Integer type:** int32**** int64**** nag_int** show int32 show int32 show int64 show int64 show nag_int show nag_int

nag_ode_ivp_2nd_rkn (d02la) is a function for integrating a non-stiff system of second-order ordinary differential equations using Runge–Kutta–Nystrom techniques.

Mark 22: lrwork has been removed from the interface

.Given the initial values x,y_{1},y_{2}, … ,y_{neq},y_{1}^{ ′ },y_{2}^{ ′ }, … ,y_{neq}^{ ′ }$x,{y}_{1},{y}_{2},\dots ,{y}_{{\mathbf{neq}}},{y}_{1}^{\prime},{y}_{2}^{\prime},\dots ,{y}_{{\mathbf{neq}}}^{\prime}$ nag_ode_ivp_2nd_rkn (d02la) integrates a non-stiff system of second-order differential equations of the type

from x = t$x={\mathbf{t}}$ to x = tend$x={\mathbf{tend}}$ using a Runge–Kutta–Nystrom formula pair. The system is defined by fcn, which evaluates f_{i}${f}_{i}$ in terms of x$x$ and y_{1},y_{2}, … ,y_{neq}${y}_{1},{y}_{2},\dots ,{y}_{{\mathbf{neq}}}$, where y_{1},y_{2}, … ,y_{neq}${y}_{1},{y}_{2},\dots ,{y}_{{\mathbf{neq}}}$ are supplied at x$x$.

$${y}_{i}^{\prime \prime}={f}_{i}(x,{y}_{1},{y}_{2},\dots ,{y}_{{\mathbf{neq}}})\text{, \hspace{1em}}i=1,2,\dots ,{\mathbf{neq}}\text{,}$$ |

There are two Runge–Kutta–Nystrom formula pairs implemented in this function. The lower order method is intended if you have moderate accuracy requirements and may be used in conjunction with the interpolation function nag_ode_ivp_2nd_rkn_interp (d02lz) to produce solutions and derivatives at user-specified points. The higher order method is intended if you have high accuracy requirements.

In one-step mode the function returns approximations to the solution, derivative and f_{i}${f}_{i}$ at each integration point. In interval mode these values are returned at the end of the integration range. You select the order of the method, the mode of operation, the error control and various optional inputs by a prior call to nag_ode_ivp_2nd_rkn_setup (d02lx).

For a description of the Runge–Kutta–Nystrom formula pairs see Dormand et al. (1986a) and Dormand et al. (1986b) and for a description of their practical implementation see Brankin et al. (1989).

Brankin R W, Dormand J R, Gladwell I, Prince P J and Seward W L (1989) Algorithm 670: A Runge–Kutta–Nystrom Code *ACM Trans. Math. Software* **15** 31–40

Dormand J R, El–Mikkawy M E A and Prince P J (1986a) Families of Runge–Kutta–Nystrom formulae *Mathematical Report TPMR 86-1* Teesside Polytechnic

Dormand J R, El–Mikkawy M E A and Prince P J (1986b) High order embedded Runge–Kutta–Nystrom formulae *Mathematical Report TPMR 86-2* Teesside Polytechnic

- 1: fcn – function handle or string containing name of m-file
- fcn must evaluate the functions f
_{i}${f}_{i}$ (that is the second derivatives y_{i}^{ ′ ′ }${y}_{i}^{\prime \prime}$) for given values of its arguments x$x$, y_{1},y_{2}, … ,y_{neq}${y}_{1},{y}_{2},\dots ,{y}_{{\mathbf{neq}}}$.[f] = fcn(neq, t, y)**Input Parameters**- 1: neq – int64int32nag_int scalar
- The number of differential equations.
- 2: t – double scalar
- x$x$, the value of the argument.
- 3: y(neq) – double array
- y
_{i}${y}_{\mathit{i}}$, for i = 1,2, … ,neq$\mathit{i}=1,2,\dots ,{\mathbf{neq}}$, the value of the argument.

