GSL sample: Initial Value Problem for Ordinary Differential Equation

Last Update: Oct. 4, 2007

Sample code

linear_ode.c

linear_ode.c
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
/***********************************************/
/* T.Kouya's GSL sample program collection     */
/* Stiff and Non-stiff Linear Ordinary         */
/*                      Differential Equations */
/*                   Written by Tomonori Kouya */
/*                                             */
/* Version 0.01: 2007-08-13                    */
/***********************************************/
#include <stdio.h>
#include <gsl/gsl_errno.h>// GSL_SUCCESS ...
#include <gsl/gsl_math.h> // cos(x), sin(x) ...
#include <gsl/gsl_odeiv.h>// ODE solver
 
/* Return relative and absolute errors */
void vec_rel_error(double relerr[], double approx_val[], double true_val[], int dimension)
{
    static int i;
    static double abs_error, rel_error;
 
    for(i = 0; i < dimension; i++)
    {
        abs_error = fabs(approx_val[i] - true_val[i]);
        rel_error = abs_error;
        if(true_val[i] >= 1.0e-15)
            rel_error /= fabs(true_val[i]);
        relerr[i] = rel_error;
    }
 
    return;
}
 
/* Dimension of ODEs */
#define DIM 2
 
/* Common True Solution */
void true_solution(double ret_y[], double x)
{
    ret_y[0] = exp(-x);
    ret_y[1] = exp(-x) + cos(x);
}
 
/* Definition of ODE system */
/* Non-stiff Linear ODE (cf. http://na-inet.jp/nasoft/chap16.pdf) */
/* dydx[] := func(x, y[]) */
int nonstiff_lode_func(double x, const double y[], double dydx[], void *params)
{
    dydx[0] = -2.0 * y[0] +       y[1] - cos(x);
    dydx[1] =  2.0 * y[0] - 3.0 * y[1] + 3.0 * cos(x) - sin(x);
 
    return GSL_SUCCESS;
}
 
/* Jacobian matrix of nonstiff_lode_func */
int jac_nonstiff_lode_func(double x, const double y[], double *dfdy, double dfdx[], void *params)
{
    /* Jacobian matrix -> dfdy */
    *(dfdy + 0 * DIM + 0) = -2.0;
    *(dfdy + 0 * DIM + 1) =  1.0;
    *(dfdy + 1 * DIM + 0) =  2.0;
    *(dfdy + 1 * DIM + 1) = -3.0;
 
    /* df/dt */
    dfdx[0] = sin(x);
    dfdx[1] = -3.0 * sin(x) - cos(x);
 
    return GSL_SUCCESS;
}
 
/* Definition of ODE system */
/* Stiff Linear ODE (cf. http://na-inet.jp/nasoft/chap16.pdf) */
/* dydx[] := func(x, y[]) */
int stiff_lode_func(double x, const double y[], double dydx[], void *params)
{
    dydx[0] =    -2.0 * y[0] +          y[1] - cos(x);
    dydx[1] =  1998.0 * y[0] - 1999.0 * y[1] + 1999.0 * cos(x) - sin(x);
 
    return GSL_SUCCESS;
}
 
/* Jacobian matrix of stiff_lode_func */
int jac_stiff_lode_func(double x, const double y[], double *dfdy, double dfdx[], void *params)
{
    /* Jacobian matrix -> dfdy */
    *(dfdy + 0 * DIM + 0) = -2.0;
    *(dfdy + 0 * DIM + 1) =  1.0;
    *(dfdy + 1 * DIM + 0) =  1998.0;
    *(dfdy + 1 * DIM + 1) = -1999.0;
 
    /* df/dt */
    dfdx[0] = sin(x);
    dfdx[1] = -1999.0 * sin(x) - cos(x);
 
    return GSL_SUCCESS;
}
 
int main(void)
{
    int i;
 
    /* Integration Interval: [0, 10] */
    /* Initial min stepsize: 1.0e-5  */
    /* Initial value       : y(0) = [1, 2]^T */
    double x_start = 0.0, x_end = 10.0;
    double h_init = 1.0e-5;
    double y_init[DIM] = {1.0, 2.0};
    double relerr_y[DIM], true_y[DIM];
 
    /* Variables used for evolving */
    int status_nonstiff, status_stiff;
    double h_nonstiff, x_nonstiff, y_nonstiff[DIM];
    double h_stiff, x_stiff, y_stiff[DIM];
 
    /* Definitions to determine ODE solvers */
    const gsl_odeiv_step_type *solver_nonstiff = gsl_odeiv_step_rkf45; // Runge-Kutta Felberg (4, 5)
    const gsl_odeiv_step_type *solver_stiff = gsl_odeiv_step_rk4imp; // Fully implicit RK Gauss 4th order
 
