前言
写在前面
因为编程底子差，加之数值分析虽然上课还比较认真，吧（至少在大力老师面前我还是一个乖孩子）。最不能接受的就是课程设计当理论报告写去了，所以索性把数值分析这本书重学了！！
重学的原因有几点：
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这是一本友善的英文书籍，虽然大创看过很多英文论文，但是多半是强大的DeepL翻译软件帮我克服语言障碍
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未来我需要使用**《机器学习》**里面的东西，大量依赖数值分析的基本方法！什么最速梯度下降啊啥的，太吃数值分析了！！
-
因为报告是水过去的嘛(bushi)，所以写一点忏悔一点
所以下面开始了！！
单变量方程的解
二分法
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| function zero_point = Bisection(F, endpoint_a, endpoint_b, TOL, N)
if F(endpoint_a) * F(endpoint_b) < 0
i = 0;
FA = F(endpoint_a);
while i < N
p = (endpoint_a + endpoint_b) / 2;
FP = F(p);
if FP == 0 || p - endpoint_a < TOL
zero_point = p;
break;
else
i = i + 1;
if FA * FP > 0
endpoint_a = p;
else
endpoint_b = p;
end
end
fprintf("第%d次迭代区间为:[ %f , %f ]\n", i, endpoint_a, endpoint_b);
end
else
disp('无法判断是否存在零点');
end
end
|
例如：
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| >> F = @(x)log(x) - x^0.5 + x - 3
>> Bisection(F, 1, 10, 0.00001, 100000)
第1次迭代区间为:[ 1.000000 , 5.500000 ]
第2次迭代区间为:[ 3.250000 , 5.500000 ]
第3次迭代区间为:[ 3.250000 , 4.375000 ]
第4次迭代区间为:[ 3.250000 , 3.812500 ]
第5次迭代区间为:[ 3.531250 , 3.812500 ]
第6次迭代区间为:[ 3.531250 , 3.671875 ]
第7次迭代区间为:[ 3.601563 , 3.671875 ]
第8次迭代区间为:[ 3.601563 , 3.636719 ]
第9次迭代区间为:[ 3.601563 , 3.619141 ]
第10次迭代区间为:[ 3.610352 , 3.619141 ]
第11次迭代区间为:[ 3.614746 , 3.619141 ]
第12次迭代区间为:[ 3.614746 , 3.616943 ]
第13次迭代区间为:[ 3.615845 , 3.616943 ]
第14次迭代区间为:[ 3.615845 , 3.616394 ]
第15次迭代区间为:[ 3.616119 , 3.616394 ]
第16次迭代区间为:[ 3.616119 , 3.616257 ]
第17次迭代区间为:[ 3.616188 , 3.616257 ]
第18次迭代区间为:[ 3.616188 , 3.616222 ]
第19次迭代区间为:[ 3.616205 , 3.616222 ]
ans =
3.6162
|
定点迭代法
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| function zero_point = Fixed_Point(g, p_0, TOL, N)
i = 1;
while i < N
p = g(p_0);
fprintf('第%d次迭代结果为:%f\n', i, p);
if abs(p - p_0) < TOL
zero_point = p;
break;
else
i = i + 1;
p_0 = p;
end
end
if i >= N
disp('超出最大迭代次数，无法判断零点')
end
end
|
例如：
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| >> g = @(x) 0.5 * (10 - x^3)^0.5
>> Fixed_Point(g, 1.5, 0.0001, 100000)
第1次迭代结果为:1.286954
第2次迭代结果为:1.402541
第3次迭代结果为:1.345458
第4次迭代结果为:1.375170
第5次迭代结果为:1.360094
第6次迭代结果为:1.367847
第7次迭代结果为:1.363887
第8次迭代结果为:1.365917
第9次迭代结果为:1.364878
第10次迭代结果为:1.365410
第11次迭代结果为:1.365138
第12次迭代结果为:1.365277
第13次迭代结果为:1.365206
ans =
1.3652
|
牛顿法及其拓展
牛顿迭代法
pn=pn−1−f′(pn−1)f(pn−1)
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| function zero_point = Newton(F, p_0, TOL, N)
i = 1;
f = sym(F);
df = diff(f);
dF = matlabFunction(df);
while i < N
p = p_0 - F(p_0) / dF(p_0);
fprintf('第%d次迭代结果为:%f\n', i, p);
if abs(p - p_0) < TOL
zero_point = p;
break;
else
i = i + 1;
p_0 = p;
end
end
if i >= N
disp('超出最大迭代次数，无法判断零点')
end
end
|
例如：
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| >> F = @(x)log(x) - x^0.5 + x - 3
>> Newton(F, 1, 0.00001, 100000)
第1次迭代结果为:3.000000
第2次迭代结果为:3.606360
第3次迭代结果为:3.616205
第4次迭代结果为:3.616207
ans =
3.6162
|
割线法
pn=pn−1−f(pn−1)−f(pn−2)f(pn−1)(pn−1−pn−2)
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| function zero_point = Secant(F, p_0, p_1, TOL, N)
i = 2;
q_0 = F(p_0);
q_1 = F(p_1);
while i < N
p = p_1 - q_1 * (p_1 - p_0) / (q_1 - q_0);
fprintf('第%d次迭代结果为:%f\n', i - 1, p);
if abs(p - p_1) < TOL
zero_point = p;
break;
else
i = i + 1;
p_0 = p_1;
q_0 = q_1;
p_1 = p;
q_1 = F(p);
end
end
if i >= N
disp('fail')
end
end
|
例如：
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| >> F = @(x)log(x) - x^0.5 + x - 3
>> Secant(F, 1, 10, 0.00001, 100000)
第1次迭代结果为:3.953949
第2次迭代结果为:3.599312
第3次迭代结果为:3.616313
第4次迭代结果为:3.616207
第5次迭代结果为:3.616207
ans =
3.6162
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试位法
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| function zero_point = False_Position(F, p_0, p_1, TOL, N)
i = 2;
q_0 = F(p_0);
q_1 = F(p_1);
while i < N
fprintf("第%d次迭代后，根所在的区间为:[ %f , %f ]\n", i - 1, p_0, p_1);
p = p_1 - q_1 * (p_1 - p_0) / (q_1 - q_0);
if abs(p - p_1) < TOL
zero_point = p;
break;
else
i = i + 1;
q = F(p);
if q_0 * q < 0
p_0 = p;
p_1 = p_1;
q_0 = q;
q_1 = q_1;
else
p_0 = p_0;
p_1 = p;
q_0 = q_0;
q_1 = q_1;
end
end
end
if i >= N
disp('fail')
end
end
|
例如：
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| >> F = @(x)log(x) - x^0.5 + x - 3
>> False_Position(F, 1, 10, 0.00001, 100000)
第1次迭代后，根所在的区间为:[ 1.000000 , 10.000000 ]
第2次迭代后，根所在的区间为:[ 3.953949 , 10.000000 ]
第3次迭代后，根所在的区间为:[ 3.599312 , 10.000000 ]
第4次迭代后，根所在的区间为:[ 3.617119 , 10.000000 ]
第5次迭代后，根所在的区间为:[ 3.616158 , 10.000000 ]
第6次迭代后，根所在的区间为:[ 3.616210 , 10.000000 ]
第7次迭代后，根所在的区间为:[ 3.616207 , 10.000000 ]
第8次迭代后，根所在的区间为:[ 3.616207 , 10.000000 ]
第9次迭代后，根所在的区间为:[ 3.616207 , 10.000000 ]
第10次迭代后，根所在的区间为:[ 3.616207 , 10.000000 ]
第11次迭代后，根所在的区间为:[ 3.616207 , 10.000000 ]
第12次迭代后，根所在的区间为:[ 3.616207 , 10.000000 ]
第13次迭代后，根所在的区间为:[ 3.616207 , 10.000000 ]
第14次迭代后，根所在的区间为:[ 3.616207 , 10.000000 ]
第15次迭代后，根所在的区间为:[ 3.616207 , 3.616207 ]
ans =
3.6162
|
重根问题
pn=pn−1−[f′(pn)]2−f(pn)f′′(pn)f(pn)f′(pn)
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| function zero_point = Modified_Newton(F, p_0, TOL, N)
i = 1;
f = sym(F);
df = diff(f);
ddf = diff(df);
dF = matlabFunction(df);
ddF = matlabFunction(ddf);
while i < N
p = p_0 - F(p_0) * dF(p_0) / dF(p_0)^2 - F(p_0) * ddF(p_0);
fprintf('第%d次迭代结果为:%f\n', i, p);
if abs(p - p_0) < TOL
zero_point = p;
break;
else
i = i + 1;
p_0 = p;
end
end
if i >= N
disp('超出最大迭代次数，无法判断零点')
end
end
|
例如：
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| >> F = @(x) exp(x) - x - 1
>> Modified_Newton(F, 10, 0.0000001, 10000000)
第1次迭代结果为:-484922895.285593
第2次迭代结果为:-1.000000
第3次迭代结果为:-0.553359
第4次迭代结果为:-0.325108
第5次迭代结果为:-0.188120
第6次迭代结果为:-0.104895
第7次迭代结果为:-0.056316
第8次迭代结果为:-0.029365
第9次迭代结果为:-0.015025
第10次迭代结果为:-0.007604
第11次迭代结果为:-0.003826
第12次迭代结果为:-0.001919
第13次迭代结果为:-0.000961
第14次迭代结果为:-0.000481
第15次迭代结果为:-0.000241
第16次迭代结果为:-0.000120
第17次迭代结果为:-0.000060
第18次迭代结果为:-0.000030
第19次迭代结果为:-0.000015
第20次迭代结果为:-0.000008
第21次迭代结果为:-0.000004
第22次迭代结果为:-0.000002
第23次迭代结果为:-0.000001
第24次迭代结果为:-0.000000
第25次迭代结果为:-0.000000
第26次迭代结果为:-0.000000
第27次迭代结果为:-0.000000
ans =
-5.8954e-08
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加速收敛法
Steffensen法
Δpn=pn+1−pnΔkpn=Δ(Δk−1pn)pn=pn−1−Δ2pn(Δpn)2
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| function zero_point = Steffensen(g, p_0, TOL, N)
i = 1;
while i < N
p_1 = g(p_0);
p_2 = g(p_1);
p = p_0 - (p_1 - p_0)^2 / (p_2 - 2 * p_1 + p_0);
fprintf('第%d次迭代结果为:%f\n', i, p);
if abs(p - p_0) < TOL
zero_point = p;
break;
else
i = i + 1;
p_0 = p;
end
end
if i >= N
disp('fail')
end
end
|
例如：
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| >> g = @(x) 0.5 * (10 - x^3)^0.5
>> Steffensen(g, 1.5, 0.0001, 100000)
第1次迭代结果为:1.361886
第2次迭代结果为:1.365228
第3次迭代结果为:1.365230
ans =
1.3652
>> Fixed_Point(g, 1.5, 0.0001, 100000)
第1次迭代结果为:1.286954
第2次迭代结果为:1.402541
第3次迭代结果为:1.345458
第4次迭代结果为:1.375170
第5次迭代结果为:1.360094
第6次迭代结果为:1.367847
第7次迭代结果为:1.363887
第8次迭代结果为:1.365917