**Output Parameters**- 1: f(neq) – double array

- 2: t – double scalar
- The initial value of the independent variable x$x$.
- 3: tend – double scalar
- 4: y(neq) – double array
- The initial values of the solution y
_{1},y_{2}, … ,y_{neq}${y}_{1},{y}_{2},\dots ,{y}_{{\mathbf{neq}}}$. - 5: yp(neq) – double array
- The initial values of the derivatives y
_{1}^{ ′ },y_{2}^{ ′ }, … ,y_{neq}^{ ′ }${y}_{1}^{\prime},{y}_{2}^{\prime},\dots ,{y}_{{\mathbf{neq}}}^{\prime}$. - 6: ydp(neq) – double array
- Must be unchanged from a previous call to nag_ode_ivp_2nd_rkn (d02la).
- 7: rwork(lrwork) – double array
- This
**must**be the same parameter rwork as supplied to nag_ode_ivp_2nd_rkn_setup (d02lx). It is used to pass information from nag_ode_ivp_2nd_rkn_setup (d02lx) to nag_ode_ivp_2nd_rkn (d02la), and from nag_ode_ivp_2nd_rkn (d02la) to both nag_ode_ivp_2nd_rkn_diag (d02ly) and nag_ode_ivp_2nd_rkn_interp (d02lz). Therefore the contents of this array**must not**be changed before the call to nag_ode_ivp_2nd_rkn (d02la) or calling either of the functions nag_ode_ivp_2nd_rkn_diag (d02ly) and nag_ode_ivp_2nd_rkn_interp (d02lz).

- 1: neq – int64int32nag_int scalar
*Default*: The dimension of the arrays y, yp, ydp. (An error is raised if these dimensions are not equal.)The number of second-order ordinary differential equations to be solved by nag_ode_ivp_2nd_rkn (d02la). It must contain the same value as the parameter neq used in a prior call to nag_ode_ivp_2nd_rkn_setup (d02lx).

- lrwork

- 1: t – double scalar
- 2: y(neq) – double array
- 3: yp(neq) – double array
- 4: ydp(neq) – double array
- 5: rwork(lrwork) – double array
- 6: ifail – int64int32nag_int scalar
- ifail = 0${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

Errors or warnings detected by the function:

- Illegal input detected, i.e., one of the following conditions:
- on any call, t = tend${\mathbf{t}}={\mathbf{tend}}$, or the value of neq or lrwork has been altered;
- on a continuation call, the direction of integration has been changed;
- nag_ode_ivp_2nd_rkn_setup (d02lx) had not been called previously, or the previous call to nag_ode_ivp_2nd_rkn_setup (d02lx) resulted in an error exit.

This error exit can be caused if elements of rwork have been overwritten.

- The maximum number of steps has been attempted. (See parameter maxstp in nag_ode_ivp_2nd_rkn_setup (d02lx).) If integration is to be continued then you need only reset ifail and call the function again and a further maxstp steps will be attempted.

- In order to satisfy the error requirements, the step size needed is too small for the machine precision being used.

- The code has detected two successive error exits at the current value of x$x$ and cannot proceed. Check all input variables.

- The code has detected inefficient use of the integration method. The step size has been reduced by a significant amount too often in order to hit the output points specified by tend. (Of the last 100$100$ or more successful steps more than 10%$10\%$ are steps with sizes that have had to be reduced by a factor of greater than a half.)

The accuracy of integration is determined by the parameters tol, thres and thresp in a prior call to nag_ode_ivp_2nd_rkn_setup (d02lx). Note that only the local error at each step is controlled by these parameters. The error estimates obtained are not strict bounds but are usually reliable over one step. Over a number of steps the overall error may accumulate in various ways, depending on the system. The code is designed so that a reduction in tol should lead to an approximately proportional reduction in the error. You are strongly recommended to call nag_ode_ivp_2nd_rkn (d02la) with more than one value for tol and to compare the results obtained to estimate their accuracy.

The accuracy obtained depends on the type of error test used. If the solution oscillates around zero a relative error test should be avoided, whereas if the solution is exponentially increasing an absolute error test should not be used. For a description of the error test see the specifications of the parameters tol, thres and thresp in function document nag_ode_ivp_2nd_rkn_setup (d02lx).

