修正根误差太大的问题
+        if abs(dfx(x1)) < 1e-10:
+            return None, 0
+        x1 = x0 - fx(x0) / dfx(x0)
+        count += 1
+        if count > N0:
+            return None, -1
+        x0 = x1
+    return x1, 1
+
+def FindRootZone(fx,start,stop,step):
+    x = start
+    while x < stop:
+        if fx(x) * fx(x+step) < 0:
+            return x
+        x += step
+    return None
+
+if __name__ == "__main__":
+    err = 1e-5
+    N0 = 100
+
+    x0 = 1.6
+    result,status = NewtonSolve(f1, df1, x0, err,N0)
+    if status == 1:
+        print(f"f1收敛 解为: {result}")
+    elif status == -1:
+        print("f1不收敛")
+    else:
+        print("f1导数为0，无法收敛")
+
+
+    x0 = FindRootZone(f2, -5, 5, 0.01)
+    result, status = NewtonSolve(f2, df2, x0, err,N0)
+    if status == 1:
+        print(f"f2收敛 解为: {result}")
+    elif status == -1:
+        print("f2不收敛")
+    else:
+        print("f2导数为0，无法收敛")

159-6.py

@@ -0,0 +1,38 @@
+import math
+
+
+def f(x):
+    return x**2 - 30
+
+def df(x):
+    return 2*x
+
+
+def NewtonSolve(fx, dfx, x0, err, N0):
+    count = 0
+    x1 = x0 + 1 + err
+    while abs(x1 - x0) > err or abs(fx(x1)) > err: # 添加条件修正根误差太大的问题
+        if abs(dfx(x1)) < 1e-10:
+            return None, 0
+        x1 = x0 - fx(x0) / dfx(x0)
+        count += 1
+        if count > N0:
+            return None, -1
+        x0 = x1
+    return x1, 1
+
+def FindRootZone(fx,start,stop,step):
+    x = start
+    while x < stop:
+        if fx(x) * fx(x+step) < 0:
+            return x
+        x += step
+    return None
+
+if __name__ == "__main__":
+    err = 1e-4
+    N0 = 100
+
+    x0 = FindRootZone(f, 5, 6, 0.01)
+    result,status = NewtonSolve(f, df, x0, err,N0)
+    print(f"sqrt(30) = {result:.3f}")

159-7.py

@@ -0,0 +1,36 @@
+def FindRootZone(fx,start,stop,step):
+    x = start
+    while x < stop:
+        if fx(x) * fx(x+step) < 0:
+            return x
+        x += step
+    return None
+
+# 求n次方根迭代过程如下
+def GetNthRoot(a,n):
+    if a < 0 and n % 2 == 0:
+        print("Cannot compute even root of negative number")
+        return None
+    fx = lambda x: x**n - a
+    dfx = lambda x: n * x**(n-1)
+    err = 1e-10
+    N0 = 100
+    x0 = 0
+    if a > 0:
+        x0 = FindRootZone(fx, 0, a, 0.01)
+    else:
+        x0 = FindRootZone(fx, a, 0, 0.01)
+
+    count = 0
+    x1 = x0 + 1 + err
+    while abs(x1 - x0) > err or abs(fx(x1)) > err: # 添加条件修正根误差太大的问题
+        x1 = x0 - fx(x0) / dfx(x0)
+        count += 1
+        if count > N0:
+            return None
+        x0 = x1
+    return x1
+
+if __name__ == "__main__":
+    re = GetNthRoot(30, 5)
+    print(re)