第9次迭代结果为:1.364878
第10次迭代结果为:1.365410
第11次迭代结果为:1.365138
第12次迭代结果为:1.365277
第13次迭代结果为:1.365206
ans =
1.3652
|
多项式的零点及Muller方法
Hoener
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| function [y, z] = Horner(n, aa, x_0)
disp('多项式为：')
poly2sym(fliplr(aa))
y = aa(n + 1);
z = aa(n + 1);
for j = n - 1:-1:1
y = x_0 * y + aa(j + 1);
z = x_0 * z + y;
end
y = x_0 * y + aa(1);
end
|
例如：
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| >> aa = [-4, 3, -3, 0, 2]
>> [p, p_1] = Horner(4, aa, -2)
|
Muller
其实书上没讲多细，连证明收敛性都没有，讲真书上代码很牵强
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| function zero_point = Muller(F, p_0, p_1, p_2, TOL, N)
i = 3;
h1 = p_1 - p_0;
h2 = p_2 - p_1;
f1 = (F(p_1) - F(p_0)) / h1;
f2 = (F(p_2) - F(p_1)) / h2;
a = (f2 - f1) / (h2 + h1);
while i < N
b = f2 + h2 * a;
c = F(p_2);
delta = (b^2 - 4 * a * c)^0.5;
if abs(b - delta) < abs(b + delta)
e = b + delta;
else
e = b - delta;
end
h = -2 * c / e;
p = p_2 + h;
fprintf('第%d次迭代结果为:%f\n', i - 2, p);
if abs(h) < TOL
zero_point = p;
break;
end
p_0 = p_1;
p_1 = p_2;
p_2 = p;
h1 = p_1 - p_0;
h2 = p_2 - p_1;
f1 = (F(p_1) - F(p_0)) / h1;
f2 = (F(p_2) - F(p_1)) / h2;
a = (f2 - f1) / (h2 + h1);
i = i + 1;
end
if i >= N
disp('fail')
end
end
|
例如：
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| >> F = @(x)x^4 - 3 * x^3 + x^2 + x + 1
>> Muller(F, 0.5, -0.5, 0, 0.00001, 100000)
第1次迭代结果为:-0.100000
第2次迭代结果为:-0.492146
第3次迭代结果为:-0.352226
第4次迭代结果为:-0.340229
第5次迭代结果为:-0.339095
第6次迭代结果为:-0.339093
第7次迭代结果为:-0.339093
ans =
-0.3391 - 0.4466i
|
插值多项式逼近
Lagrange插值法
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| function P = Lagrange(F, xx, endpoint_a, endpoint_b)
syms x;
P = 0;
n = length(xx);
n = n - 1;
for k = 0:n
m = 1;
d = 1;
yy(k + 1) = F(xx(k + 1));
for i = 0:n
if i == k
continue;
end
m = m * (x - xx(i + 1));
d = d * (xx(k + 1) - xx(i + 1));
end
L = m / d;
P = P + L * yy(k + 1);
end
simplify(P)
p = matlabFunction(P);
fplot(F, [endpoint_a, endpoint_b]); hold on;
plot(xx, yy, '>'); hold on;
fplot(p, [endpoint_a, endpoint_b]);
end
|
例如：
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| >> xx = [2, 2.75, 4]
>> yy = 1 ./ xx
>> plot(xx, yy, '>')
>> hold on
>> P = Lagrange(xx, yy)
>> fplot(P, [0.5, 5])
>> hold on
>> fplot(@(x)1 ./ x, [0.5, 5])
>> legend('插值点', '拉格朗日插值多项式P(x)', 'y=1/x')
|
Neville插值法
Neville插值多项式
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| function Q = Neville(xx, yy)
syms x;
n = length(xx);
n = n - 1;
QQ(1, 1) = x;
for k = 0:n
QQ(k + 1, 1) = yy(k + 1);
end
for i = 1:n
for j = 1:i
QQ(i + 1, j + 1) = ((x - xx(i - j +1)) * QQ(i + 1, j) - (x - xx(i +1)) * QQ(i, j)) / (xx(i + 1) - xx(i - j + 1));
end
end
Q = QQ(n + 1, n + 1);
simplify(Q);
Q = matlabFunction(Q);
disp('Neville插值表为:')
QQ
end
|
例如：
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| >> xx = [1, 1.3, 1.6, 1.9, 2.2]
>> yy = [0.7651977, 0.6200860, 0.4554022, 0.2818186, 0.1103623]
>> plot(xx, yy, '>')
>> hold on
>> Q = Neville(xx, yy)
>> fplot(Q, [0.5, 5])
>> legend('插值点', 'Neville插值多项式Q(x)')
|
Neville插值表求值
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| function QQ = NevilleTable(xx, yy, TOL, x)
n = length(xx);
QQ = zeros(n, n);
for k = 1:n
QQ(k, 1) = yy(k);
end
for i = 2:n
for j = 2:i
QQ(i, j) = ((x - xx(i - j + 1)) * QQ(i, j - 1) - (x - xx(i)) * QQ(i - 1, j - 1)) / (xx(i) - xx(i - j + 1));
end
if abs(QQ(i, i) - QQ(i - 1, i - 1)) < TOL
break
end
end
end
|
例如：
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| >> xx = [1, 1.3, 1.6, 1.9, 2.2]
>> yy = [0.7651977, 0.6200860, 0.4554022, 0.2818186, 0.1103623]
>> NevilleTable(xx, yy, 0.0001, 1.5)
ans =
0.7652 0 0 0 0
0.6201 0.5233 0 0 0
0.4554 0.5103 0.5125 0 0
0.2818 0.5133 0.5113 0.5118 0
0.1104 0.5104 0.5137 0.5118 0.5118
|
牛顿插值法
牛顿插值多项式
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| function F = Newton_Divided_Difference(xx, ff)
syms x;
n = length(xx);
n = n - 1;
FF = zeros(n, n);
for k = 0:n
FF(k + 1, 1) = ff(k + 1);
end
for i = 1:n
for j = 1:i
FF(i + 1, j + 1) = (FF(i + 1, j) - FF(i, j)) / (xx(i + 1) - xx(i - j + 1));
end
end
FF
F = FF(1, 1);
for i = 1:n
f = FF(i + 1, i + 1);
for j = 1:i
f = f * (x - xx(j));
end
F = F + f;
end
simplify(F);
F = matlabFunction(F);
end
|
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| >> xx = [1, 1.3, 1.6, 1.9, 2.2]
>> yy = [0.7651977, 0.6200860, 0.4554022, 0.2818186, 0.1103623]
>> plot(xx, yy, '>')
>> hold on
>> F = Newton_Divided_Difference(xx, yy)
>> fplot(F, [0, 3])
>> legend('插值点', '牛顿插值多项式F(x)')
|
牛顿前插公式
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| function F_x = Newton_Forward_Difference(F, x_0, n, step_length, TOL, x)
FF = zeros(n + 1, n + 1);
s = (x - x_0) / step_length;
for k = 0:n
xx(k + 1) = x_0 + k * step_length;
yy(k + 1) = F(xx(k + 1));
end
for k = 0:n
FF(k + 1, 1) = yy(k + 1);
end
for i = 1:n
for j = 1:i
FF(i + 1, j + 1) = (FF(i + 1, j) - FF(i, j)) / (xx(i + 1) - xx(i - j + 1));
end
if abs(FF(i + 1, i + 1)) < TOL
break;
end
end
FF
F_x = FF(1, 1);
for i = 1:n
ss = 1;
for j = 1:i
ss = ss * (s - j + 1);
end
f = FF(i + 1, i + 1) * step_length^i * ss;
F_x = F_x + f;
end
fprintf('比较：F(x) = %f', F(x));
end
|
牛顿后插公式
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| function F_x = Newton_Backward_Difference(F, x_n, n, step_length, TOL, x)
FF = zeros(n + 1, n + 1);
s = (x - x_n) / step_length;
for k = n:-1:0
xx(k + 1) = x_n - (n - k) * step_length;
yy(k + 1) = F(xx(k + 1));
end
for k = 0:n
FF(k + 1, 1) = yy(k + 1);
end
for i = 1:n
for j = 1:i
FF(n - i + j + 1, j + 1) = (FF(n - i + j + 1, j) - FF(n - i + j, j)) / (xx(n - i + j + 1) - xx(n - i + 1));
end
if abs(FF(n + 1, i + 1)) < TOL
break;
end
end
FF
F_x = FF(n + 1, 1);
for i = 1:n
ss = 1;
for j = 1:i
ss = ss * (s + j - 1);
end
f = FF(n + 1, i + 1) * step_length^i * ss;
F_x = F_x + f;
end
fprintf('比较：F(x) = %f', F(x));
end
|
Stirling
书上没写原理，可以理解成耍流氓吗？
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| function F_x = Stirling(F, x_0, n, step_length, TOL, x)
if mod(n, 2) == 1
n = n - 1;
end
FF = zeros(n + 1, n + 1);
s = (x - x_0) / step_length;
m = n / 2;
xx(m + 1) = x_0;
for k = 1:m
xx(m + 1 + k) = x_0 + k * step_length;
xx(m + 1 - k) = x_0 - k * step_length;
end
for k = 1:n + 1
FF(k, 1) = F(xx(k));
end
mark = n + 1;
for i = 2:n + 1
if mod(i, 2) == 0
for j = 2:i
FF((m + 1) + i / 2, j) = (FF((m + 1) + i / 2, j - 1) - FF((m + 1) + i / 2 - 1, j - 1)) / (xx((m + 1) + i / 2) - xx((m + 1) + i / 2 - (j - 1)));
FF((m + 1) - i / 2 + j - 1, j) = (FF((m + 1) - i / 2 + j - 1, j - 1) - FF((m + 1) - i / 2 + j - 2, j - 1)) / (xx((m + 1) - i / 2 + j - 1) - xx((m + 1) - i / 2));
end
else
FF((m + 1) + (i - 1) / 2, i) = (FF((m + 1) + (i - 1) / 2, i - 1) - FF((m + 1) + (i - 1) / 2 - 1, i - 1)) / (xx((m + 1) + (i - 1) / 2) - xx((m + 1) - (i - 1) / 2));
if abs(FF((m + 1) + (i - 1) / 2, i)) < TOL
mark = i;
break;
end
end
end
F_x = FF(m + 1, 1) + s * step_length * (FF(m + 1, 2) + FF(m + 2, 2)) / 2 + s^2 * step_length^2 * FF(m + 2, 3);
if mark > 3
