aaa

按方法整理/数值积分-龙贝格算法.py

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+import math
+
+# 龙贝格方法 积分
+def Romberg(a, b, fx, err):
+    table = [[],[],[],[]]
+    t00 = (b-a)*(fx(a)+fx(b))/2
+    table[0].append(t00)
+    print(f"t0(0)={t00}")
+    t01 = t10 = t11 = t20 = t21 = t30 = t31 = 0
+    k = 1
+    while True:
+        tmp = 0
+        for i in range(1, 2**(k-1)+1):
+            tmp += fx(a + (b-a)*(2*i-1)/(2**k))
+        t01 = t00/2+(b-a)*tmp/(2**k)
+        print(f"t0({k})={t01}")
+        table[0].append(t01)
+        
+        if k>1:
+            t11 = (4*t01-t00)/3
+            print(f"t1({k-1})={t11}")
+            table[1].append(t11)
+            if k>2:
+                t21 = (16*t11-t10)/15
+                print(f"t2({k-2})={t21}")
+                table[2].append(t21)
+                if k>3:
+                    t31 = (64*t21-t20)/63
+                    print(f"t3({k-3})={t31}")
+                    table[3].append(t31)
+                    if abs(t31-t30) < err:
+                        break
+                    t30 = t31
+                else:
+                    t30 = (64*t21-t20)/63
+                    print(f"t3(0)={t30}")
+                    table[3].append(t30)
+                t20 = t21
+            else:
+                t20 = (16*t11-t10)/15
+                print(f"t2(0)={t20}")
+                table[2].append(t20)
+            t10 = t11
+        else:
+            t10 = (4*t01-t00)/3
+            print(f"t1(0)={t10}")
+            table[1].append(t10)
+        t00 = t01
+        k += 1
+    print("Romberg table:")
+    for i in range(len(table)):
+        print(f"t{i}: {table[i]}")
+    return t31, k
+
+
+# 积分原函数 ##############################################################
+def fx(x):
+    if x == 0:
+        x = 1e-10  # Avoid division by zero  #如果x能为0，注释掉这行##############
+        pass
+    return math.sin(x)/x   #把函数改成题干的形式###################
+
+
+if __name__ == "__main__":
+    ##############################################################################################################
+    x_start = 0       # 积分下限
+    x_end = 1         # 积分上限
+    err = 0.5e-6      # 精度要求   P113
+
+    result, k = Romberg(x_start, x_end, fx, err)
+    print(f"Result: {result}, k={k}")