R 中的数值积分：错误的被积函数行为错误
Numerical integration in R : bad integrand behaviour error
这是我的矩阵
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-3.70907277842741, -3.61955988253518, -3.55951762156004, -3.27305364378077,
-3.15716747633446, -3.08436906473098, -2.94699678899349, -2.67819176248474,
-2.51950911610342, -2.45801068672049, -2.38080987129559, -2.23660157184075,
-2.03595688651075, -1.58836441712333, -1.47187389747016, -1.39555037998287,
-1.27113029247092, -1.15673682064109, -1.0426545765059, -0.943946623196274,
-0.835837575946843, -0.805837349555374, -0.689648730659341, -0.648094429117022,
-0.584092303333525, -0.394149224677363, -0.222824659020402, -0.0906541511144734,
0, 0.0906541511144734, 0.222824659020402, 0.394149224677363,
0.584092303333525, 0.648094429117022, 0.689648730659341, 0.805837349555374,
0.835837575946843, 0.943946623196274, 1.0426545765059, 1.15673682064109,
1.27113029247092, 1.39555037998287, 1.47187389747016, 1.58836441712333,
2.03595688651075, 2.23660157184075, 2.38080987129559, 2.45801068672049,
2.51950911610342, 2.67819176248474, 2.94699678899349, 3.08436906473098,
3.15716747633446, 3.27305364378077, 3.55951762156004, 3.61955988253518,
3.70907277842741, 3.77843705689023, 3.93194755639431, 4.26474198682156,
4.37919276705198, 4.60547103236952, 4.78689576531572, 4.95085939235108,
5.09102141962306, 5.43365721329602, 5.85854042138457, 6.15110072929745,
6.36994876479492, 6.48897104737938, 7.13162491301438, 7.50817793772083,
7.93147284818737, 8.30440651819497, 10.1862843945321, 11.1957507602953,
12.2966556390103, 12.6000848203634, -12.7233949898779, -12.425009683628,
-11.2494681735225, -10.2166322197219, -8.3755092456272, -8.00338523998922,
-7.57923229725886, -7.19608684570266, -6.5405512372713, -6.40269304276938,
-6.17884044516668, -5.8896632980985, -5.47132446137006, -5.13601542532135,
-4.9932306291456, -4.82484171541241, -4.64182091659811, -4.41774407285591,
-4.29324182862204, -3.95837714178017, -3.80090594237893, -3.73788575810611,
-3.6386462286676, -3.57417931862452, -3.29515190899568, -3.18022337150066,
-3.10777656095811, -2.97137842876953, -2.69668587115743, -2.54143464829308,
-2.47346988209172, -2.39673461879726, -2.24821478269518, -2.05469079214647,
-1.61210502089736, -1.50444445385474, -1.42335954097761, -1.29707578786137,
-1.18226432293022, -1.06498427720905, -0.963826675880699, -0.85422851115692,
-0.817708608597364, -0.698957479868998, -0.647234505242058, -0.580874151897233,
-0.392280434624987, -0.221324908501217, -0.0951930012794266,
0, 0.0951930012794266, 0.221324908501217, 0.392280434624987,
0.580874151897233, 0.647234505242058, 0.698957479868998, 0.817708608597364,
0.85422851115692, 0.963826675880699, 1.06498427720905, 1.18226432293022,
1.29707578786137, 1.42335954097761, 1.50444445385474, 1.61210502089736,
2.05469079214647, 2.24821478269518, 2.39673461879726, 2.47346988209172,
2.54143464829308, 2.69668587115743, 2.97137842876953, 3.10777656095811,