S1 = 1;
S2 = 1;
for k = 4:n + 1
if mod(k, 2) == 0
S1 = S1 * (s^2 - (k / 2 - 1)^2);
F_x = F_x + (FF((m + 1) + k / 2, k) +FF((m + 1) + k / 2 - 1, k)) * (s * S1) * step_length^(k - 1) / 2;
else
S2 = S2 * (s^2 - (k / 2 - 1/2)^2);
F_x = F_x + FF((m + 1) + (k - 1) / 2, k) * (s^2 * S2) * step_length^(k - 1);
end
end
end
end
|
比较一下下面的两个差距，明白前插与后插的区别，就近原则嘛~
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| >> F = @(x) sin(x)
>> Newton_Forward_Difference(F, 0, 8, 0.3, 0.01, 0.2)
>> Newton_Forward_Difference(F, 0, 8, 0.3, 0.01, 2)
>> Newton_Backward_Difference(F, 2, 8, 0.3, 0.01, 1.8)
>> Newton_Backward_Difference(F, 2, 8, 0.3, 0.01, 0.2)
|
Hermite插值
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| function H = Hermite(xx, ff, dff)
syms x;
n = length(xx);
Q = zeros(2 * n, 2 * n);
n = n - 1;
for k = 1:n + 1
z(2 * k - 1) = xx(k);
z(2 * k) = xx(k);
Q(2 * k - 1, 1) = ff(k);
Q(2 * k, 1) = ff(k);
Q(2 * k, 2) = dff(k);
if k > 1
Q(2 * k - 1, 2) = (Q(2 * k - 1, 1) - Q(2 * k - 2, 1)) / (z(2 * k - 1) - z(2 * k - 2));
end
end
for i = 2:2 * n + 1
for j = 2:i
Q(i + 1, j + 1) = (Q(i + 1, j) - Q(i, j)) / (z(i + 1) - z(i - j + 1));
end
end
H = Q(1, 1);
omega = 1;
for k = 1:2 * n + 1
omega = omega * (x - z(k));
H = H + Q(k + 1, k + 1) * omega;
end
simplify(H);
H = matlabFunction(H);
end
|
例如：
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| >> xx = 1.3:0.3:1.9
>> ff = [0.6200860, 0.4554022, 0.2818186]
>> dff = [-0.5220232, -0.5698959, -0.5811571]
>> plot(xx, ff, '>');
>> hold on;
>> H = Hermite(xx, ff, dff)
>> fplot(H, [0, 3])
>> legend('插值点', 'Hermite插值H(x)')
|
三次样条插值
分段插值
平时用的plot函数就是这样画图的
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| function y = piecewise_liner(xx, yy)
syms x;
n = length(xx);
n = n - 1;
y = 0;
for k = 1:n
a = (yy(k + 1) - yy(k)) / (xx(k + 1) - xx(k));
y_k(k) = a * (x - xx(k)) + yy(k);
end
for k = 1:n
y = piecewise(xx(k) <= x <= xx(k + 1), y_k(k), y);
end
end
|
例如：
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| >> F = @(x)sin(x)
>> xx = -4:0.5:4
>> yy = F(xx)
>> plot(xx, yy, '>');
>> hold on;
>> y = piecewise_liner(xx, yy)
>> fplot(y, [-4, 4])
>> legend('插值点', '分段插值y(x)')
|
自然二次样条
条件为：
Sj=ai+bi(x−xi)+ci(x−xi)2,其中x∈[xj,xj+1],对于j=0,1,…,n−1成立Sj(xj)=f(xj)且Sj(xj+1)=f(xj+1),对于j=0,1,…,n−1成立Sj+1(xj+1)=Sj(xj+1),对于j=0,1,…,n−2成立Sj+1′(xj+1)=Sj′(xj+1),对于j=0,1,…,n−2成立
附加边界条件：(自然边界)S0′=Sn−1′=0
但是不需要使用Sn−1′=0，使用Sn−1(xn)=f(xn)才能保证最后一个函数过最后一个点
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| function S = Quadratic_Splines(xx, yy)
syms x;
n0 = length(xx);
S = 0;
for k = 1:n0
a(k) = yy(k);
end
n = n0 - 1;
for k = 1:n
h(k) = xx(k + 1) - xx(k);
end
b(1) = 0;
for k = 1:n - 1
b(k + 1) = 2 * (a(k + 1) - a(k)) / h(k) - b(k);
c(k) = (b(k + 1) - b(k)) / (2 * h(k));
end
c(n) = (a(n + 1) - a(n) - b(n) * h(n)) / h(n)^2;
for k = 1:n
s(k) = a(k) + b(k) * (x - xx(k)) + c(k) * (x - xx(k))^2;
S = piecewise(xx(k) <= x <= xx(k + 1), s(k), S);
end
end
|
例如：说实话不咋地
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| >> F = @(x)exp(x)
>> xx = 1:3
>> yy = F(xx)
>> plot(xx, yy, '>');
>> hold on;
>> S = Quadratic_Splines(xx, yy)
>> fplot(S, [1, 3])
>> hold on;
>> fplot(@(x)exp(x), [1, 3])
>> legend('插值点', '自然二次样条S(x)','e^x')
|
自然三次样条
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| function S = Natural_Cubic_Spline(xx, yy)
syms x;
n0 = length(xx);
S = 0;
for k = 1:n0
a(k) = yy(k);
end
n = n0 - 1;
for k = 1:n
h(k) = xx(k + 1) - xx(k);
end
for i = 1:n - 1
b(i + 1) = 3 * (a(i + 2) - a(i + 1)) / h(i + 1) - 3 * (a(i + 1) - a(i)) / h(k);
end
l(1) = 1;
u(1) = 0;
z(1) = 0;
for i = 1:n - 1
l(i + 1) = 2 * (xx(i + 2) - xx(i)) - h(i) * u(i);
u(i + 1) = h(i + 1) / l(i + 1);
z(i + 1) = (b(i + 1) - h(i) * z(i)) / l(i + 1);
end
l(n + 1) = 1;
z(n + 1) = 0;
c(n + 1) = 0;
for j = n - 1:-1:0
c(j + 1) = z(j + 1) - u(j + 1) * c(j + 2);
b(j + 1) = (a(j + 2) - a(j + 1)) / h(j + 1) - h(j + 1) * (c(j + 2) + 2 * c(j + 1)) / 3;
d(j + 1) = (c(j + 2) - c(j + 1)) / (3 * (h(j + 1)));
end
for k = 1:n
s(k) = a(k) + b(k) * (x - xx(k)) + c(k) * (x - xx(k))^2 +d(k) * (x - xx(k))^3;
S = piecewise(xx(k) < x <= xx(k + 1), s(k), S);
end
end
|
例如：
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| >> F = @(x)exp(x)
>> xx = 1:3
>> yy = F(xx)
>> plot(xx, yy, '>');
>> hold on;
>> S = Natural_Cubic_Spline(xx, yy)
>> fplot(S, [1, 3])
>> hold on;
>> fplot(@(x)exp(x), [1, 3])
>> legend('插值点', '自然三次样条S(x)','e^x')
|
固支三次样条(Clamped Cubic Spline)
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| function S = Clamped_Cubic_Spline(xx, yy, df_0, df_n)
syms x;
n0 = length(xx);
for k = 1:n0
a(k) = yy(k);
end
n = n0 - 1;
for k = 1:n
h(k) = xx(k + 1) - xx(k);
end
b(1) = 3 * (a(2) - a(1)) / h(1) -3 * df_0;
b(n + 1) = 3 * df_n - 3 * (a(n + 1) - a(n)) / h(n);
for i = 1:n - 1
b(i + 1) = 3 * (a(i + 2) - a(i + 1)) / h(i + 1) - 3 * (a(i + 1) - a(i)) / h(k)
end
l(1) = 2 * h(1);
u(1) = 0.5;
z(1) = b(1) / l(1);
for i = 1:n - 1
l(i + 1) = 2 * (xx(i + 2) - xx(i)) - h(i) * u(i);
u(i + 1) = h(i + 1) / l(i + 1);
z(i + 1) = (b(i + 1) - h(i) * z(i)) / l(i + 1);
end
l(n + 1) = h(n) * (2 - u(n));
z(n + 1) = (b(n + 1) - h(n) * z(n)) / l(n + 1);
c(n + 1) = z(n + 1);
for j = n - 1:-1:0
c(j + 1) = z(j + 1) - u(j + 1) * c(j + 2);
b(j + 1) = (a(j + 2) - a(j + 1)) / h(j + 1) - h(j + 1) * (c(j + 2) + 2 * c(j + 1)) / 3;
d(j + 1) = (c(j + 2) - c(j + 1)) / (3 * (h(j + 1)));
end
for k = 1:n
S(k) = a(k) + b(k) * (x - xx(k)) + c(k) * (x - xx(k))^2 +d(k) * (x - xx(k))^3;
simply = simplify(S(k));
s_k = matlabFunction(simply);
fplot(s_k, [xx(k), xx(k + 1)]); hold on;
end
plot(xx, yy, '>'); hold on;
end
|
例如：
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| >> F = @(x)exp(x)
>> xx = 1:3
>> yy = F(xx)
>> plot(xx, yy, '>');
>> hold on;
>> f = sym(F)
>> df = diff(f, 1)
>> dF = matlabFunction(df)
>> df_0 = dF(0)
>> df_n = dF(3)
>> S = Clamped_Cubic_Spline(xx, yy, df_0, df_n)
>> fplot(S, [1, 3])
>> hold on;
>> fplot(@(x)exp(x), [1, 3])
>> legend('插值点', '固支三次样条S(x)','e^x')
|
贝塞尔曲线
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| function C = Bezier(xx, yy, xxleft, yyleft, xxright, yyright)
syms t;
n = length(xx);
n = n -1;
for k = 0:n - 1
a_0(k + 1) = xx(k + 1);
b_0(k + 1) = yy(k + 1);
a_1(k + 1) = 3 * (xxleft(k + 1) - xx(k + 1));
b_1(k + 1) = 3 * (yyleft(k + 1) - yy(k + 1));
a_2(k + 1) = 3 * (xx(k + 1) + xxright(k + 1) - 2 * xxleft(k + 1));
b_2(k + 1) = 3 * (yy(k + 1) + yyright(k + 1) - 2 * yyleft(k + 1));
a_3(k + 1) = xx(k + 2) - xx(k + 1) + 3 * xxleft(k + 1) - 3 * xxright(k + 1);
b_3(k + 1) = yy(k + 2) - yy(k + 1) + 3 * yyleft(k + 1) - 3 * yyright(k + 1);
x(k + 1) = a_0 + a_1 * t + a_2 * t^2 + a_3 * t^3;
C(k + 1, 1) = x(k + 1);
line([xx(k+1),xxleft(k+1)],[yy(k+1),yyleft(k+1)]); hold on;
line([xx(k+2),xxright(k+1)],[yy(k+2),yyright(k+1)]); hold on;
y(k + 1) = b_0 + b_1 * t + b_2 * t^2 + b_3 * t^3;
C(k + 1, 2) = y(k + 1);
fplot(x(k+1),y(k+1),[0,1])
end
end
|
觉得这种拟合曲线的方法还不如直接对两个坐标三次样条算了，拉格朗日也比上面的算法强的不知道哪里去了
例如：什么牛马，没有例子！！
数值微分与数值积分
数值微分