If nag_ode_ivp_2nd_rkn (d02la) fails with ifail = 3${\mathbf{ifail}}={\mathbf{3}}$ then the value of tol may be so small that a solution cannot be obtained, in which case the function should be called again with a larger value for tol. If the accuracy requested is really needed then you should consider whether there is a more fundamental difficulty. For example:

nag_ode_ivp_2nd_rkn (d02la) can be used for producing results at short intervals (for example, for tabulation), in two ways. By far the less efficient is to call nag_ode_ivp_2nd_rkn (d02la) successively over short intervals, t + (i − 1) × h$t+(i-1)\times h$ to t + i × h$t+i\times h$, although this is the only way if the higher order method has been selected and precisely **not** what it is intended for. A more efficient way, **only** for use when the lower order method has been selected, is to use nag_ode_ivp_2nd_rkn (d02la) in one-step mode. The output values of parameters y, yp, ydp, t and rwork are set correctly for a call to nag_ode_ivp_2nd_rkn_interp (d02lz) to compute the solution and derivative at the required points.

(a) | in the region of a singularity the solution components will usually be of a large magnitude. nag_ode_ivp_2nd_rkn (d02la) could be used in one-step mode to monitor the size of the solution with the aim of trapping the solution before the singularity. In any case numerical integration cannot be continued through a singularity, and analytical treatment may be necessary; |

(b) | if the solution contains fast oscillatory components, the function will require a very small step size to preserve stability. This will usually be exhibited by excessive computing time and sometimes an error exit with ifail = 3${\mathbf{ifail}}={\mathbf{3}}$. The Runge–Kutta–Nystrom methods are not efficient in such cases and you should consider reposing your problem as a system of first-order ordinary differential equations and then using a function from sub-chapter D02M–N with the Blend formulae (see nag_ode_ivp_stiff_dassl (d02mv)). |

This example solves the following system (the two body problem)

over the range [0,20]$[0,20]$ with initial conditions y_{1} = 1.0 − ε${y}_{1}=1.0-\epsilon $, y_{2} = 0.0${y}_{2}=0.0$, y_{1}^{ ′ } = 0.0${y}_{1}^{\prime}=0.0$ and y_{2}^{ ′ } = sqrt(
((1 + ε)/(1 − ε))
)${y}_{2}^{\prime}=\sqrt{\left(\frac{1+\epsilon}{1-\epsilon}\right)}$ where ε$\epsilon $, the eccentricity, is 0.5$0.5$. The system is solved using the lower order method with relative local error tolerances 1.0e−4$\text{1.0e\u22124}$ and 1.0e−5$\text{1.0e\u22125}$ and default threshold tolerances. nag_ode_ivp_2nd_rkn (d02la) is used in one-step mode (onestp = true${\mathbf{onestp}}=\mathbf{true}$) and nag_ode_ivp_2nd_rkn_interp (d02lz) provides solution values at intervals of 2.0$2.0$.

$$\begin{array}{ccc}\hfill {y}_{1}^{\prime \prime}& =& -{y}_{1}/{({y}_{1}^{2}+{y}_{2}^{2})}^{3/2}\hfill \\ \hfill {y}_{2}^{\prime \prime}& =& -{y}_{2}/{({y}_{1}^{2}+{y}_{2}^{2})}^{3/2}\end{array}$$ |