3.18022337150066, 3.29515190899568, 3.57417931862452, 3.6386462286676,
3.73788575810611, 3.80090594237893, 3.95837714178017, 4.29324182862204,
4.41774407285591, 4.64182091659811, 4.82484171541241, 4.9932306291456,
5.13601542532135, 5.47132446137006, 5.8896632980985, 6.17884044516668,
6.40269304276938, 6.5405512372713, 7.19608684570266, 7.57923229725886,
8.00338523998922, 8.3755092456272, 10.2166322197219, 11.2494681735225,
12.425009683628, 12.7233949898779, -12.9701665831908, -12.6818752140478,
-11.3569688906404, -10.2773650952434, -8.51780191626772, -8.14868819459235,
-7.72003821070641, -7.32508978114711, -6.64377488616525, -6.46822176338875,
-6.23435390289731, -5.95194722735932, -5.54670516078474, -5.22605862710606,
-5.07802507599128, -4.90078016073291, -4.71456527242172, -4.49489397213016,
-4.35027647059818, -4.01126873151645, -3.85381415345849, -3.78760418054142,
-3.67684233255883, -3.6035206970325, -3.33937554552141, -3.22636344257365,
-3.15462026543263, -3.02017161524238, -2.73369677368098, -2.58531260689116,
-2.50440723533775, -2.42860364735808, -2.27145544935968, -2.09218158273497,
-1.66032132882236, -1.56891953844307, -1.47901197414181, -1.34899860381407,
-1.23335063996306, -1.10967106859385, -1.00361116644947, -0.894069184830091,
-0.838429443534018, -0.717586396553344, -0.645513602695347, -0.574433901587114,
-0.388540562231589, -0.218323567844169, -0.104276269037671, 0,
0.104276269037671, 0.218323567844169, 0.388540562231589, 0.574433901587114,
0.645513602695347, 0.717586396553344, 0.838429443534018, 0.894069184830091,
1.00361116644947, 1.10967106859385, 1.23335063996306, 1.34899860381407,
1.47901197414181, 1.56891953844307, 1.66032132882236, 2.09218158273497,
2.27145544935968, 2.42860364735808, 2.50440723533775, 2.58531260689116,
2.73369677368098, 3.02017161524238, 3.15462026543263, 3.22636344257365,
3.33937554552141, 3.6035206970325, 3.67684233255883, 3.78760418054142,
3.85381415345849, 4.01126873151645, 4.35027647059818, 4.49489397213016,
4.71456527242172, 4.90078016073291, 5.07802507599128, 5.22605862710606,
5.54670516078474, 5.95194722735932, 6.23435390289731, 6.46822176338875,
6.64377488616525, 7.32508978114711, 7.72003821070641, 8.14868819459235,
8.51780191626772, 10.2773650952434, 11.3569688906404, 12.6818752140478,
12.9701665831908), .Dim = c(99L, 11L))
每一列都是我想要集成的函数，并将值放入我的 integr 矩阵中：
integr=matrix(0,11)
for (t in 1:11){
integrating = approxfun(thau,func[,t],rule=2)
integr[t,1]=integrate(integrating, lower = 0.01, upper = 0.5,subdivisions = 1000)$value
}
我收到此错误消息：
Error in integrate(integrating, lower = 0.01, upper = 0.5, subdivisions = 1000) :
extremely bad integrand behaviour
我如何解决这个问题并继续使用 R 中的函数 integrate。我需要做一些非线性方法吗？如果可以，我该怎么做？
非常感谢。
首先，最近10天左右看到不少关于分位数函数的数值积分题，非常有趣。例如：
注意这个问题与第一个问题有何相似之处。虽然你没有提到 thau 是什么，但我相信它是 thau <- seq(0.01, 0.99, 0.01)。让我们根据 thau:
绘制您的 func 矩阵
matplot(thau, func, type = "l")
此外，让我们验证 func 的所有列都是单调递增的：