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| function Numerical_Differentiation(F, x_0, h)
f = sym(F);
df = diff(f, 1);
dF = matlabFunction(df);
df0 = dF(x_0)
df1 = (F(x_0 + h) - F(x_0)) / h;
fprintf('前插公式：df(x_0)=%f\r\n', df1);
error1 = abs(df0 - df1);
fprintf('误差为：error=%f\r\n', error1);
df2 = (-3 * F(x_0) + 4 * F(x_0 + h) - F(x_0 + 2 * h)) / (2 * h);
fprintf('三点端点公式：df(x_0)=%f\r\n', df2);
error2 = abs(df0 - df2);
fprintf('误差为：error=%f\r\n', error2);
df3 = (F(x_0 + h) - F(x_0 - h)) / (2 * h);
fprintf('三点中点公式：df(x_0)=%f\r\n', df3);
error3 = abs(df0 - df3);
fprintf('误差为：error=%f\r\n', error3);
df4 = (F(x_0 - 2 * h) - 8 * F(x_0 - h) + 8 * F(x_0 + h) - F(x_0 + 2 * h)) / (12 * h);
fprintf('五点中点公式：df(x_0)=%f\r\n', df4);
error4 = abs(df0 - df4);
fprintf('误差为：error=%f\r\n', error4);
df5 = (-25 * F(x_0) + 48 * F(x_0 +h) - 36 * F(x_0 + 2 * h) + 16 * F(x_0 + 3 * h) - 3 * F(x_0 + 4 * h)) / (12 * h);
fprintf('五点端点公式：df(x_0)=%f\r\n', df5);
error5 = abs(df0 - df5);
fprintf('误差为：error=%f\r\n', error5);
end
|
例如：
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| >> F = @(x)exp(x)
>> Numerical_Differentiation(F, 1, 0.01)
前插公式：df(x_0)=2.731919
误差为：error=0.013637
三点端点公式：df(x_0)=2.718191
误差为：error=0.000091
三点中点公式：df(x_0)=2.718327
误差为：error=0.000045
五点中点公式：df(x_0)=2.718282
误差为：error=0.000000
五点端点公式：df(x_0)=2.718282
误差为：error=0.000000
|
理查德外推
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| function Richardson_Extrapolation(F, x_0, h, n, TOL)
f = sym(F);
df = diff(f, 1);
dF = matlabFunction(df);
df0 = dF(x_0)
for k = 1:n
N1(k, 1) = (F(x_0 + h / k) - F(x_0 - h / k)) / (2 * h / k);
N2(k, 1) = (F(x_0 + h / k) - F(x_0 - h / k)) / (2 * h / k);
end
for i = 2:n
for j = i:n
N1(j, i) = N1(j, i - 1) + (N1(j, i - 1) - N1(j - 1, i - 1)) / (2^(i - 1) - 1);
N2(j, i) = N1(j, i - 1) + (N1(j, i - 1) - N1(j - 1, i - 1)) / (4^(i - 1) - 1);
end
if abs(N1(i, i) - N1(i - 1, i - 1)) < TOL
break;
end
end
N1
N2
end
|
例如：
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| >> F = @(x)exp(x)
>> Richardson_Extrapolation(F, 1, 0.1, 6, 0.0001)
N1 =
2.7228 0 0 0 0
2.7194 2.7160 0 0 0
2.7188 2.7182 2.7189 0 0
2.7186 2.7183 2.7184 2.7183 0
2.7185 2.7184 2.7184 2.7184 2.7184
2.7184 2.7184 2.7183 2.7183 2.7183
N2 =
2.7228 0 0 0 0
2.7194 2.7183 0 0 0
2.7188 2.7186 2.7183 0 0
2.7186 2.7185 2.7184 2.7184 0
2.7185 2.7184 2.7184 2.7184 2.7184
2.7184 2.7184 2.7184 2.7183 2.7183
|
数值积分
闭合式牛顿科特斯公式
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| function I = Closed_Newton_Cotes(F, n, endpoint_a, endpoint_b)
syms x;
I = 0;
P = 0;
h = (endpoint_b - endpoint_a) / n;
for i = 0:n
L = 1;
for j = 0:n
if j == i
continue;
end
L = L * (x - (endpoint_a + j * h)) / ((endpoint_a + i * h) - (endpoint_a + j * h));
end
p = sym2poly(L);
q = polyint(p);
a = diff(polyval(q, [endpoint_a endpoint_b]));
P = P + L * F(endpoint_a + i * h);
I = I + a * F(endpoint_a + i * h);
end
end
|
例如：
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| >> F = @(x)sin(x)
>> Closed_Newton_Cotes(F, 1, 0, pi / 4)
ans =
0.2777
>> Closed_Newton_Cotes(F, 2, 0, pi / 4)
ans =
0.2929
>> Closed_Newton_Cotes(F, 3, 0, pi / 4)
ans =
0.2929
>> Closed_Newton_Cotes(F, 4, 0, pi / 4)
ans =
0.2929
|
开放式牛顿科特斯公式
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| function I = Open_Newton_Cotes(F, n, endpoint_a, endpoint_b)
syms x;
I = 0;
P = 0;
h = (endpoint_b - endpoint_a) / (n + 2);
for i = 0:n
L = 1;
for j = 0:n
if j == i
continue;
end
L = L * (x - (endpoint_a + (j + 1) * h)) / ((endpoint_a + (i + 1) * h) - (endpoint_a + (j + 1) * h));
end
if L ~= 1
p = sym2poly(L);
q = polyint(p);
a = diff(polyval(q, [endpoint_a endpoint_b]));
else
a = endpoint_b - endpoint_a;
end
P = P + L * F(endpoint_a + (i + 1) * h);
I = I + a * F(endpoint_a + (i + 1) * h);
end
end
|
例如：
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| >> F = @(x)sin(x)
>> Open_Newton_Cotes(F, 0, 0, pi / 4)
ans =
0.3006
>> Open_Newton_Cotes(F, 1, 0, pi / 4)
ans =
0.2980
>> Open_Newton_Cotes(F, 2, 0, pi / 4)
ans =
0.2929
>> Open_Newton_Cotes(F, 3, 0, pi / 4)
ans =
0.2929
|
复合数值积分
复化Simpson’s Rule
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| function XI = Composite_Simpson(F, n, endpoint_a, endpoint_b)
if mod(n, 2) == 1
n = n + 1;
end
h = (endpoint_b - endpoint_a) / n;
XI0 = F(endpoint_a) + F(endpoint_b);
XI1 = 0;
XI2 = 0;
for i = 1:n - 1
X = endpoint_a + i * h;
if mod(i, 2) == 0
XI2 = XI2 + F(X);
else
XI1 = XI1 + F(X);
end
end
XI = h / 3 * (XI0 + 2 * XI2 + 4 * XI1);
end
|
例如：
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| >> F = @(x)sin(x)
>> Composite_Simpson(F, 8, 0, 4)
ans =
1.6542
|
复化梯形公式
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| function XI = Composite_Trapezoidal(F, n, endpoint_a, endpoint_b)
h = (endpoint_b - endpoint_a) / n;
XI0 = F(endpoint_a) + F(endpoint_b);
XI1 = 0;
for i = 1:n - 1
XI1 = XI1 + F(endpoint_a + i * h);
end
XI = h / 2 * (XI0 + 2 * XI1);
end
|
例如：
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| >> F = @(x)sin(x)
>> Composite_Trapezoidal(F, 8, 0, 4)
ans =
1.6190
|
复化中点公式
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| function XI = Composite_Midpoint(F, n, endpoint_a, endpoint_b)
if mod(n, 2) == 1
n = n + 1;
end
h = (endpoint_b - endpoint_a) / (n + 2);
XI1 = 0;
for i = 0:n / 2
XI1 = XI1 + F(endpoint_a + (2 * i + 1) * h);
end
XI = 2 * h * XI1;
end
|
例如：
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| >> F = @(x)sin(x)
>> Composite_Midpoint(F, 8, 0, 4)
ans =
1.6986
|
龙贝格积分
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| function R = Romberg(F, n, endpoint_a, endpoint_b, TOL)
h(1) = endpoint_b - endpoint_a;
R(1, 1) = h / 2 * (F(endpoint_a) + F(endpoint_b));
for i = 2:n
sum_f = 0;
K = 2^(i - 2);
h(i) = h(i - 1) / 2;
for k = 1:K
sum_f = sum_f + F(endpoint_a + (2 * k - 1) * h(i));
end
R(i, 1) = 1/2 * (R(i - 1, 1) + h(i - 1) * sum_f);
for j = 2:i
R(i, j) = R(i, j - 1) + (R(i, j - 1) - R(i - 1, j - 1)) / (4^(j - 1) - 1);
end
if abs(R(i, i) - R(i - 1, i - 1)) < TOL
break;
end
end
end
|
例如：
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| >> F = @(x)sin(x)
>> Romberg(F, 6, 0, pi, 0.00000001)
ans =
0.0000 0 0 0 0 0
1.5708 2.0944 0 0 0 0
1.8961 2.0046 1.9986 0 0 0
1.9742 2.0003 2.0000 2.0000 0 0
1.9936 2.0000 2.0000 2.0000 2.0000 0
1.9984 2.0000 2.0000 2.0000 2.0000 2.0000
|
自适应求积
堆栈法
算法本质上是属于迭代，书上把迭代转换成了堆栈
理解：在数据结构我们学习了二叉树，其中三种遍历方法：先序遍历，中序遍历以及后序遍历。例如先序遍历是在每一个节点上，先访问该节点，然后依次访问左节点与右节点。
而在这个算法中，我们对每一个积分区间进行折半，先对右半区间进行积分计算，再对左半区间进行积分计算。如果在其中一个积分区间不满足条件∣∣Si(a,b)−Si(a,2a+b)−Si(2a+b,b)∣∣<TOLi，则继续细分子区间，并且优先计算右半区间。
因此取名为右半序遍历自适应积分算法不是不行，当然稍微改一下也可以成为左半序区间遍历算法。
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| function APP = Adaptive_Quadrature(F, TOL, N, endpoint_a, endpoint_b)
APP = 0;
i = 1;
tol(i) = 10 * TOL;
a(i) = endpoint_a;
h(i) = (endpoint_b - endpoint_a) / 2;
FA(i) = F(a(i));
FC(i) = F(a(i) + h(i));
FB(i) = F(endpoint_b);
S(i) = h(i) / 3 * (FA(i) + 4 * FC(i) + FB(i));
mark(i) = 1;
while i > 0
