matlab - 使用 quad 仅针对一个变量对函数进行积分

Question

有什么办法可以整合两个变量的函数，比如说

f=@(x) x^2 + x*y

超过 x

试过四边形（f，a，b）

但不起作用，正在寻找替代解决方案

Answer 1

看起来你想要这样的东西：

y = 100; % whatever y is
a = 0;
b = 2;
% you'll need to vectorize the integrand function
f = @(x) x.*x + x.*y
val = quad(f, a, b);

但是，如果您正在寻找代数答案，则需要使用 Symbolic Toolbox，或其他一些软件，或您的微积分书。:-)

整个“矢量化”的东西来自Mathworks quad 文档，上面写着：

函数 y = fun(x) 应该接受向量参数 x 并返回向量结果 y，即在 x 的每个元素处计算的被积函数。

Answer 2

抱歉，quad 不能解决符号问题。它只进行数值积分。

syms x y
int(x^2 + x*y,x)
ans =
    (x^2*(2*x + 3*y))/6

解决符号问题的自然方法是使用符号工具。

从后续来看，安雅想要介于两者之间。借用一位名叫米克的老摇滚明星的话，“你不能总是得到你想要的。”

同样，如果您只想在 x 上进行积分，则不能使用 quad，因为 quad 是一种自适应工具。

在一些简单的情况下，您可以使用像辛普森规则这样的简单工具来完成这项工作。例如，假设您想解决上述问题，x 在区间 [0 1] 上积分。出于比较的目的，我将首先象征性地进行。

syms x y
res = int(x^2 + x*y,x);

subs(res,x,1) - subs(res,0)
ans =
   y/2 + 1/3

现在，让我们尝试使用 x 上的数值积分。

syms y
x = 0:.01:1;
coef = mod((0:100)',2)*2 + 2;
coef([1 end]) = 1;
coef = 0.01*coef/3;

(x.^2 + x.*y)*coef
ans =
   y/2 + 1/3

所以在这个非常简单的情况下，它确实有效。稍微复杂一点的东西怎么样？在区间 [-1 1] 上积分 x*exp(x*y)。同样，已知形式可以象征性地访问。

syms x y
res = int(x*exp(x*y),x);

res = subs(res,x,1) - subs(res,-1)
res =
   (exp(-y)*(y + 1))/y^2 + (exp(y)*(y - 1))/y^2

稍后测试它，在 y = 1/2 时它取什么值？

vpa(subs(res,y,1/2))
ans =
   0.34174141687554424792549563431876

让我们尝试同样的技巧，使用辛普森法则。

syms y
x = -1:.01:1;
coef = mod((-100:100)',2)*2 + 2;
coef([1 end]) = 1;
coef = 0.01*coef/3;