Open in the MATLAB editor: nag_ode_ivp_2nd_rkn_example

function nag_ode_ivp_2nd_rkn_example% Initialize variables and arrays. ecc = 0.5; tend = 20; tstart = 0; tinc = 2; neq = 2; lrwork = 16+20*neq; nwant = neq; thres = zeros(neq, 1); thresp = zeros(neq, 1); fprintf('nag_ode_ivp_2nd_rkn example program results \n\n'); for itol = 4:5 tol = 10^(-itol); thres(1) = 0.0; thresp(1) = 0.0; h = 0.0; maxstp = 0; start = true; high = false; onestp = true; rwork = zeros(56,1); t = 0; y = [1-ecc; 0]; yp = [0; sqrt((1+ecc)/(1-ecc))]; ydp = zeros(neq, 1); % nag_ode_ivp_2nd_rkn_setup is a setup routine to be called prior to nag_ode_ivp_2nd_rkn. [start, rwork, ifail] = ... nag_ode_ivp_2nd_rkn_setup(h, tol, thres, thresp, int64(maxstp),... start, onestp, high, rwork, 'neq', int64(neq)); if ifail ~= 0 % Illegal input. Print message and exit. error('Warning: nag_ode_ivp_2nd_rkn_setup returned with ifail = %1d ',ifail); end fprintf('Calculation with tol = %1.1e \n\n',tol); fprintf(' T Y(1) Y(2) '); t = tstart; tnext = t + tinc; fprintf('\n %4.1f ',t); fprintf(' %10.5f',y); fprintf('\n'); if itol == 4 ncall = 1; tkeep = tnext; ykeep = y'; end while (t < tend) [t, y, yp, ydp, rwork, ifail] = nag_ode_ivp_2nd_rkn(@fcn, t, tend, y,... yp, ydp, rwork, 'neq', int64(neq)); if ifail ~= 0 % Illegal input or integration problems. Print message and exit. error('Warning: nag_ode_ivp_2nd_rkn returned with ifail = %1d ',ifail); end while (tnext <= t) % nag_ode_ivp_2nd_rkn_interp interpolates components of solution provided by nag_ode_ivp_2nd_rkn. [ywant, ypwant, ifail] = ... nag_ode_ivp_2nd_rkn_interp(t, y, yp, int64(neq), tnext,... rwork, 'neq', int64(neq)); if ifail ~= 0 % Illegal input or interpolation problems. Print message and exit. error('Warning: nag_ode_ivp_2nd_rkn_interp returned with ifail = %1d ',ifail); end ncall = ncall+1; fprintf(' %4.1f ',tnext); fprintf(' %10.5f',ywant); fprintf('\n'); if itol == 4 tkeep(ncall) = tnext; ykeep(ncall,:) = ywant'; end tnext = tnext + tinc; end end % nag_ode_ivp_2nd_rkn_diag is a diagnostic routine for use after calling nag_ode_ivp_2nd_rkn. [hnext, hused, hstart, nsucc, nfail, natt, thres, thresp, ifail] =... nag_ode_ivp_2nd_rkn_diag(int64(neq), rwork); if ifail ~= 0 % Illegal input. Print message and exit. error('Warning: nag_ode_ivp_2nd_rkn_diag returned with ifail = %1d ',ifail); end fprintf('\nNumber of successful steps = %1.0f\n',nsucc); fprintf('Number of failed steps = %1.0f\n',nfail); fprintf('\n\n'); end % Now repeat the calculation with a smaller t increment, and store % the trajectory in order to plot it. tol = 10^(-4); tinc = 0.1; y = [1-ecc; 0]; yp = [0; sqrt((1+ecc)/(1-ecc))]; % nag_ode_ivp_2nd_rkn_setup is a setup routine to be called prior to