FD = F(a(i) + h(i) / 2);
FE = F(a(i) + 3/2 * h(i));
S1 = h(i) / 6 * (FA(i) + 4 * FD + FC(i));
S2 = h(i) / 6 * (FC(i) + 4 * FE + FB(i));
v1 = a(i);
v2 = FA(i);
v3 = FC(i);
v4 = FB(i);
v5 = h(i);
v6 = tol(i);
v7 = S(i);
v8 = mark(i);
i = i - 1;
if abs(S1 +S2 -v7) < v6
APP = APP + S1 + S2;
else
if v8 > N
disp('超过最大迭代次数');
break;
else
i = i + 1;
a(i) = v1 + v5;
FA(i) = v3;
FC(i) = FE;
FB(i) = v4;
h(i) = v5 / 2;
tol(i) = v6 / 2;
S(i) = S2;
mark(i) = v8 + 1;
i = i + 1;
a(i) = v1;
FA(i) = v2;
FC(i) = FD;
FB(i) = v3;
h(i) = h(i - 1);
tol(i) = tol(i - 1);
S(i) = S1;
mark(i) = mark(i - 1);
end
end
end
end
|
例如：
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| >> F = @(x)(100 / x^2) * sin(10 / x)
>> Adaptive_Quadrature(F, 0.000001, 1000000, 1, 3)
ans =
-1.4260
|
递归法
显然递归法代码简洁(不是一点两点的简洁)，且方便理解，但是这里没有设置迭代次数上限
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| function I = Adaptive_Quadrature_Recursion(F, TOL, endpoint_a, endpoint_b)
h = (endpoint_b - endpoint_a) / 2;
FA = F(endpoint_a);
FC = F(endpoint_a + h);
FB = F(endpoint_b);
S_0 = h / 3 * (FA +4 * FC + FB);
FD = F(endpoint_a + h / 2);
FE = F(endpoint_a + 3 * h / 2);
S_1 = h / 6 * (FA + 4 * FD + FC);
S_2 = h / 6 * (FC + 4 * FE + FB);
if abs(S_0 - S_1 - S_2) < TOL
I = S_0;
else
I = Adaptive_Quadrature_Recursion(F, TOL / 2, endpoint_a, endpoint_a + h) + Adaptive_Quadrature_Recursion(F, TOL / 2, endpoint_a + h, endpoint_b);
end
end
|
例如：
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| >> F = @(x)(100 / x^2) * sin(10 / x)
>> Adaptive_Quadrature_Recursion(F, 0.00001, 1, 3)
ans =
-1.4260
|
多重积分
Simpson二重积分
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| function I = Simpson_Double_Integral(F, c, d, n, m, endpoint_a, endpoint_b)
h = (endpoint_b - endpoint_a) / n;
J1 = 0;
J2 = 0;
J3 = 0;
for i = 0:n
x_i = endpoint_a + i * h;
k = (d(x_i) - c(x_i)) / m;
K1 = F(x_i, c(x_i)) + F(x_i, d(x_i));
K2 = 0;
K3 = 0;
for j = 1:m - 1
y_i = c(x_i) + j * k;
Q = F(x_i, y_i);
if mod(j, 2) == 0
K2 = K2 + Q;
else
K3 = K3 + Q;
end
end
L = k / 3 * (K1 + 2 * K2 +4 * K3);
if i == 0 || i == n
J1 = J1 + L;
elseif mod(i, 2) == 0
J2 = J2 + L;
else
J3 = J3 + L;
end
end
I = h / 3 * (J1 +2 * J2 +4 * J3);
end
|
例如：
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| >> F = @(x, y)exp(y / x)
>> c = @(x)x^3
>> d = @(x)x^2
>> Simpson_Double_Integral(F, c, d, 10, 10, 0.1, 0.5)
ans =
0.0333
|
瑕积分
方法只局限于类似于$$\int_{a}^{b} \frac{dx}{(x-a)^p}$$
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| function I = Improper(F, endpoint_a, endpoint_b)
syms x;
f = F(x);
[numerator, denominator] = numden(sym(f));
dg = numerator;
dG = matlabFunction(dg);
p = simplify(log(denominator) / log(x - endpoint_a));
I = dG(endpoint_a) / (1 - p) * (endpoint_b - endpoint_a)^(1 - p);
Taylor = dG(endpoint_a);
for k = 1:4
dg = diff(numerator, k);
dG = matlabFunction(dg);
Taylor = Taylor + dG(endpoint_a) * (x - endpoint_a)^k / factorial(k);
I = I + dG(endpoint_a) / (factorial(k) * (k + 1 - p)) * (endpoint_b - endpoint_a)^(k + 1 - p);
end
G(x) = piecewise(endpoint_a < x <= endpoint_b, (numerator - Taylor) / denominator, x == endpoint_a, 0);
I = I + Adaptive_Quadrature(G, 0.00000001, 10000000, endpoint_a, endpoint_b);
I = double(I);
end
|
例如：
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| >> F=@(x)exp(x)/x^0.5
>> Improper(F, 0, 1)
ans =
2.9253
|
常微分方程的初值问题
欧拉步长法
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| function [t, omega] = Euler(F, N, y_0, endpoint_a, endpoint_b)
h = (endpoint_b - endpoint_a) / N;
t(1) = endpoint_a;
omega(1) = y_0;
for i = 1:N
omega(i + 1) = omega(i) + h * F(t(i), omega(i));
t(i + 1) = endpoint_a + i * h;
end
end
|
例如：
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| >> y = dsolve('Dy - y + t^2 - 1 = 0', 'y(0)=0.5', 't')
>> fplot(y,[0, 2])
>> hold on
>> F = @(t, y)y - t^2 + 1
>> [t, omega] = Euler(F, 10, 0.5, 0, 2)
>> plot(t, omega);
>> legend('微分方程的精确解2t - e^t/2 + t^2 + 1', '欧拉步长法近似解')
|
高阶泰勒方法
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| function [t, u] = Taylor(F, n, N, y_0, endpoint_a, endpoint_b)
syms x y;
h = (endpoint_b - endpoint_a) / N;
t(1) = endpoint_a;
u(1) = y_0;
f = F(x, y);
df = f;
T_n = f;
for k = 1:n - 1
df = diff(df, x) + diff(df, y) * f;
T_n = T_n + h^k * df / factorial(k + 1);
end
T = matlabFunction(T_n);
for i = 1:N
u(i + 1) = u(i) + h * T(t(i), u(i));
t(i + 1) = endpoint_a + i * h;
end
end
|
例如：
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| >> y = dsolve('Dy - y + t^2 - 1 = 0', 'y(0)=0.5', 't')
>> fplot(y,[0, 2])
>> hold on
>> F = @(t, y)y - t^2 + 1
>> [t, u] = Taylor(F, 4, 10, 0.5, 0, 2)
>> plot(t, u);
>> legend('微分方程的精确解2t - e^t/2 + t^2 + 1', '高阶泰勒方法近似解')
|
龙格库达方法
Midpoint Method
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| function [t, omega] = Midpoint_Method(F, N, y_0, endpoint_a, endpoint_b)
h = (endpoint_b - endpoint_a) / N;
t(1) = endpoint_a;
omega(1) = y_0;
for i = 1:N
omega(i + 1) = omega(i) + h * F(t(i) + h / 2, omega(i) + h * F(t(i), omega(i)) / 2);
t(i + 1) = endpoint_a + i * h;
end
end
|
例如：
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| >> y = dsolve('Dy - y + t^2 - 1 = 0', 'y(0)=0.5', 't')
>> fplot(y,[0, 2])
>> hold on
>> F = @(t, y)y - t^2 + 1
>> [t, u] = Midpoint_Method(F, 10, 0.5, 0, 2)
>> plot(t, u);
>> legend('微分方程的精确解2t - e^t/2 + t^2 + 1', 'Midpoint Method近似解')
|
Modified Euler
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| function [t, omega] = Modified_Euler(F, N, y_0, endpoint_a, endpoint_b)
h = (endpoint_b - endpoint_a) / N;
t(1) = endpoint_a;
omega(1) = y_0;
for i = 1:N
omega(i + 1) = omega(i) + h / 2 * (F(t(i), omega(i)) + F(t(i) + h, omega(i) + h * F(t(i), omega(i))));
t(i + 1) = endpoint_a + i * h;
end
plot(t, omega);
end
|
例如：
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| >> y = dsolve('Dy - y + t^2 - 1 = 0', 'y(0)=0.5', 't')
>> fplot(y,[0, 2])
>> hold on
>> F = @(t, y)y - t^2 + 1
>> [t, u] = Modified_Euler(F, 10, 0.5, 0, 2)
>> plot(t, u);
>> legend('微分方程的精确解2t - e^t/2 + t^2 + 1', 'Modified Euler Method近似解')
|
Heun’s Method
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| function [t, omega] = Heun(F, N, y_0, endpoint_a, endpoint_b)
h = (endpoint_b - endpoint_a) / N;
t(1) = endpoint_a;
omega(1) = y_0;
for i = 1:N
omega(i + 1) = omega(i) + h / 4 * (F(t(i), omega(i)) + 3 * F(t(i) + 2 * h / 3, omega(i) + 2 * h / 3 * F(t(i) + h / 3, omega(i) + h / 3 * F(t(i), omega(i)))));
t(i + 1) = endpoint_a + i * h;
end
end
|
例如：
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| >> y = dsolve('Dy - y + t^2 - 1 = 0', 'y(0)=0.5', 't')
>> fplot(y,[0, 2])
>> hold on
>> F = @(t, y)y - t^2 + 1
>> [t, u] = Heun(F, 10, 0.5, 0, 2)
>> plot(t, u);
>> legend('微分方程的精确解2t - e^t/2 + t^2 + 1', 'Heuns Method近似解')
|
Runge-Kutta
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| function [t, omega] = Runge_Kutta(F, N, y_0, endpoint_a, endpoint_b)
h = (endpoint_b - endpoint_a) / N;
t(1) = endpoint_a;
omega(1) = y_0;
for i = 1:N