all(diff(func) > 0)
# [1] TRUE
基本上，您的问题是使用我在第一个链接问题中提供的答案（不过，rule = 2 的理由在第三个链接问题中给出）。但多亏了你的问题；我现在意识到背后有一些潜在的数字缺陷。
由于我不是该领域的专家，因此在有限的时间内了解自适应正交背后的数学知识对我来说很复杂。但令我惊讶的是，它有时会在如此微不足道的任务上失败。
正如我在第二个相关问题中提到的，我们甚至可以使用梯形法则。
当我测试 integrate 时，它是 func 的第 1 列，而不是您报告的第 5 列失败。
## get interpolation function for all columns in a list
flst <- lapply(1:ncol(func), function (i) approxfun(thau, func[,i], rule = 2))
## all OK excluding the 1st column
sapply(flst[-1], function (fun) integrate(fun, 0.01, 0.5)$value)
# [1] -2.010421 -2.088981 -2.114083 -2.000653 -2.015932 -1.986130 -1.912076
# [8] -1.877459 -1.892291 -1.921983
## the 1st one fails
integrate(flst[[1]], 0.01, 0.5)
# extremely bad integrand behaviour
如前所述，由于问题的简单性，我认为此失败是人为的。事实上，让我们考虑
integrate(flst[[1]], 0.01 + .Machine$double.eps ^ 0.25, 0.5)
# -2.13653 with absolute error < 8.7e-05
integrate(flst[[1]], 0, 0.5)
# -2.286034 with absolute error < 0.00027
它们都有效。
据我所知，integrate 函数使用了两个 Fortran 子例程：
integrate 的 R 文档对这些例程的错误处理没有太多解释，但 Fortran 页面做了一些解释。不幸的是，"bad behaviour" 是什么仍然不是很清楚。但是很明显，无论错误代码是什么，那些Fortran子程序总是return集成结果。
查看 integrate 的源代码可以验证这一点。积分结果存储在变量wk中，然后使用swtich语句解释存储在wk$ierr中的整数错误代码：
res$message <- switch(wk$ierr + 1L, "OK", "maximum number of subdivisions reached",
"roundoff error was detected", "extremely bad integrand behaviour",
"roundoff error is detected in the extrapolation table",
"the integral is probably divergent", "the input is invalid")
if (wk$ierr == 6L || (wk$ierr > 0L && stop.on.error))
stop(res$message)
此 switch 之后的 if 语句决定我们是否要忽略任何错误。 注意 integrate 中有一个 stop.on.error 参数；如果我们设置它 FALSE 而不是默认的 TRUE，integrate 将始终有效。 因此，让我们做
z <- integrate(flst[[1]], 0.01, 0.5, stop.on.error = FALSE)
str(z)
# $ value : num -2.14
# $ abs.error : num 0.000446
# $ subdivisions: int 69
# $ message : chr "extremely bad integrand behaviour"
# $ call : language integrate(f = flst[[1]], lower = 0.01, upper = 0.5, stop.on.error = FALSE)
# - attr(*, "class")= chr "integrate"
z$value
# [1] -2.138348
目前我只能做这些了。我相信在不久的将来我会有机会阅读自适应正交。
这是我的矩阵
func=structure(c(-14.7690673280818, -14.5543581356252, -12.1406211639974,
-10.7200919648493, -9.55507848352318, -9.20790894914246, -8.74647670464071,
-8.26548763467919, -7.3962484443768, -6.94590909664862, -6.63903257406218,
-6.40598087538929, -6.09621085755895, -5.88245015332806, -5.69615454359131,
-5.4543516801245, -5.24485270997044, -5.05729667581875, -4.7660441793448,
-4.39683466679171, -4.30668996362841, -4.0828498417172, -3.95528199743041,
-3.81741171570944, -3.66175435590196, -3.56271256245504, -3.49609868983868,
-3.37586125647396, -3.0034966287813, -2.90517151506904, -2.72993249556762,
-2.66092057836039, -2.4408738658527, -2.36547969703899, -2.09546783234301,
-1.95526398213212, -1.88470374007828, -1.72750243012653, -1.60575660667247,
-1.43542615745848, -1.29362975715702, -1.18449733575892, -0.98947894198681,
-0.853386354928181, -0.63296867063245, -0.527486151545778, -0.361277864998242,