    /* Memories of stepsizes */
    gsl_odeiv_step *step_nonstiff, *step_stiff;
 
    /* Constants to control errors */
    gsl_odeiv_control *tol_nonstiff, *tol_stiff;
 
    /* Memories needed to evolve ODE solvers */
    gsl_odeiv_evolve *evol_nonstiff, *evol_stiff;
 
    /* Definitions of ODE systems */
    gsl_odeiv_system sys_nonstiff, sys_stiff;
 
/* Non-stiff Problem */
 
    /* ODE system */
    sys_nonstiff.function = nonstiff_lode_func;
    sys_nonstiff.jacobian = jac_nonstiff_lode_func;
    sys_nonstiff.dimension = (size_t)(DIM);
    sys_nonstiff.params = NULL;
 
    /* ODE solver */
    /* Determine constans for error controling */
    /* Preparing evolution */
    step_nonstiff = gsl_odeiv_step_alloc(solver_nonstiff, DIM);
    tol_nonstiff = gsl_odeiv_control_standard_new(1.0e-10, 1.0e-5, 1.0, 0.0);
    evol_nonstiff = gsl_odeiv_evolve_alloc(DIM);
 
    /* Initialize for integration */
    h_nonstiff = h_init;
    x_nonstiff = x_start;
    for(i = 0; i < DIM; i++)
        y_nonstiff[i] = y_init[i];
 
    /* Integration with stepsize control */
    printf("Non-stiff Problem:\n");
    printf("  Solver: %s, Controling: %s\n", gsl_odeiv_step_name(step_nonstiff), gsl_odeiv_control_name(tol_nonstiff));
    printf("     x                    y[0]                     y[1]           Relerr y[0] Relerr y[1]\n");
    while(x_nonstiff < x_end)
    {
        status_nonstiff = gsl_odeiv_evolve_apply(evol_nonstiff, tol_nonstiff, step_nonstiff, &sys_nonstiff, &x_nonstiff, x_end, &h_nonstiff, y_nonstiff);
        if(status_nonstiff != GSL_SUCCESS)
        {
            fprintf(stderr, "ERROR: Non-stiff Problem at x = %25.17e\n", x_nonstiff);
            break;
        }
     
        true_solution(true_y, x_nonstiff);
        vec_rel_error(relerr_y, y_nonstiff, true_y, DIM);
        printf("%15.7e %25.17e %25.17e %10.3e %10.3e\n", x_nonstiff, y_nonstiff[0], y_nonstiff[1], relerr_y[0], relerr_y[1]);
    }
 
    /* Free */
    gsl_odeiv_evolve_free(evol_nonstiff);
    gsl_odeiv_control_free(tol_nonstiff);
    gsl_odeiv_step_free(step_nonstiff);
 
/* Stiff Problem */
 
    /* ODE system */
    sys_stiff.function = stiff_lode_func;
    sys_stiff.jacobian = jac_stiff_lode_func;
    sys_stiff.dimension = (size_t)(DIM);
    sys_stiff.params = NULL;
 
    /* ODE solver */
    /* Determine constans for error controling */
    /* Preparing evolution */
    step_stiff = gsl_odeiv_step_alloc(solver_stiff, DIM);
    tol_stiff = gsl_odeiv_control_standard_new(1.0e-10, 1.0e-5, 1.0, 0.0);
    evol_stiff = gsl_odeiv_evolve_alloc(DIM);
 
    /* Initialize for integration */
    h_stiff = h_init;
    x_stiff = x_start;
    for(i = 0; i < DIM; i++)
        y_stiff[i] = y_init[i];
 
    /* Integration with stepsize control */
    printf("Stiff Problem:\n");
    printf("  Solver: %s, Controling: %s\n", gsl_odeiv_step_name(step_stiff), gsl_odeiv_control_name(tol_stiff));
    printf("     x                    y[0]                     y[1]           Relerr y[0] Relerr y[1]\n");
    while(x_stiff < x_end)
    {
        status_stiff = gsl_odeiv_evolve_apply(evol_stiff, tol_stiff, step_stiff, &sys_stiff, &x_stiff, x_end, &h_stiff, y_stiff);
        if(status_stiff != GSL_SUCCESS)
        {
            fprintf(stderr, "ERROR: Stiff Problem at x = %25.17e\n", x_stiff);
            break;
        }
     
        true_solution(true_y, x_stiff);
        vec_rel_error(relerr_y, y_stiff, true_y, DIM);
        printf("%15.7e %25.17e %25.17e %10.3e %10.3e\n", x_stiff, y_stiff[0], y_stiff[1], relerr_y[0], relerr_y[1]);
    }
 