k1 = h * F(t(i), omega(i));
k2 = h * F(t(i) + h / 2, omega(i) + k1 / 2);
k3 = h * F(t(i) + h / 2, omega(i) + k2 / 2);
k4 = h * F(t(i) + h, omega(i) + k3);
omega(i + 1) = omega(i) + (k1 + 2 * k2 + 2 * k3 + k4) / 6;
t(i + 1) = endpoint_a + i * h;
end
end
|
例如：
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| >> y = dsolve('Dy - y + t^2 - 1 = 0', 'y(0)=0.5', 't')
>> fplot(y,[0, 2])
>> hold on
>> F = @(t, y)y - t^2 + 1
>> [t, u] = Runge_Kutta(F, 10, 0.5, 0, 2)
>> plot(t, u);
>> legend('微分方程的精确解2t - e^t/2 + t^2 + 1', 'Runge-Kutta方法近似解')
|
Runge Kutta Fehlberg
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| function [t, omega] = Runge_Kutta_Fehlberg(F, y_0, TOL, hmax, hmin, endpoint_a, endpoint_b)
h = hmax
t(1) = endpoint_a;
omega(1) = y_0;
flag = 1;
i = 1;
while flag == 1
k1 = h * F(t(i), omega(i));
k2 = h * F(t(i) + h / 4, omega(i) + k1 / 4);
k3 = h * F(t(i) + 3 * h / 8, omega(i) +3 * k1 / 32 + 9 * k2 / 32);
k4 = h * F(t(i) +12 * h / 13, omega(i) +1932 * k1 / 2197 - 7200 * k2 / 2197 +7296 * k3 / 2197);
k5 = h * F(t(i) +h, omega(i) +439 * k1 / 216 - 8 * k2 + 3680 * k3 / 513 -845 * k4 / 4104);
k6 = h * F(t(i) +h / 2, omega(i) -8 * k1 / 27 + 2 * k2 - 3544 * k3 / 2565 +1859 * k4 / 4104 - 11 * k5 / 40);
R = abs(k1 / 360 - 128 * k3 / 4275 - 2197 * k4 / 75240 + k5 / 50 + 2 * k6 / 55) / h;
if R < TOL
omega(i + 1) = omega(i) + 25 * k1 / 216 + 1408 * k3 / 2565 + 2197 * k4 / 4104 - k5 / 5;
t(i + 1) = t(i) + h;
i = i + 1;
end
q = (TOL / (2 * R))^0.25;
if q <= 0.1
h = h / 10;
elseif q >= 4
h = 4 * h;
else
h = q * h;
end
if h > hmax
h = hmax;
end
if t(i) >= endpoint_b
flag = 0;
elseif t(i) + h > endpoint_b
h = endpoint_b - t(i);
elseif h < hmin
flag = 0;
disp('h小于hmin，无法进行控制');
end
end
end
|
例如：
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| >> y = dsolve('Dy - y + t^2 - 1 = 0', 'y(0)=0.5', 't')
>> fplot(y,[0, 2])
>> hold on
>> F = @(t, y)y - t^2 + 1
>> [t, u] = Runge_Kutta_Fehlberg(F,0.5,0.00001,0.25,0.01,0,2)
>> plot(t, u);
>> legend('微分方程的精确解2t - e^t/2 + t^2 + 1', 'Runge Kutta Fehlberg方法近似解')
|
多步法
四阶阿达姆斯预测矫正法
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| function [t, omega] = Adams_Fourth_Order_Predictor_Corrector(F, N, y_0, endpoint_a, endpoint_b)
h = (endpoint_b - endpoint_a) / N;
t(1) = endpoint_a;
omega(1) = y_0;
for i = 1:3
k1 = h * F(t(i), omega(i));
k2 = h * F(t(i) + h / 2, omega(i) + k1 / 2);
k3 = h * F(t(i) + h / 2, omega(i) + k2 / 2);
k4 = h * F(t(i) + h, omega(i) + k3);
omega(i + 1) = omega(i) + (k1 + 2 * k2 + 2 * k3 + k4) / 6;
t(i + 1) = endpoint_a + i * h;
end
for i = 4:N
t(i + 1) = endpoint_a + i * h;
omega_predictor = omega(i) + h * (55 * F(t(i), omega(i)) - 59 * F(t(i - 1), omega(i - 1)) + 37 * F(t(i - 2), omega(i - 2)) - 9 * F(t(i - 3), omega(i - 3))) / 24;
omega(i + 1) = omega(i) + h * (9 * F(t(i + 1), omega_predictor) + 19 * F(t(i), omega(i)) - 5 * F(t(i - 1), omega(i - 1)) + F(t(i - 2), omega(i - 2))) / 24;
end
end
|
例如：
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| >> y = dsolve('Dy - y + t^2 - 1 = 0', 'y(0)=0.5', 't')
>> fplot(y,[0, 2])
>> hold on
>> F = @(t, y)y - t^2 + 1
>> [t, u] = Adams_Fourth_Order_Predictor_Corrector(F, 10, 0.5, 0, 2)
>> plot(t, u);
>> legend('微分方程的精确解2t - e^t/2 + t^2 + 1', '四阶阿达姆斯预测矫正法近似解')
|
可变步长的多步法
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| function [t, omega] = Adams_Variable_Step_Size(F, y_0, TOL, hmax, hmin, endpoint_a, endpoint_b)
flag = 1;
h = hmax;
omega(1) = y_0;
t(1) = endpoint_a;
[tt, uu] = RK4(F, h, omega(1), t(1));
for j = 2:4
t(j) = tt(j);
omega(j) = uu(j);
end
i = 5;
ti = t(4) + h;
if ti > endpoint_b
disp('hmax过大');
end
while ti <= endpoint_b
t(i) = ti;
omega_predictor = omega(i - 1) + h * (55 * F(t(i - 1), omega(i - 1)) - 59 * F(t(i - 2), omega(i - 2)) + 37 * F(t(i - 3), omega(i - 3)) - 9 * F(t(i - 4), omega(i - 4))) / 24;
omega_corrector = omega(i - 1) + h * (9 * F(t(i), omega_predictor) + 19 * F(t(i - 1), omega(i - 1)) - 5 * F(t(i - 2), omega(i - 2)) + F(t(i - 3), omega(i - 3))) / 24;
delta = 19 * abs(omega_predictor - omega_corrector) / (270 * h);
if delta < TOL
omega(i) = omega_corrector;
if ti == endpoint_b
break;
end
i = i + 1;
if delta < TOL / 10
q = (TOL / (2 * delta))^0.25;
if q > 4
h = 4 * h;
else
h = q * h;
end
if h > hmax
h = hmax;
end
if t(i - 1) + 4 * h > endpoint_b
h = (endpoint_b - t(i - 1)) / 4;
end
[tt, uu] = RK4(F, h, omega(i - 1), t(i - 1));
for j = i:i + 2
t(j) = tt(j - i + 2);
omega(j) = uu(j - i + 2);
end
i = i + 3;
end
if t(i - 1) + h > endpoint_b
h = (endpoint_b - t(i - 1)) / 4;
[tt, uu] = RK4(F, h, omega(i - 1), t(i - 1));
for j = i:i + 2
t(j) = tt(j - i + 2);
omega(j) = uu(j - i + 2);
end
i = i + 3;
end
else
q = (TOL / (2 * delta))^0.25;
if q < 0.1
h = 0.1 * h;
else
h = q * h;
end
if h < hmin
disp('h小于hmin，无法进行控制');
flag = 0;
break;
else
i = i - 3;
[tt, uu] = RK4(F, h, omega(i - 1), t(i - 1));
for j = i:i + 2
t(j) = tt(j - i + 2);
omega(j) = uu(j - i + 2);
end
i = i + 3;
end
end
ti = t(i - 1) + h;
end
end
function [t, omega] = RK4(F, h, y_0, endpoint_a)
t(1) = endpoint_a;
omega(1) = y_0;
for i = 1:3
k1 = h * F(t(i), omega(i));
k2 = h * F(t(i) + h / 2, omega(i) + k1 / 2);
k3 = h * F(t(i) + h / 2, omega(i) + k2 / 2);
k4 = h * F(t(i) + h, omega(i) + k3);
omega(i + 1) = omega(i) + (k1 + 2 * k2 + 2 * k3 + k4) / 6;
t(i + 1) = endpoint_a + i * h;
end
end
|
例如：
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| >> y = dsolve('Dy - y + t^2 - 1 = 0', 'y(0)=0.5', 't')
>> fplot(y,[0, 2])
>> hold on
>> F = @(t, y)y - t^2 + 1
>> [t, u] = Adams_Variable_Step_Size(F,0.5,0.00001,0.25,0.01,0,2)
>> plot(t, u);
>> legend('微分方程的精确解2t - e^t/2 + t^2 + 1', '可变步长的多步法近似解')
|
外推法
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| function [t, omega] = Extrapolation(F, y_0, TOL, hmax, hmin, endpoint_a, endpoiny_b)
NK = [2, 4, 6, 8, 12, 16, 24, 32];
t(1) = endpoint_a;
omega(1) = y_0;
n = 1;
T0 = endpoint_a;
W0 = y_0;
h = hmax;
flag = 1;
for i = 1:7
for j = 1:i
Q(i, j) = (NK(i + 1) / NK(i - j + 1))^2;
end
end
while flag == 1
k = 1;
nflag = 0;
while k <= 8 && nflag == 0
h_k = h / NK(k);
T = T0;
W2 = W0;
W3 = W2 + h_k * F(T, W2);
T = T0 + h_k;
for j = 1:NK(k) - 1
W1 = W2;
W2 = W3;
W3 = W1 + 2 * h_k * F(T, W2);
T = T0 + (j + 1) * h_k;
end
y(k, 1) = (W3 + (W2 + h_k * F(T, W3))) / 2;
if k >=2
for i = 2:k
y(k, i) = y(k, i - 1) + (y(k, i - 1) - y(k - 1, i - 1)) / (Q(k - 1, i - 1) - 1);
end
if abs(y(k, k) - y(k - 1, k - 1)) < TOL
nflag = 1;
end
end
k = k + 1;
end
k = k - 1;
if nflag == 0
h = h / 2;
if h < hmin
disp('h小于hmin');
flag = 0;
end
else
W0 = y(k, k);
T0 = T0 + h;
n = n + 1;
t(n) = T0;
omega(n) = W0;
if T0 >= endpoiny_b
flag = 0;
elseif T0 + h > endpoiny_b
h = endpoiny_b - T0;
else
if k <= 3 && h < 0.5 * hmax
h = 2 * h;
end
end
end
end
end
|
例如：
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| >> y = dsolve('Dy - y + t^2 - 1 = 0', 'y(0)=0.5', 't')
>> fplot(y,[0, 2])
>> hold on
>> F = @(t, y)y - t^2 + 1
>> [t, u] = Extrapolation(F, 0.5, 0.000000000000001, 0.2, 0.001, 0, 2)