-0.257665212366851, -0.109270291541468, 0, 0.109270291541468,
0.257665212366851, 0.361277864998242, 0.527486151545778, 0.63296867063245,
0.853386354928181, 0.98947894198681, 1.18449733575892, 1.29362975715702,
1.43542615745848, 1.60575660667247, 1.72750243012653, 1.88470374007828,
1.95526398213212, 2.09546783234301, 2.36547969703899, 2.4408738658527,
2.66092057836039, 2.72993249556762, 2.90517151506904, 3.0034966287813,
3.37586125647396, 3.49609868983868, 3.56271256245504, 3.66175435590196,
3.81741171570944, 3.95528199743041, 4.0828498417172, 4.30668996362841,
4.39683466679171, 4.7660441793448, 5.05729667581875, 5.24485270997044,
5.4543516801245, 5.69615454359131, 5.88245015332806, 6.09621085755895,
6.40598087538929, 6.63903257406218, 6.94590909664862, 7.3962484443768,
8.26548763467919, 8.74647670464071, 9.20790894914246, 9.55507848352318,
10.7200919648493, 12.1406211639974, 14.5543581356252, 14.7690673280818,
-13.7079821475458, -13.4498703618915, -11.67838231268, -10.4589486406842,
-8.9432388382521, -8.58312548137122, -8.14102991846389, -7.71079209128989,
-6.95240042017031, -6.66414427349685, -6.40033205538623, -6.13816822549321,
-5.77208382991545, -5.49527630669887, -5.33154964830415, -5.12782641892739,
-4.93206161073643, -4.72556232300793, -4.52080276980657, -4.1694078333758,
-4.03956026669583, -3.90869860153294, -3.79104380757601, -3.69124767313376,
-3.47159857371935, -3.3643163654408, -3.29467696235366, -3.16605701449292,
-2.84435464789616, -2.71650210221017, -2.59690597248341, -2.52388797476942,
-2.34094207608544, -2.20427426100811, -1.83879580218285, -1.72737794398069,
-1.6454056454373, -1.5042411957216, -1.38609220687676, -1.2432788707001,
-1.1225617148797, -1.01318771357089, -0.90038209504245, -0.773284479514442,
-0.640368323748707, -0.555178375238413, -0.377358821158776, -0.230347860323025,
-0.110436162251717, 0, 0.110436162251717, 0.230347860323025,
0.377358821158776, 0.555178375238413, 0.640368323748707, 0.773284479514442,
0.90038209504245, 1.01318771357089, 1.1225617148797, 1.2432788707001,
1.38609220687676, 1.5042411957216, 1.6454056454373, 1.72737794398069,
1.83879580218285, 2.20427426100811, 2.34094207608544, 2.52388797476942,
2.59690597248341, 2.71650210221017, 2.84435464789616, 3.16605701449292,
3.29467696235366, 3.3643163654408, 3.47159857371935, 3.69124767313376,
3.79104380757601, 3.90869860153294, 4.03956026669583, 4.1694078333758,
4.52080276980657, 4.72556232300793, 4.93206161073643, 5.12782641892739,
5.33154964830415, 5.49527630669887, 5.77208382991545, 6.13816822549321,
6.40033205538623, 6.66414427349685, 6.95240042017031, 7.71079209128989,
8.14102991846389, 8.58312548137122, 8.9432388382521, 10.4589486406842,
11.67838231268, 13.4498703618915, 13.7079821475458, -14.358609077878,
-14.1271105147046, -11.9618138886078, -10.6190742201334, -9.31840132054895,
-8.96622474353695, -8.51227248333579, -8.05091545299969, -7.22455526958365,