res = (x.*exp(x*y))*coef
res =
   exp(y/2)/300 - exp(-y/2)/300 - exp(-y)/300 - exp(-y/4)/300 + exp(y/4)/300 - exp(-y/5)/750 + exp(y/5)/750 - exp(-(3*y)/4)/100 - exp(-(2*y)/5)/375 + exp((2*y)/5)/375 + exp((3*y)/4)/100 - exp(-(3*y)/5)/250 + exp((3*y)/5)/250 - (2*exp(-(4*y)/5))/375 + (2*exp((4*y)/5))/375 - exp(-y/10)/1500 + exp(y/10)/1500 - exp(-(3*y)/10)/500 + exp((3*y)/10)/500 - (7*exp(-(7*y)/10))/1500 + (7*exp((7*y)/10))/1500 - (3*exp(-(9*y)/10))/500 + (3*exp((9*y)/10))/500 - exp(-y/20)/1500 + exp(y/20)/1500 - exp(-(3*y)/20)/500 + exp((3*y)/20)/500 - exp(-y/25)/3750 + exp(y/25)/3750 - (7*exp(-(7*y)/20))/1500 - exp(-(2*y)/25)/1875 + exp((2*y)/25)/1875 + (7*exp((7*y)/20))/1500 - exp(-(3*y)/25)/1250 + exp((3*y)/25)/1250 - (3*exp(-(9*y)/20))/500 - (2*exp(-(4*y)/25))/1875 + (2*exp((4*y)/25))/1875 + (3*exp((9*y)/20))/500 - (11*exp(-(11*y)/20))/1500 - exp(-(6*y)/25)/625 + exp((6*y)/25)/625 + (11*exp((11*y)/20))/1500 - (7*exp(-(7*y)/25))/3750 + (7*exp((7*y)/25))/3750 - (13*exp(-(13*y)/20))/1500 - (4*exp(-(8*y)/25))/1875 + (4*exp((8*y)/25))/1875 + (13*exp((13*y)/20))/1500 - (3*exp(-(9*y)/25))/1250 + (3*exp((9*y)/25))/1250 - (11*exp(-(11*y)/25))/3750 + (11*exp((11*y)/25))/3750 - (17*exp(-(17*y)/20))/1500 - (2*exp(-(12*y)/25))/625 + (2*exp((12*y)/25))/625 + (17*exp((17*y)/20))/1500 - (13*exp(-(13*y)/25))/3750 + (13*exp((13*y)/25))/3750 - (19*exp(-(19*y)/20))/1500 - (7*exp(-(14*y)/25))/1875 + (7*exp((14*y)/25))/1875 + (19*exp((19*y)/20))/1500 - (8*exp(-(16*y)/25))/1875 + (8*exp((16*y)/25))/1875 - (17*exp(-(17*y)/25))/3750 + (17*exp((17*y)/25))/3750 - (3*exp(-(18*y)/25))/625 + (3*exp((18*y)/25))/625 - (19*exp(-(19*y)/25))/3750 + (19*exp((19*y)/25))/3750 - (7*exp(-(21*y)/25))/1250 + (7*exp((21*y)/25))/1250 - (11*exp(-(22*y)/25))/1875 + (11*exp((22*y)/25))/1875 - (23*exp(-(23*y)/25))/3750 + (23*exp((23*y)/25))/3750 - (4*exp(-(24*y)/25))/625 + (4*exp((24*y)/25))/625 - exp(-y/50)/7500 + exp(y/50)/7500 - exp(-(3*y)/50)/2500 + exp((3*y)/50)/2500 - (7*exp(-(7*y)/50))/7500 + (7*exp((7*y)/50))/7500 - (3*exp(-(9*y)/50))/2500 + (3*exp((9*y)/50))/2500 - (11*exp(-(11*y)/50))/7500 + (11*exp((11*y)/50))/7500 - (13*exp(-(13*y)/50))/7500 + (13*exp((13*y)/50))/7500 - (17*exp(-(17*y)/50))/7500 + (17*exp((17*y)/50))/7500 - (19*exp(-(19*y)/50))/7500 + (19*exp((19*y)/50))/7500 - (7*exp(-(21*y)/50))/2500 + (7*exp((21*y)/50))/2500 - (23*exp(-(23*y)/50))/7500 + (23*exp((23*y)/50))/7500 - (9*exp(-(27*y)/50))/2500 + (9*exp((27*y)/50))/2500 - (29*exp(-(29*y)/50))/7500 + (29*exp((29*y)/50))/7500 - (31*exp(-(31*y)/50))/7500 + (31*exp((31*y)/50))/7500 - (11*exp(-(33*y)/50))/2500 + (11*exp((33*y)/50))/2500 - (37*exp(-(37*y)/50))/7500 + (37*exp((37*y)/50))/7500 - (13*exp(-(39*y)/50))/2500 + (13*exp((39*y)/50))/2500 - (41*exp(-(41*y)/50))/7500 + (41*exp((41*y)/50))/7500 - (43*exp(-(43*y)/50))/7500 + (43*exp((43*y)/50))/7500 - (47*exp(-(47*y)/50))/7500 + (47*exp((47*y)/50))/7500 - (49*exp(-(49*y)/50))/7500 + (49*exp((49*y)/50))/7500 - exp(-y/100)/7500 + exp(y/100)/7500 - exp(-(3*y)/100)/2500 + exp((3*y)/100)/2500 - (7*exp(-(7*y)/100))/7500 + (7*exp((7*y)/100))/7500 - (3*exp(-(9*y)/100))/2500 + (3*exp((9*y)/100))/2500 - (11*exp(-(11*y)/100))/7500 + (11*exp((11*y)/100))/7500 - (13*exp(-(13*y)/100))/7500 + (13*exp((13*y)/100))/7500 - (17*exp(-(17*y)/100))/7500 + (17*exp((17*y)/100))/7500 - (19*exp(-(19*y)/100))/7500 + (19*exp((19*y)/100))/7500 - (7*exp(-(21*y)/100))/2500 + (7*exp((21*y)/100))/2500 - (23*exp(-(23*y)/100))/7500 + (23*exp((23*y)/100))/7500 - (9*exp(-(27*y)/100))/2500 + (9*exp((27*y)/100))/2500 - (29*exp(-(29*y)/100))/7500 + (29*exp((29*y)/100))/7500 - (31*exp(-(31*y)/100))/7500 + (31*exp((31*y)/100))/7500 - (11*exp(-(33*y)/100))/2500 + (11*exp((33*y)/100))/2500 - (37*exp(-(37*y)/100))/7500 + (37*exp((37*y)/100))/7500 - (13*exp(-(39*y)/100))/2500 + (13*exp((39*y)/100))/2500 - (41*exp(-(41*y)/100))/7500 + (41*exp((41*y)/100))/7500 - (43*exp(-(43*y)/100))/7500 + (43*exp((43*y)/100))/7500 - (47*exp(-(47*y)/100))/7500 + (47*exp((47*y)/100))/7500 - (49*exp(-(49*y)/100))/7500 + (49*exp((49*y)/100))/7500 - (17*exp(-(51*y)/100))/2500 + (17*exp((51*y)/100))/2500 - (53*exp(-(53*y)/100))/7500 + (53*exp((53*y)/100))/7500 - (19*exp(-(57*y)/100))/2500 + (19*exp((57*y)/100))/2500 - (59*exp(-(59*y)/100))/7500 + (59*exp((59*y)/100))/7500 - (61*exp(-(61*y)/100))/7500 + (61*exp((61*y)/100))/7500 - (21*exp(-(63*y)/100))/2500 + (21*exp((63*y)/100))/2500 - (67*exp(-(67*y)/100))/7500 + (67*exp((67*y)/100))/7500 - (23*exp(-(69*y)/100))/2500 + (23*exp((69*y)/100))/2500 - (71*exp(-(71*y)/100))/7500 + (71*exp((71*y)/100))/7500 - (73*exp(-(73*y)/100))/7500 + (73*exp((73*y)/100))/7500 - (77*exp(-(77*y)/100))/7500 + (77*exp((77*y)/100))/7500 - (79*exp(-(79*y)/100))/7500 + (79*exp((79*y)/100))/7500 - (27*exp(-(81*y)/100))/2500 + (27*exp((81*y)/100))/2500 - (83*exp(-(83*y)/100))/7500 + (83*exp((83*y)/100))/7500 - (29*exp(-(87*y)/100))/2500 + (29*exp((87*y)/100))/2500 - (89*exp(-(89*y)/100))/7500 + (89*exp((89*y)/100))/7500 - (91*exp(-(91*y)/100))/7500 + (91*exp((91*y)/100))/7500 - (31*exp(-(93*y)/100))/2500 + (31*exp((93*y)/100))/2500 - (97*exp(-(97*y)/100))/7500 + (97*exp((97*y)/100))/7500 - (33*exp(-(99*y)/100))/2500 + (33*exp((99*y)/100))/2500 + exp(y)/300

所以我得到了一个结果，但它不是我想要的分析结果，而且有点令人讨厌。这是正确的吗？

vpa(subs(res,y,1/2))
ans =
   0.34174141693463006644516447861307

我将复制上面的分析结果，以便我们进行比较...

   0.34174141687554424792549563431876

如您所见，辛普森规则在 [-1,1] 上的步长为 0.01 时，表现相当不错，同意大约 9 位小数。

不能保证这种技术在任何更通用的内核上也能正常工作，但它可能会给你你想要的东西。