>> plot(t, u);
>> legend('微分方程的精确解2t - e^t/2 + t^2 + 1', '外推法近似解')
|
高阶方程与微分方程组
微分方程组
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| function [t, omega] = Runge_Kutta_Equations(A, f, N, y_0, endpoint_a, endpoint_b)
h = (endpoint_b - endpoint_a) / N;
t(1) = endpoint_a;
omega(1, :) = y_0;
for i = 1:N
k1 = h * fun(f(t(i)), omega(i, :), A);
k2 = h * fun(f(t(i) + h / 2), omega(i, :) + 0.5 * k1, A);
k3 = h * fun(f(t(i) + h / 2), omega(i, :) + 0.5 * k2, A);
k4 = h * fun(f(t(i) + h), omega(i, :) + k3, A);
omega(i + 1, :) = omega(i, :) + (k1 +2 * k2 + 2 * k3 + k4) / 6;
t(i + 1) = endpoint_a + i * h;
end
end
function fx = fun(f, w, A)
[m, n] = size(A);
if n == length(w) && n == length(rot90(f))
fx = rot90(f + mtimes(A, w'));
else
disp('输入错误');
end
end
|
例如：
⎩⎨⎧x(t)=−4x+3y+6y(t)=−2.4x+1.6y+3.6x(0)=0,y(0)=0
通解为：
{x(t)=−3.375e−2t+1.875e−0.4t+1.5y(t)=−2.25e−2t+2.25e−0.4t
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| >> syms f1(x) f2(x)
>> f1(x) = 6
>> f2(x) = 3.6
>> A = [-4 3; -2.4 1.6]
>> f(x) = [f1(x); f2(x)]
>> y_0 = [0, 0]
>> [t, omega] = Runge_Kutta_Equations(A, f, 5, y_0, 0, 0.5)
>> plot(t, omega(:, 1))
>> hold on
>> plot(t, omega(:, 2))
>> hold on
>> [x y] = dsolve('Dx=-4*x+3*y+6,Dy=-2.4*x+1.6*y+3.6', 'x(0)=0,y(0)=0')
>> fplot(x, [0, 0.5])
>> hold on
>> fplot(y, [0, 0.5])
>> legend('微分方程组f1(x)的精确解','微分方程组f2(x)的精确解','微分方程组f1(x)的近似解','微分方程组f2(x)的近似解')
|
高阶微分方程
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| function [t, omega] = Higher_Order_Differential_Equations(a, f, N, y_0, endpoint_a, endpoint_b)
m = length(a);
syms x;
for i = 1:m - 1
for j = 1:m
if j == i + 1
A(i, j) = 1;
else
A(i, j) = 0;
end
end
end
for j = 1:m
A(m, j) = -1 * a(m - j + 1);
if j ~= m
F(x) = [0, f(x)];
end
end
F = rot90(F);
h = (endpoint_b - endpoint_a) / N;
t(1) = endpoint_a;
omega(1, :) = y_0;
for i = 1:N
k1 = h * fun(F(t(i)), omega(i, :), A);
k2 = h * fun(F(t(i) + h / 2), omega(i, :) + 0.5 * k1, A);
k3 = h * fun(F(t(i) + h / 2), omega(i, :) + 0.5 * k2, A);
k4 = h * fun(F(t(i) + h), omega(i, :) + k3, A);
omega(i + 1, :) = omega(i, :) + (k1 +2 * k2 + 2 * k3 + k4) / 6;
t(i + 1) = endpoint_a + i * h;
end
end
function fx = fun(f, w, A)
[m, n] = size(A);
if n == length(w) && n == length(rot90(f))
fx = rot90(f + mtimes(A, w'));
else
disp('输入错误');
end
end
|
例如：
y′′−2y′+2y=e2tsinty(0)=−0.4,y′(0)=−0.6
通解：
y(t)=51e2t(sin(t)−2cos(t))
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| >> y = dsolve('D2y-2*Dy+2*y=exp(2*x)*sin(x)', 'y(0)=-0.4,Dy(0)=-0.6', 'x')
>> fplot(y, [0, 1])
>> hold on
>> a = [-2 2]
>> syms f(x)
>> f(x) = exp(2 * x) * sin(x)
>> y_0 = [-0.4 -0.6]
>> [t, omega] = Higher_Order_Differential_Equations(a, f, 10, y_0, 0, 1)
>> plot(t, omega(:, 1))
>> legend('高阶微分方程的精确解','高阶微分方程的近似解')
|
刚性微分方程
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| function [t, omega] = Trapezoidal_Newton_Iteration(F, N, y_0, TOL, M, endpoint_a, endpoint_b)
syms t y;
f(t, y) = F(t, y);
dfy = diff(f, y);
dFy = matlabFunction(dfy);
h = (endpoint_b - endpoint_a) / N;
t(1) = endpoint_a;
omega(1) = y_0;
for i = 1:N
K1 = omega(i) + h / 2 * F(t(i), omega(i));
w0 = K1;
mark = 1;
flag = 0;
while flag == 0
omega(i + 1) = w0 - (w0 - h / 2 * F(t(i) + h, w0) - K1) / (1 - h / 2 * dFy(t(i) + h, w0));
if abs(omega(i + 1) - w0) < TOL
flag = 1;
else
mark = mark + 1;
w0 = omega(i + 1);
if mark > M
disp('超出最大迭代次数');
break;
end
end
end
t(i + 1) = endpoint_a + i * h;
end
end
|
例如：
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| >> y = dsolve('Dy=5*exp(5*t)*(y-t)^2+1', 'y(0)=-1')
>> fplot(y, [0, 1])
>> hold on
>> F = @(t, y)5 * exp(5 * t) * (y - t)^2 + 1
>> [t, omega] = Trapezoidal_Newton_Iteration(F, 5, -1, 0.000001, 10, 0, 1)
>> plot(t, omega)
>> legend('刚性微分方程的精确解','刚性微分方程的近似解')
|
逼近论
多项式逼近
离散最小二乘法
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| function P = Polynomial_Least_Squares(xx, yy, n)
syms x;
m = length(xx);
A = zeros(n + 1, n + 1);
b = zeros(n + 1, 1);
for i = 1:n + 1
for j = i:n + 1
sum_x_n = sum(xx .^ (i + j - 2));
A(i, j) = sum_x_n;
A(j, i) = sum_x_n;
end
sum_yx_n = sum(yy .* xx .^ (i - 1));
b(i) = sum_yx_n;
end
a = A \ b;
P = a(1);
for k = 1:n
P = P + a(k + 1) * x ^ k;
end
end
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例如：
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| >> xx = 0:0.25:1
>> yy = [1.0000, 1.2840, 1.6487, 2.1170, 2.7183]
>> plot(xx, yy, '>')
>> hold on
>> fplot(P, [0, 1])
|
连续最小二乘法
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| function P = Polynomial_Least_Squares_Function(F, endpoint_a, endpoint_b, n)
syms x;
f = F(x);
A = zeros(n + 1, n + 1);
b = zeros(n + 1, 1);
for i = 1:n + 1
for j = i:n + 1
A(j, i) = (endpoint_b ^ (i + j - 1) - endpoint_a ^ (i + j - 1)) / (i + j - 1);
A(i, j) = A(j, i);
end
b(i) = int(f * x ^ (i - 1), x, endpoint_a, endpoint_b);
end
a = A \ b;
P = a(1);
for k = 1:n
P = P + a(k + 1) * x ^ k;
end
end
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例如：
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| >> F = @(x)sin(pi * x)
>> P = Polynomial_Least_Squares_Function(F, 0, 1, 2)
>> fplot(F, [0, 1])
>> hold on
>> fplot(P, [0, 1])
>> legend('原函数', '最小二乘逼近函数')
|
正交多项式
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| function [Phi] = Gram_Schmidt_Orthogonal_Polynomials(Omega, endpoint_a, endpoint_b, n)
syms x;
omega = Omega(x);
Phi = sym('x', [1 n + 1]);
Phi(1) = 1;
B = int(x * omega * Phi(1) ^ 2, x, endpoint_a, endpoint_b) / int(omega * Phi(1) ^ 2, x, endpoint_a, endpoint_b);
Phi(2) = x - B;
for k = 3:n + 1
B = int(x * omega * Phi(k - 1) ^ 2, x, endpoint_a, endpoint_b) / int(omega * Phi(k - 1) ^ 2, x, endpoint_a, endpoint_b);
C = int(x * omega * Phi(k - 1) * Phi(k - 2), x, endpoint_a, endpoint_b) / int(omega * Phi(k - 2) ^ 2, x, endpoint_a, endpoint_b);
Phi(k) = simplify((x - B) * Phi(k - 1) - C * Phi(k - 2));
end
end
|
Legendre多项式
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| >> Omega = @(x)1
>> [Phi] = Gram_Schmidt_Orthogonal_Polynomials(Omega, -1, 1, 5)
>> figure;
>> hold on;
>> for k = 1:length(Phi)
fplot(Phi(k), [-1, 1], 'DisplayName', sprintf('P_%d(x)', k-1));
end
>> legend('show');
>> grid on;
>> title('Legendre Polynomials');
>> xlabel('x');
>> ylabel('P(x)');
>> line([-1, 1], [0, 0], 'Color', 'k', 'LineStyle', '--');
>> line([0, 0], [min(ylim), max(ylim)], 'Color', 'k', 'LineStyle', '--');
>> hold off;
|
切比雪夫多项式
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| function [Tn] = Chebyshev_Polynomial(endpoint_a, endpoint_b, n)
syms x;
Tn = sym('x', [1 n + 1]);
syms x_prime;
x_prime = ((endpoint_b - endpoint_a) * x + endpoint_a + endpoint_b) / 2;
Tn(1) = 1;
Tn(2) = simplify(x_prime);
Tn(3) = simplify(x_prime * Tn(2) - Tn(1) / 2);
for k = 4:n+1
Tn(k) = simplify(x_prime * Tn(k - 1) - Tn(k - 2) / 4);
end
end
|
例如：
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| >> [Tn] = Chebyshev_Polynomial(-1, 1, 5)
>> figure;
>> hold on;
>> for k = 1:length(Tn)
