数值微分方程求解器算法意外出现段错误

Question

我试图用倍频程求解一个微分方程，但我选择的微分单位需要很长时间，所以我决定用 C 对其进行编码。算法如下：

#include <stdio.h>

double J = 5.78e-5; // (N.m)/(rad/s^2)
double bo = 6.75e-4; // (N.m)/(rad/s)
double ko = 5.95e-4; // (N.m)/rad
double Ka = 1.45e-3; // (N.m)/A
double Kb = 1.69e-3; // V/(rad/s)
double L = 0.311e-3; // mH
double R = 150; // ohms
double E = 5; // V

// Simulacion
int tf = 2;
double h = 1e-6;

double dzdt, dwdt, didt;

void solver(double t, double z, double w, double i) {
    printf("%f %f %f\n", z, w, i);
    if (t >= tf) {
        printf("Finished!\n");
        return; // End simulation
    }
    else {
        dzdt = w;
        dwdt = 1/J*( Ka*i - ko*z - bo*w );
        didt = 1/L*( E - R*i - Kb*w );
        // Solve next step with newly calculated "initial conditions"
        solver(t+h, z+h*dzdt, w+h*dwdt, i+h*didt);
    }
}

int main() {
    solver(0, 0, 0, 0);
    // Solve data
    // Write data to file
    return 0;
}

定义为h（如您所见）的微分单位必须那么小，否则值会失控并且解决方案将不正确。现在，h 的数值更大，程序从头到尾都没有错误（nan 值除外），但是我选择了h，我得到了分段错误；是什么原因造成的？

替代 Octave 解决方案

在我的一个朋友告诉我他能够使用 MATLAB 使用 1e-3 的微分步长求解方程后，我发现 MATLAB 有一个“僵硬”版本的 ode23 模块——“僵硬”对于求解那些需要极小步长的微分方程有特殊意义。后来我在 Octave 中搜索了“僵硬”的 ODE 求解器，发现 lsode 属于该类别。在第一次尝试中，lsode 在微秒内解决了方程（都比 MATLAB 和我的 C 实现快），并且得到了完美的解决方案。 FOSS 万岁！

Answer 1

求解器函数递归调用自身。有多深？大概是几百万次。你的堆栈空间用完了吗？每个递归都需要四个双精度数和一个堆栈帧的空间，这很快就会加起来。

我建议你将求解器重写为迭代而不是递归函数。

Answer 2

你的递归没有足够快地终止，所以你正在耗尽你的堆栈。

要解决这个问题，只需让它成为一个循环，看起来你实际上并没有在做任何需要递归的事情。

我认为这样做：

void solver(double t, double z, double w, double i) {
    while (!(t >= tf)) {
        printf("%f %f %f\n", z, w, i);
        dzdt = w;
        dwdt = 1/J*( Ka*i - ko*z - bo*w );
        didt = 1/L*( E - R*i - Kb*w );
        // Solve next step with newly calculated "initial conditions"
        t = t+h;
        z = z+h*dzdt;
        w = w+h*dwdt;
        i = i+h*didt;
    } 
    printf("Finished!\n");
}

---

附带说明一下，您的函数有资格进行尾递归优化，因此，如果您在打开一些优化的情况下编译它（例如 -O2），任何体面的编译器实际上都足够聪明，可以进行尾递归调用，并且您的程序不会出现段错误。

Answer 3

您正在对求解器进行 1e6 次递归调用。我猜你的堆栈用完了。尝试在求解器中使用循环更新变量而不是调用函数。

伪代码：

 while t < tf
      do dt step
      t = t + dt

等等

Answer 4

正如@hexist 所说，这里不需要递归。

堆栈溢出：

==4734== Memcheck, a memory error detector
==4734== Copyright (C) 2002-2011, and GNU GPL'd, by Julian Seward et al.
==4734== Using Valgrind-3.7.0 and LibVEX; rerun with -h for copyright info
==4734== Command: ./demo
==4734== 
==4734== Stack overflow in thread 1: can't grow stack to 0x7fe801ff8
==4734== 
==4734== Process terminating with default action of signal 11 (SIGSEGV)
==4734==  Access not within mapped region at address 0x7FE801FF8
==4734==    at 0x40054E: solver (demo.c:18)
==4734==  If you believe this happened as a result of a stack
==4734==  overflow in your program's main thread (unlikely but
==4734==  possible), you can try to increase the size of the
==4734==  main thread stack using the --main-stacksize= flag.
==4734==  The main thread stack size used in this run was 8388608.
==4734== Stack overflow in thread 1: can't grow stack to 0x7fe801fe8
==4734== 
==4734== Process terminating with default action of signal 11 (SIGSEGV)
==4734==  Access not within mapped region at address 0x7FE801FE8
==4734==    at 0x4A226E0: _vgnU_freeres (vg_preloaded.c:58)
==4734==  If you believe this happened as a result of a stack
==4734==  overflow in your program's main thread (unlikely but
==4734==  possible), you can try to increase the size of the
==4734==  main thread stack using the --main-stacksize= flag.
==4734==  The main thread stack size used in this run was 8388608.
==4734== 
==4734== HEAP SUMMARY:
==4734==     in use at exit: 0 bytes in 0 blocks
==4734==   total heap usage: 0 allocs, 0 frees, 0 bytes allocated
==4734== 
==4734== All heap blocks were freed -- no leaks are possible
==4734== 
==4734== For counts of detected and suppressed errors, rerun with: -v
==4734== ERROR SUMMARY: 0 errors from 0 contexts (suppressed: 4 from 4)

Answer 5

嗯，你已经得到了很多关于你的实际问题的答案。我只是想提请您注意另一个库。使用boost.odeint，如果您需要一个快速且易于使用的 ode 求解器，这基本上是最先进的库。忘掉 GSL、Matlab 等，Odeint 的表现都胜过所有这些。

您的程序将如下所示：

#include <boost/numeric/odeint.hpp>
using namespace boost::numeric::odeint;

typedef boost::array<double,3> State;

const double J  = 5.78e-5; // (N.m)/(rad/s^2)
const double bo = 6.75e-4; // (N.m)/(rad/s)
const double ko = 5.95e-4; // (N.m)/rad
const double Ka = 1.45e-3; // (N.m)/A
const double Kb = 1.69e-3; // V/(rad/s)
const double L  = 0.311e-3; // mH
const double R  = 150; // ohms
const double E  = 5; // V

void my_ode( State const &s , State &dsdt , double t ) {
   double const &z = s[0],    // this is just a name
                &w = s[1],    // forwarding for better
                &i = s[2];    // readability of the ode
   dsdt[0] = w;
   dsdt[1] = 1./J * ( Ka*i - ko*z - bo*w );
   dsdt[2] = 1./L * ( E - R*i - Kb*w );
}

void printer( State const &s , double t ) {
   std::cout << s[0] << " " << s[1] << " " << s[2] << std::endl;
}

int main() {
   State s = {{ 0, 0, 0 }};
   integrate_const(
      euler<State>() , my_ode , s , 0. , 2. , 1e-6 , printer
   );
}