    /* Free */
    gsl_odeiv_evolve_free(evol_stiff);
    gsl_odeiv_control_free(tol_stiff);
    gsl_odeiv_step_free(step_stiff);
 
    return 0;
}

Example execution

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
$ ./linear_ode
Non-stiff Problem:
  Solver: rkf45, Controling: standard
     x                    y[0]                     y[1]           Relerr y[0] Relerr y[1]
  1.0000000e-05   9.99990000049999828e-01   1.99998999999999993e+00  0.000e+00  0.000e+00
  6.0000000e-05   9.99940001799964007e-01   1.99993999999996408e+00  0.000e+00  0.000e+00
  3.1000000e-04   9.99690048045035251e-01   1.99968999999503572e+00  0.000e+00  0.000e+00
  1.5600000e-03   9.98441216167510692e-01   1.99843999936775751e+00  0.000e+00  0.000e+00
  7.8100000e-03   9.92220418808186233e-01   1.99218992091321856e+00  3.692e-15  3.567e-15
  3.9060000e-02   9.61693005778894716e-01   1.96093026112794799e+00  5.827e-11  5.610e-11
(OMIT)
  9.7572849e+00   5.78719406467666030e-05  -9.45169137882558519e-01  7.013e-06  8.551e-10
  9.7874581e+00   5.61518489106037504e-05  -9.34893173693603030e-01  7.137e-06  8.440e-10
  9.8176313e+00   5.44828825914175032e-05  -9.23766026470360435e-01  7.265e-06  8.332e-10
  9.8478044e+00   5.28635220633079945e-05  -9.11797828859521320e-01  7.396e-06  8.227e-10
  9.8779776e+00   5.12922928725537602e-05  -8.98999479068398610e-01  7.529e-06  8.122e-10
  9.9081507e+00   4.97677643944452827e-05  -8.85382630946418447e-01  7.664e-06  8.018e-10
  9.9383239e+00   4.82885485301170624e-05  -8.70959683378632055e-01  7.798e-06  7.913e-10
  9.9684971e+00   4.68532984421406228e-05  -8.55743769000906318e-01  7.932e-06  7.807e-10
  9.9986702e+00   4.54607073277795066e-05  -8.39748742247069080e-01  8.065e-06  7.699e-10
  1.0000000e+01   4.54002943932760116e-05  -8.39026129912536334e-01  8.032e-06  7.658e-10
Stiff Problem:
  Solver: rk4imp, Controling: standard
     x                    y[0]                     y[1]           Relerr y[0] Relerr y[1]
  1.0000000e-05   9.99990000049999828e-01   1.99998999999999971e+00  0.000e+00  1.110e-16
  6.0000000e-05   9.99940001799964007e-01   1.99993999999996386e+00  0.000e+00  1.110e-16
  3.1000000e-04   9.99690048045035362e-01   1.99968999999464692e+00  1.111e-16  1.944e-13
  1.5600000e-03   9.98441216170407153e-01   1.99843999358035185e+00  2.901e-12  2.896e-09
  4.3639627e-03   9.95645547812364029e-01   1.99563145641214712e+00  2.296e-09  2.289e-06
  5.8180308e-03   9.94198861343948082e-01   1.99418160531753896e+00  1.667e-10  1.661e-07
  8.0221772e-03   9.92009914828045125e-01   1.99197728977807276e+00  2.257e-10  2.246e-07
(OMIT)
  9.9804552e+00   4.62958761739525062e-05  -8.49496859624095046e-01  2.480e-06  2.294e-07
  9.9825539e+00   4.61988174664507395e-05  -8.48387990757782595e-01  2.482e-06  2.291e-07
  9.9846526e+00   4.61019622419043659e-05  -8.47275384776054685e-01  2.484e-06  2.288e-07
  9.9867512e+00   4.60053100737143624e-05  -8.46159046580215457e-01  2.486e-06  2.285e-07
  9.9888499e+00   4.59088605361768437e-05  -8.45038981088006569e-01  2.488e-06  2.282e-07
  9.9909486e+00   4.58126132044794909e-05  -8.43915193233583549e-01  2.490e-06  2.279e-07
  9.9930473e+00   4.57165676547016873e-05  -8.42787687967497146e-01  2.492e-06  2.276e-07
  9.9951460e+00   4.56207234638109064e-05  -8.41656470256668010e-01  2.494e-06  2.273e-07
  9.9972447e+00   4.55250802096614788e-05  -8.40521545084366828e-01  2.496e-06  2.270e-07
  9.9993434e+00   4.54296374709929523e-05  -8.39382917450192223e-01  2.498e-06  2.267e-07
  1.0000000e+01   4.53998991727971499e-05  -8.39026068028467908e-01  6.738e-07  6.112e-08

<- Go back