fplot(Tn(k), [-1, 1], 'DisplayName', sprintf('P_%d(x)', k-1));
end
>> legend('show');
>> grid on;
>> title('Legendre Polynomials');
>> xlabel('x');
>> ylabel('P(x)');
>> line([-1, 1], [0, 0], 'Color', 'k', 'LineStyle', '--');
>> line([0, 0], [min(ylim), max(ylim)], 'Color', 'k', 'LineStyle', '--');
>> hold off;
|
正交多项式逼近
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| function P = Orthogonal_Polynomial_Least_Squares_Function(F, Omega, n, endpoint_a, endpoint_b)
syms x;
f = F(x);
omega = Omega(x);
omega = subs(omega, x, (2 * x - (endpoint_a + endpoint_b)) / (endpoint_b - endpoint_a));
[Phi] = Gram_Schmidt_Orthogonal_Polynomials(matlabFunction(omega), endpoint_a, endpoint_b, n);
a = sym(zeros(n + 1, 1));
for k = 1:n + 1
a(k) = integral(matlabFunction(omega * Phi(k) * f), endpoint_a, endpoint_b) / int(omega * Phi(k) ^ 2, x, endpoint_a, endpoint_b);
end
P = a(1);
for k = 1:n
P = P + a(k + 1) * Phi(k + 1);
end
P = simplify(P);
end
|
例如：
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| >> F = @(x)sin(x)
>> Omega = @(x)1 / sqrt(1 - x ^ 2)
>> P = Orthogonal_Polynomial_Least_Squares_Function(F, Omega, 5, -3, 3)
>> fplot(F, [-pi, pi])
>> hold on
>> fplot(P, [-pi, pi])
|
多项式函数逼近
Pade逼近
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| function r = Pade_Approximation(F, numerator_n, denominator_m)
N = denominator_m + numerator_n;
syms x;
f = F(x);
maclaurin_series = taylor(f, x, 'Order', N + 1);
[coeffs_list, terms] = coeffs(maclaurin_series, x);
coeffs_list_reversed = fliplr(coeffs_list);
A = zeros(N, N);
b = zeros(N, 1);
for i = 1:N
for k = 1:i
if i - k + 1 <= denominator_m
A(i, i - k + 1) = coeffs_list_reversed(k);
end
end
if i <= numerator_n
A(i, denominator_m + i) = -1;
end
b(i) = -coeffs_list_reversed(i + 1);
end
pq = A \ b;
p = coeffs_list_reversed(1);
for k = 1:numerator_n;
p = p + x ^ k * pq(denominator_m + k);
end
q = 1;
for k = 1:denominator_m
q = q + x ^ k * pq(k);
end
r = simplify(p / q);
end
|
例如：
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| >> F = @(x)exp(-x)
>> r = Pade_Approximation(F, 3, 2)
>> fplot(r, [-2, 2])
>> hold on
>> fplot(F, [-2, 2])
>> legend('多项式函数r', '原函数F')
|
切比雪夫多项式逼近
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| function r = Chebyshev_Rational_Approximation(F, numerator_n, denominator_m)
N = denominator_m + numerator_n;
[Tn] = Chebyshev_Polynomial(-1, 1, N);
syms theta;
a = zeros(N + 1, 1);
a(1) = 1 / pi * int(F(cos(theta)), theta, 0, pi);
for k = 1:N
a(k + 1) = 2 / pi * int(F(cos(theta)) * cos(k * theta), theta, 0, pi);
end
A = zeros(N + 1, N + 1);
b = zeros(N + 1, 1);
for i = 1:N + 1
for j = 1:denominator_m
if i ~= 1
A(i, j) = 0;
if (1 <= i - j) && (i - j <= N + 1)
A(i, j) = A(i, j) + a(i - j);
end
if (1 <= i + j) && (i + j <= N + 1)
A(i, j) = A(i, j) + a(i + j);
end
if (-1 <= j - i) && (j - i <= N - 1)
A(i, j) = A(i, j) + a(j - i + 2);
end
else
A(i, j) = a(j + 1) / 2;
end
end
if i == 1
A(i, denominator_m + i) = -1;
elseif i <= numerator_n + 1
A(i, denominator_m + i) = -2;
end
if i ~= 1
b(i) = -2 * a(i);
else
b(i) = -a(i);
end
end
pq = A \ b;
p = pq(denominator_m + 1);
for k = 1:numerator_n
p = p + 2 ^ (k - 1) * Tn(k + 1) * pq(denominator_m + k + 1);
end
q = 1;
for k = 1:denominator_m
q = q + 2 ^ (k - 1) * Tn(k + 1) * pq(k);
end
r = simplify(p / q);
end
|
例如：
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| >> F = @(x)exp(-x)
>> r = Chebyshev_Rational_Approximation(F, 3, 2)
>> fplot(r, [-3, 5])
>> hold on
>> fplot(F, [-3, 5])
>> legend('多项式函数r', '原函数F')
|
三角多项式逼近
连续函数逼近
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| function S = Trigonometric_Polynomial_Approximation(F, n)
syms x;
f = F(x);
a = zeros(n + 1, 1);
b = zeros(n, 1);
for k = 0:n
a(k + 1) = int(f * cos(k * x), x, -pi, pi) / pi;
end
for k = 1:n - 1
b(k) = int(f * sin(k * x), x, -pi, pi) / pi;
end
S = a(1) / 2;
for k = 1:n - 1
S = S + a(k + 1) * cos(k * x) + b(k) * sin(k * x);
end
S = S + a(n + 1) * cos(n * x);
end
|
例如：
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| >> sawtoothWave = @(x) mod(x, 2 * pi);
>> S1 = Trigonometric_Polynomial_Approximation(sawtoothWave, 1)
>> S2 = Trigonometric_Polynomial_Approximation(sawtoothWave, 2)
>> S3 = Trigonometric_Polynomial_Approximation(sawtoothWave, 3)
>> S4 = Trigonometric_Polynomial_Approximation(sawtoothWave, 4)
>> S5 = Trigonometric_Polynomial_Approximation(sawtoothWave, 5)
>> figure;
>> hold on;
>> fplot(sawtoothWave, [-2 * pi, 2 * pi]);
>> fplot(S1, [-2 * pi, 2 * pi]);
>> fplot(S2, [-2 * pi, 2 * pi]);
>> fplot(S3, [-2 * pi, 2 * pi]);
>> fplot(S4, [-2 * pi, 2 * pi]);
>> fplot(S5, [-2 * pi, 2 * pi]);
>> legend('锯齿波', '1阶逼近S1', '2阶逼近S2', '3阶逼近S3', '4阶逼近S4', '5阶逼近S5');
>> hold off;
|
离散点逼近
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| function S = Discrete_Trigonometric_Polynomial_Approximation(xx, yy, n)
m = length(yy);
syms x;
a = zeros(n + 1, 1);
b = zeros(n, 1);
for k = 0:n
a(k + 1) = (2/m) * sum(yy .* cos(k * xx));
end
for k = 1:n - 1
b(k) = (2/m) * sum(yy .* sin(k * xx));
end
S = a(1) / 2;
for k = 1:n - 1
S = S + a(k + 1) * cos(k * x) + b(k) * sin(k * x);
end
S = S + a(n + 1) * cos(n * x);
end
|
例如：
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| >> sawtoothWave = @(x) mod(x, 2 * pi);
>> m = 20
>> xx = linspace(-pi, (1 - 2 / m) * pi, m)
>> yy = sawtoothWave(xx)
>> S1 = Discrete_Trigonometric_Polynomial_Approximation(xx, yy, 1)
>> S2 = Discrete_Trigonometric_Polynomial_Approximation(xx, yy, 2)
>> S3 = Discrete_Trigonometric_Polynomial_Approximation(xx, yy, 3)
>> S4 = Discrete_Trigonometric_Polynomial_Approximation(xx, yy, 4)
>> S5 = Discrete_Trigonometric_Polynomial_Approximation(xx, yy, 5)
>> figure; % Create a new figure
>> hold on; % Hold the current plot
>> fplot(sawtoothWave, [-pi, pi]);
>> fplot(S1, [-pi, pi]);
>> fplot(S2, [-pi, pi]);
>> fplot(S3, [-pi, pi]);
>> fplot(S4, [-pi, pi]);
>> fplot(S5, [-pi, pi]);
>> legend('锯齿波', '1阶逼近S1', '2阶逼近S2', '3阶逼近S3', '4阶逼近S4', '5阶逼近S5');
>> hold off; % Release the hold
|
快速傅立叶变换
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| function y = FFT(a)
n = length(a);
y = zeros(n, 1);
if n ~= 2 ^ nextpow2(n)
error('输入数组的长度非2的整数次幂');
end
if n == 1
y(1) = a(1);
return;
end
a_odd = a(1:2:end);
a_even = a(2:2:end);
y_odd = FFT(a_odd);
y_even = FFT(a_even);
for k = 1:n / 2
omega_k = exp(2 * pi * 1i * (k - 1) / n);
y(k) = y_odd(k) + omega_k * y_even(k);
y(k + n / 2) = y_odd(k) - omega_k * y_even(k);
end
end
|
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| function a = DFT(y)
N = length(y);
a = recursiveDFT(y) / N;
end
function a = recursiveDFT(y)
n = length(y);
a = zeros(n, 1);
if n ~= 2 ^ nextpow2(n)
error('输入数组的长度非2的整数次幂');
end
if n == 1
a(1) = y(1);
return;
end
y_odd = y(1:2:end);
y_even = y(2:2:end);
a_odd = recursiveDFT(y_odd);
a_even = recursiveDFT(y_even);
for j = 1:n / 2
omega_j = exp(-2 * pi * 1i * (j - 1) / n);
a(j) = (a_odd(j) + omega_j * a_even(j));
a(j + n / 2) = (a_odd(j) - omega_j * a_even(j));
end
end
|
