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发表于 2013-6-3 09:09:18
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本帖最后由 zouzhichao 于 2013-6-3 09:13 编辑
无级电工 发表于 2013-6-3 08:33
看起来效果不错啊。
m代码分四个:
(1)求解一阶导数GetYijieDaoshu.m
%得到一阶导数的值,公式如下
%2 * exp(b * x) * (c - y + a * exp(b * x))
%2 * a * x * exp(b * x) * (c - y + a * exp(b * x))
%2 * c - 2 * y + 2 * a * exp(b * x)
function Res = GetYijieDaoshu(Xy, Para)
Res = zeros(3, 1);
for i = 1 : size(Xy, 1)
Res(1, 1) = Res(1, 1) + 2 * exp(Para(2) * Xy(i, 1)) * (Para(3) - Xy(i, 2) + Para(1) * exp(Para(2) * Xy(i, 1)));
Res(2, 1) = Res(2, 1) + 2 * Para(1) * Xy(i, 1) * exp(Para(2) * Xy(i, 1)) * (Para(3) - Xy(i, 2) + Para(1) * exp(Para(2) * Xy(i, 1)));
Res(3, 1) = Res(3, 1) + 2 * Para(3) - 2 * Xy(i, 2) + 2 * Para(1) * exp(Para(2) * Xy(i, 1));
end
end
(2)求解二阶导数GetErjieDaoshu.m
%得到二阶导数的值,公式如下
%2 * exp(2 * b * x)
%2 * x * exp(b * x) * (c - y + a * exp(b * x)) + 2 * a * x * exp(2 * b * x)
%2 * exp(b * x)
%2 * x * exp(b * x) * (c - y + a * exp(b * x)) + 2 * a * x * exp(2 * b * x)
%2 * a^2 * x^2 * exp(2 * b * x) + 2 * a * x^2 * exp(b * x) * (c - y + a * exp(b * x))
%2 * a * x * exp(b * x)
%2 * exp(b * x)
%2 * a * x * exp(b * x)
%2
function Res = GetErjieDaoshu(Xy, Para)
Res = zeros(3, 3);
for i = 1 : size(Xy, 1)
Res(1, 1) = Res(1, 1) + 2 * exp(2 * Para(2) * Xy(i, 1));
Res(1, 2) = Res(1, 2) + 2 * Xy(i, 1) * exp(Para(2) * Xy(i, 1)) * (Para(3) - Xy(i, 2) + Para(1) * exp(Para(2) * Xy(i, 1))) + 2 * Para(1) * Xy(i, 1) * exp(2 * Para(2) * Xy(i, 1));
Res(1, 3) = Res(1, 3) + 2 * exp(Para(2) * Xy(i, 1));
Res(2, 1) = Res(2, 1) + 2 * Xy(i, 1) * exp(Para(2) * Xy(i, 1)) * (Para(3) - Xy(i, 2) + Para(1) * exp(Para(2) * Xy(i, 1))) + 2 * Para(1) * Xy(i, 1) * exp(2 * Para(2) * Xy(i, 1));
Res(2, 2) = Res(2, 2) + 2 * Para(1)^2 * Xy(i, 1)^2 * exp(2 * Para(2) * Xy(i, 1)) + 2 * Para(1) * Xy(i, 1)^2 * exp(Para(2) * Xy(i, 1)) * (Para(3) - Xy(i, 2) + Para(1) * exp(Para(2) * Xy(i, 1)));
Res(2, 3) = Res(2, 3) + 2 * Para(1) * Xy(i, 1) * exp(Para(2) * Xy(i, 1));
Res(3, 1) = Res(3, 1) + 2 * exp(Para(2) * Xy(i, 1));
Res(3, 2) = Res(3, 2) + 2 * Para(1) * Xy(i, 1) * exp(Para(2) * Xy(i, 1));
Res(3, 3) = Res(3, 3) + 2;
end
end
(3)迭代计算CalPara.m
%计算系数的值
function Res = CalPara(Xy, ParaInit, N)
Res = ParaInit;
for i = 1 : N
ParaN = GetYijieDaoshu(Xy, Res);
ParaNN = GetErjieDaoshu(Xy, Res);
Res = Res - 1 * (ParaNN \ ParaN);
end
end
(4)测试Test.m
clear;
clc;
%目标函数y = a * exp(b * x) + c
%预定参数
a = 1.7;
b = 0.2;
c = -4.6;
%待拟合的点,带上(rand(size(X)) - 0.5) * 0.2的随机偏差
X = -3 : 0.1 : 2.9;
Y = a * exp(b * X) + c + (rand(size(X)) - 0.5) * 0.2;
%拟合,注意需要给出a, b, c的初始值,不要离得太远,否则会发散导致迭代失败
%这里我们取[1.0, 1.0, -5.2]',跟预定参数[1.7, 0.2, -4.6]'距离不是很远,迭代200次
ParaABC = CalPara([X', Y'], [1.0, 1.0, -5.2]', 200);
%绘图比较
X1 = min(X) : (max(X)- min(X))/9999 : max(X);
Z = ParaABC(1) * exp(ParaABC(2) * X1) + ParaABC(3);
plot(X, Y, '*', X1, Z);
测试图:
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