aaa

按方法整理/常微分方程-阿达姆斯预测-校正方法.py

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+import math
+
+
+def AdamusFix(k,first_ys,x0,y0,h,xk,fxy,fx_real):
+    betas_1 = [
+        [1],
+        [3/2, -1/2],
+        [23/12, -16/12, 5/12],
+        [55/24, -59/24, 37/24, -9/24],
+        [1901/720, -2774/720, 2626/720, -1274/720, 251/720],
+        [4277/1440, -7923/1600, 9982/1440, -7298/1440, 2877/1440, -476/1440]
+    ]
+    # Bks = [1/2,5/12,3/8,251/720,95/288,10987/60480]
+    betas_2 = [
+        [1],
+        [1/2, 1/2],
+        [5/12, 8/12, -1/12],
+        [9/24, 19/24, -5/24, 1/24],
+        [251/720, 646/720, -261/720, 106/720, -19/720],
+        [475/1440, 1427/1440, -798/1440, 482/1440, -173/1440, 27/1440]
+    ]
+    result = []
+    if k<0 or k >= len(betas_1):
+        print("k超出范围")
+        return None
+    if len(first_ys) != k + 1:
+        print("first_ys(前几个y的值)与k长度不匹配")
+        return None
+    fxys = [fxy(x0+i*h, first_ys[i]) for i in range(k + 1)]
+    for i in range(k + 1):
+        result.append((x0 + i * h, first_ys[i]))
+    x1 = x0 + h*k
+    y0 = first_ys[k]
+    while x1 < xk:
+        x1 += h
+        y1_guess = y0 + h * sum(betas_1[k][i] * fxys[k-i] for i in range(k + 1))
+        fxy_guess = fxy(x1, y1_guess)
+        fxys.append(fxy_guess)
+        fxys.pop(0)
+        delta_y = h * sum(betas_2[k][i] * fxys[k-i] for i in range(k + 1))
+        y1 = y0 + delta_y
+        print(f"x={x1}, y={y1}, y_guess={y1_guess}, real_y={fx_real(x1)}, abs(real_y-y)={abs(fx_real(x1) - y1)}")
+        result.append((x1, y1))
+        fxys[k] = fxy(x1, y1)
+        y0 = y1
+    return result
+
+
+
+if __name__ == "__main__":
+    # 定义初始值和参数#####################################################################################
+    k = 3                    # 阿达姆斯k+1步显式方法，精度为k+1阶，最常用k=3，其他阶数我没试过   P253
+    first_ys = [1, 1.0954, 1.1832, 1.2649]  # 前几个y的值，可用龙格-库塔计算或者知道精确解自己算出来
+    x0 = 0.0                 # 初始x值
+    y0 = 1.0                 # 初始y值
+    h = 0.1                  # 步长
+    xk = 1.0                 # 最终x值
+    fxy = lambda x, y: y - 2*x/y          # 定义f(x,y)函数，dx/dy = f(x,y)，导函数
+    fx_real = lambda x: 0        # 实际解函数，用于验证结果，如果不知道或者不用算误差，可以直接写个 lambda x: 0
+    # 调用阿达姆斯显式方法
+    result = AdamusFix(k, first_ys, x0, y0, h, xk, fxy, fx_real)
+    print("计算结果看到有几.99999或者几.00000就自己四舍五入一下，有可能会多算一点，自己比较一下")
+    if result:
+        for xy in result:
+            print(f"(x, y): {xy}")