-6.83691436333096, -6.54669635165921, -6.30238324376159, -5.97082920578328,
-5.73268017960003, -5.55511488901834, -5.32804232266972, -5.12385612117062,
-4.92897229198519, -4.67117775232928, -4.30885942992513, -4.20335651783943,
-4.0154831357938, -3.89174994433905, -3.76860783727992, -3.58819663306074,
-3.48596721027473, -3.41818297549918, -3.29470294456033, -2.94193593581688,
-2.83218876260999, -2.67847401240583, -2.60791242809822, -2.40221737689215,
-2.3031208003073, -1.99617971268181, -1.86711111167336, -1.79213635811389,
-1.64113856670334, -1.52078410227216, -1.36109806339877, -1.22745572081485,
-1.11822984936273, -0.955013720375685, -0.822400646878094, -0.635831069171161,
-0.538198299721656, -0.367498441110635, -0.24709807532094, -0.109721283846771,
0, 0.109721283846771, 0.24709807532094, 0.367498441110635, 0.538198299721656,
0.635831069171161, 0.822400646878094, 0.955013720375685, 1.11822984936273,
1.22745572081485, 1.36109806339877, 1.52078410227216, 1.64113856670334,
1.79213635811389, 1.86711111167336, 1.99617971268181, 2.3031208003073,
2.40221737689215, 2.60791242809822, 2.67847401240583, 2.83218876260999,
2.94193593581688, 3.29470294456033, 3.41818297549918, 3.48596721027473,
3.58819663306074, 3.76860783727992, 3.89174994433905, 4.0154831357938,
4.20335651783943, 4.30885942992513, 4.67117775232928, 4.92897229198519,
5.12385612117062, 5.32804232266972, 5.55511488901834, 5.73268017960003,
5.97082920578328, 6.30238324376159, 6.54669635165921, 6.83691436333096,
7.22455526958365, 8.05091545299969, 8.51227248333579, 8.96622474353695,
9.31840132054895, 10.6190742201334, 11.9618138886078, 14.1271105147046,
14.358609077878, -14.5672256639001, -14.3442603466748, -12.0526932005212,
-10.6704167793823, -9.43869316924387, -9.08906143604798, -8.63130745231583,
-8.15997238138419, -7.31181881941087, -6.89231124997616, -6.59362650222709,
-6.3550370379524, -6.03455479085867, -5.80880119944129, -5.62679869904323,
-5.39223941682433, -5.1853529853784, -4.99419352909143, -4.7193938866676,
-4.35357309705323, -4.255876038818, -4.04972245780601, -3.92404029302206,
-3.79341254883719, -3.62558255722496, -3.52497325621722, -3.4577838613747,
-3.33595188997765, -2.97322433504632, -2.86928245642403, -2.70462793445213,
-2.63485397290467, -2.42186464857093, -2.33481488889686, -2.04664318502277,
-1.91191506158551, -1.8391839961936, -1.68503324864501, -1.56397162198931,
-1.39887553095015, -1.26108886508264, -1.15191048988921, -0.972530768371285,
-0.838149222100249, -0.634376246882453, -0.532753819606308, -0.364336815412981,
-0.252468853112474, -0.109492065710681, 0, 0.109492065710681,
0.252468853112474, 0.364336815412981, 0.532753819606308, 0.634376246882453,
0.838149222100249, 0.972530768371285, 1.15191048988921, 1.26108886508264,
1.39887553095015, 1.56397162198931, 1.68503324864501, 1.8391839961936,
1.91191506158551, 2.04664318502277, 2.33481488889686, 2.42186464857093,
2.63485397290467, 2.70462793445213, 2.86928245642403, 2.97322433504632,
