137 lines
3.7 KiB
Python
137 lines
3.7 KiB
Python
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# 列主元高斯消元法
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def SovleRowMain(A,b):
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ks = 0.00000001
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n = len(A)
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if len(A[0]) != n:
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raise ValueError("A要为方阵")
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if len(b) != n:
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raise ValueError("b与A的行数不匹配")
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p = list(range(n))
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for i in range(n):
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row_max = abs(A[i][i])
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row_max_index = i
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for j in range(i + 1, n):
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if abs(A[j][i]) > row_max:
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row_max = abs(A[j][i])
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row_max_index = j
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A[i], A[row_max_index] = A[row_max_index], A[i]
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b[i], b[row_max_index] = b[row_max_index], b[i]
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p[i], p[row_max_index] = p[row_max_index], p[i]
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if abs(A[i][i]) < ks:
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raise ValueError("A矩阵奇异，无法进行高斯消元")
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for j in range(i + 1, n):
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m = A[j][i] / A[i][i]
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A[j][i] = m
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for k in range(i + 1, n):
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A[j][k] -= m * A[i][k]
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b[j] -= m * b[i]
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if abs(A[n - 1][n - 1]) < ks:
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raise ValueError("A矩阵奇异，无法进行高斯消元")
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# 回代求解
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b[n - 1] /= A[n - 1][n - 1]
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for i in range(n - 2, -1, -1):
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for j in range(i + 1, n):
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b[i] -= A[i][j] * b[j]
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b[i] /= A[i][i]
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return b
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# 最小二乘法拟合
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def LeastSquares(list_x,list_y,n):
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m = len(list_x)
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for i in range(2*n):
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tl = [list_x[j]**(i+1) for j in range(m)]
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print(f"x^{i+1} :",tl)
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print("sum:", sum(tl))
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for i in range(n+1):
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tl = [list_y[j]*list_x[j]**i for j in range(m)]
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print(f"y*x^{i}: ",tl)
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print("sum:", sum(tl))
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x_n = []
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b = []
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for i in range(2*n+1):
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x_n.append(0)
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b.append(0)
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for j in range(m):
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x_n[i]+=(list_x[j]**i)
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b[i]+=(list_y[j]*list_x[j]**i)
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b = b[:n+1]
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A = []
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for i in range(n+1):
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tmp = []
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for j in range(n+1):
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tmp.append(x_n[i+j])
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A.append(tmp)
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print("A:", A)
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print("b:", b)
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result = SovleRowMain(A, b)
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print("result:", result)
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return result
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# 计算多项式在给定x值上的值
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def CalculateY(list_x, coeff):
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re = []
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for i in range(len(list_x)):
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re.append(0)
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for j in range(len(coeff)):
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re[i] += coeff[j]*list_x[i]**j
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print("y预测:", re)
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return re
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# 计算均方根误差
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def MeanSquareErr(list_y,list_y_approx):
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m = len(list_y)
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err = 0
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print("y差值: ")
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for i in range(m):
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print(f"y-y测={list_y[i] - list_y_approx[i]}, (y-y测)^2={(list_y[i] - list_y_approx[i])**2}")
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err += (list_y[i] - list_y_approx[i])**2
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return err**0.5
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# 计算最大误差
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def MaxErr(list_y,list_y_approx):
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m = len(list_y)
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err = 0
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for i in range(m):
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if abs(list_y[i] - list_y_approx[i]) > err:
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err = abs(list_y[i] - list_y_approx[i])
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return err
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# 打印拟合方程
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def PrintEquation(coeff):
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n = len(coeff)
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str_ = str(coeff[0]) + "+"
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for i in range(0,n-1):
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str_ += str(coeff[i+1])
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str_ += "*x^"+str(i+1)+"+"
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str_ = str_[0:len(str_)-1]
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print(str_)
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#把x和y换成题干的数值###################
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x = [2,4,6,8]
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y = [2,11,28,40]
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if __name__ == "__main__":
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print("一次拟合")
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coeff = LeastSquares(x,y,1)
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PrintEquation(coeff)
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y_approx = CalculateY(x, coeff)
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# print()
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print("均方根误差:",MeanSquareErr(y,y_approx))
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# print()
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print("最大误差:",MaxErr(y,y_approx))
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print()
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print("二次拟合")
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coeff = LeastSquares(x,y,2)
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PrintEquation(coeff)
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y_approx = CalculateY(x, coeff)
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# print()
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print("均方根误差:",MeanSquareErr(y,y_approx))
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# print()
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print("最大误差:",MaxErr(y,y_approx))
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