nag_ode_ivp_2nd_rkn. [start, rwork, ifail] = ... nag_ode_ivp_2nd_rkn_setup(h, tol, thres, thresp, int64(maxstp),... start, onestp, high, rwork, 'neq', int64(neq)); if ifail ~= 0 % Illegal input. Print message and exit. error('Warning: nag_ode_ivp_2nd_rkn_setup returned with ifail = %1d ',ifail); end t = tstart; tnext = t + tinc; nkeep = 1; ykeep = y'; while (t < tend) [t, y, yp, ydp, rwork, ifail] = nag_ode_ivp_2nd_rkn(@fcn, t, tend, y,... yp, ydp, rwork, 'neq', int64(neq)); if ifail ~= 0 % Illegal input or integration problems. Print message and exit. error('Warning: nag_ode_ivp_2nd_rkn returned with ifail = %1d ',ifail); end while (tnext <= t) % nag_ode_ivp_2nd_rkn_interp interpolates components of solution provided by nag_ode_ivp_2nd_rkn. [ywant, ypwant, ifail] = nag_ode_ivp_2nd_rkn_interp(t, y, yp, int64(neq), tnext,... rwork, 'neq', int64(neq)); if ifail ~= 0 % Illegal input or interpolation problems. Print message and exit. error('Warning: nag_ode_ivp_2nd_rkn_interp returned with ifail = %1d ',ifail); end nkeep = nkeep+1; ykeep(nkeep,:) = ywant'; tnext = tnext + tinc; end end % nag_ode_ivp_2nd_rkn_diag is a diagnostic routine for use after calling nag_ode_ivp_2nd_rkn. [hnext, hused, hstart, nsucc, nfail, natt, thres, thresp, ifail] =... nag_ode_ivp_2nd_rkn_diag(int64(neq), rwork); if ifail ~= 0 % Illegal input. Print message and exit. error('Warning: nag_ode_ivp_2nd_rkn_diag returned with ifail = %1d ',ifail); end % Find the index of the point halfway through the first orbit (i.e. the % first point at which the y coord is negative). This is used to calculate % the end points of the orbits in display_plot. nhalf = 1; while ykeep(nhalf,2) >= 0 nhalf = nhalf+1; end % Plot results. fig = figure('Number', 'off'); display_plot(ykeep, nhalf);function ydp = fcn(neq, t, y)% Evaluate second derivatives. r = sqrt(y(1)^2+y(2)^2)^3; f = zeros(2,1); ydp(1) = -y(1)/r; ydp(2) = -y(2)/r;function display_plot(ykeep, nhalf)% Formatting for title and axis labels. titleFmt = {'FontName', 'Helvetica', 'FontWeight', 'Bold', 'FontSize', 14}; labFmt = {'FontName', 'Helvetica', 'FontWeight', 'Bold', 'FontSize', 13}; set(gca, 'FontSize', 13); % for legend, axis tick labels, etc. % Find the end points of the three orbits. norb1 = 2*(nhalf-1); norb2 = 4*(nhalf-1); norb3 = 6*(nhalf-1); % Plot the orbits, including displacements. plot(ykeep(1:norb1,1), ykeep(1:norb1,2), '-+', ... ykeep(norb1+1:norb2,1), ykeep(norb1+1:norb2,2)+0.1, '--x', ... ykeep(norb2+1:norb3,1), ykeep(norb2+1:norb3,2)+0.2, ':*'); % Add title. title({'Second-order ODE Solution using Runge-Kutta-Nystrom.', ... 'The Two-body Problem (using shifts to distinguish orbits)'}, ... titleFmt{:}); % Label the axes. xlabel('x', labFmt{:}); ylabel('y', labFmt{:}); % Add a legend. legend('1st orbit', '2nd orbit+(0,0.1)', '3rd orbit+(0,0.2)', ... 'Location', 'Best');