3.33595188997765, 3.4577838613747, 3.52497325621722, 3.62558255722496,
3.79341254883719, 3.92404029302206, 4.04972245780601, 4.255876038818,
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8.51780191626772, 10.2773650952434, 11.3569688906404, 12.6818752140478,
12.9701665831908), .Dim = c(99L, 11L))
每一列都是我想要集成的函数，并将值放入我的 integr 矩阵中：
integr=matrix(0,11)
for (t in 1:11){
integrating = approxfun(thau,func[,t],rule=2)
integr[t,1]=integrate(integrating, lower = 0.01, upper = 0.5,subdivisions = 1000)$value
}
我收到此错误消息：
Error in integrate(integrating, lower = 0.01, upper = 0.5, subdivisions = 1000) :
extremely bad integrand behaviour
我如何解决这个问题并继续使用 R 中的函数 integrate。我需要做一些非线性方法吗？如果可以，我该怎么做？
非常感谢。
首先，最近10天左右看到不少关于分位数函数的数值积分题，非常有趣。例如：
注意这个问题与第一个问题有何相似之处。虽然你没有提到 thau 是什么，但我相信它是 thau <- seq(0.01, 0.99, 0.01)。让我们根据 thau:
func 矩阵
matplot(thau, func, type = "l")
此外，让我们验证 func 的所有列都是单调递增的：
all(diff(func) > 0)
# [1] TRUE
基本上，您的问题是使用我在第一个链接问题中提供的答案（不过，rule = 2 的理由在第三个链接问题中给出）。但多亏了你的问题；我现在意识到背后有一些潜在的数字缺陷。
由于我不是该领域的专家，因此在有限的时间内了解自适应正交背后的数学知识对我来说很复杂。但令我惊讶的是，它有时会在如此微不足道的任务上失败。 正如我在第二个相关问题中提到的，我们甚至可以使用梯形法则。
当我测试 integrate 时，它是 func 的第 1 列，而不是您报告的第 5 列失败。
## get interpolation function for all columns in a list
flst <- lapply(1:ncol(func), function (i) approxfun(thau, func[,i], rule = 2))
## all OK excluding the 1st column
sapply(flst[-1], function (fun) integrate(fun, 0.01, 0.5)$value)
# [1] -2.010421 -2.088981 -2.114083 -2.000653 -2.015932 -1.986130 -1.912076
# [8] -1.877459 -1.892291 -1.921983
## the 1st one fails
integrate(flst[[1]], 0.01, 0.5)
# extremely bad integrand behaviour
如前所述，由于问题的简单性，我认为此失败是人为的。事实上，让我们考虑
integrate(flst[[1]], 0.01 + .Machine$double.eps ^ 0.25, 0.5)
# -2.13653 with absolute error < 8.7e-05
integrate(flst[[1]], 0, 0.5)
# -2.286034 with absolute error < 0.00027
它们都有效。
据我所知，integrate 函数使用了两个 Fortran 子例程：
integrate 的 R 文档对这些例程的错误处理没有太多解释，但 Fortran 页面做了一些解释。不幸的是，"bad behaviour" 是什么仍然不是很清楚。但是很明显，无论错误代码是什么，那些Fortran子程序总是return集成结果。
查看 integrate 的源代码可以验证这一点。积分结果存储在变量wk中，然后使用swtich语句解释存储在wk$ierr中的整数错误代码：
res$message <- switch(wk$ierr + 1L, "OK", "maximum number of subdivisions reached",
"roundoff error was detected", "extremely bad integrand behaviour",
"roundoff error is detected in the extrapolation table",
"the integral is probably divergent", "the input is invalid")
if (wk$ierr == 6L || (wk$ierr > 0L && stop.on.error))
stop(res$message)
此 switch 之后的 if 语句决定我们是否要忽略任何错误。 注意 integrate 中有一个 stop.on.error 参数；如果我们设置它 FALSE 而不是默认的 TRUE，integrate 将始终有效。 因此，让我们做
z <- integrate(flst[[1]], 0.01, 0.5, stop.on.error = FALSE)
str(z)
# $ value : num -2.14
# $ abs.error : num 0.000446
# $ subdivisions: int 69
# $ message : chr "extremely bad integrand behaviour"
# $ call : language integrate(f = flst[[1]], lower = 0.01, upper = 0.5, stop.on.error = FALSE)
# - attr(*, "class")= chr "integrate"
z$value
# [1] -2.138348
目前我只能做这些了。我相信在不久的将来我会有机会阅读自适应正交。