nag_ode_ivp_2nd_rkn example program results Calculation with tol = 1.0e-04 T Y(1) Y(2) 0.0 0.50000 0.00000 2.0 -1.20573 0.61357 4.0 -1.33476 -0.47685 6.0 0.35748 -0.44558 8.0 -1.03762 0.73022 10.0 -1.42617 -0.32658 12.0 0.05515 -0.72032 14.0 -0.82880 0.81788 16.0 -1.48103 -0.16788 18.0 -0.26719 -0.84223 20.0 -0.57803 0.86339 Number of successful steps = 108 Number of failed steps = 16 Calculation with tol = 1.0e-05 T Y(1) Y(2) 0.0 0.50000 0.00000 2.0 -1.20573 0.61357 4.0 -1.33476 -0.47685 6.0 0.35748 -0.44558 8.0 -1.03762 0.73022 10.0 -1.42617 -0.32658 12.0 0.05516 -0.72031 14.0 -0.82880 0.81787 16.0 -1.48103 -0.16789 18.0 -0.26718 -0.84223 20.0 -0.57804 0.86338 Number of successful steps = 169 Number of failed steps = 15

Open in the MATLAB editor: d02la_example

function d02la_example% Initialize variables and arrays. ecc = 0.5; tend = 20; tstart = 0; tinc = 2; neq = 2; lrwork = 16+20*neq; nwant = neq; thres = zeros(neq, 1); thresp = zeros(neq, 1); fprintf('d02la example program results \n\n'); for itol = 4:5 tol = 10^(-itol); thres(1) = 0.0; thresp(1) = 0.0; h = 0.0; maxstp = 0; start = true; high = false; onestp = true; rwork = zeros(56,1); t = 0; y = [1-ecc; 0]; yp = [0; sqrt((1+ecc)/(1-ecc))]; ydp = zeros(neq, 1); % d02lx is a setup routine to be called prior to d02la. [start, rwork, ifail] = d02lx(h, tol, thres, thresp, int64(maxstp),... start, onestp, high, rwork, 'neq', int64(neq)); if ifail ~= 0 % Illegal input. Print message and exit. error('Warning: d02lx returned with ifail = %1d ',ifail); end fprintf('Calculation with tol = %1.1e \n\n',tol); fprintf(' T Y(1) Y(2) '); t = tstart; tnext = t + tinc; fprintf('\n %4.1f ',t); fprintf(' %10.5f',y); fprintf('\n'); if itol == 4 ncall = 1; tkeep = tnext; ykeep = y'; end while (t < tend) [t, y, yp, ydp, rwork, ifail] = d02la(@fcn, t, tend, y,... yp, ydp, rwork, 'neq', int64(neq)); if ifail ~= 0 % Illegal input or integration problems. Print message and exit. error('Warning: d02la returned with ifail = %1d ',ifail); end while (tnext <= t) % d02lz interpolates components of solution provided by d02la. [ywant, ypwant, ifail] = d02lz(t, y, yp, int64(neq), tnext,... rwork, 'neq', int64(neq)); if ifail ~= 0 % Illegal input or interpolation problems. Print message and exit. error('Warning: d02lz returned with ifail = %1d ',ifail); end ncall = ncall+1; fprintf(' %4.1f ',tnext); fprintf(' %10.5f',ywant); fprintf('\n'); if itol == 4 tkeep(ncall) = tnext; ykeep(ncall,:) = ywant'; end tnext = tnext + tinc; end end % d02ly is a diagnostic routine for use after calling d02la. [hnext, hused, hstart, nsucc, nfail, natt, thres, thresp, ifail] =... d02ly(int64(neq), rwork); if ifail ~= 0 % Illegal input. Print message and exit. error('Warning: d02ly returned with ifail = %1d ',ifail); end fprintf('\nNumber of successful steps = %1.0f\n',nsucc); fprintf('Number of failed steps = %1.0f\n',nfail); fprintf('\n\n'); end % Now repeat the calculation with a smaller t increment, and store % the trajectory in order to plot it. tol = 10^(-4); tinc = 0.1; y = [1-ecc; 0]; yp = [0; sqrt((1+ecc)/(1-ecc))]; % d02lx is a setup routine to be called prior to d02la. [start, rwork, ifail] = d02lx(h, tol, thres, thresp, int64(maxstp),... start, onestp, high, rwork, 'neq', int64(neq)); if ifail ~= 0 % Illegal input. Print message and exit. error('Warning: d02lx returned with ifail = %1d ',ifail); end t = tstart; tnext = t + tinc; nkeep = 1; ykeep = y'; while (t < tend) [t, y, yp, ydp, rwork, ifail] = d02la(@fcn, t, tend, y,... yp, ydp, rwork, 'neq', int64(neq)); if ifail ~= 0 % Illegal input or integration problems. Print message and exit. error('Warning: d02la returned with ifail = %1d ',ifail); end while (tnext <= t) % d02lz interpolates components of solution provided by d02la. [ywant, ypwant, ifail] = d02lz(t, y, yp, int64(neq), tnext,... rwork, 'neq', int64(neq)); if ifail ~= 0 % Illegal input or interpolation problems. Print message and exit. error('Warning: d02lz returned with ifail = %1d ',ifail); end nkeep = nkeep+1; ykeep(nkeep,:) = ywant'; tnext = tnext + tinc; end end % d02ly is a diagnostic routine for use after calling d02la. [hnext, hused, hstart, nsucc, nfail, natt, thres, thresp, ifail] =... d02ly(int64(neq), rwork); if ifail ~= 0 % Illegal input. Print message and exit. error('Warning: d02ly returned with ifail = %1d ',ifail); end % Find the index of the point halfway through the first orbit (i.e. the % first point at which the y coord is negative). This is used to calculate % the end points of the orbits in display_plot. nhalf = 1; while ykeep(nhalf,2) >= 0 nhalf = nhalf+1; end % Plot results. fig = figure('Number', 'off'); display_plot(ykeep, nhalf);function ydp = fcn(neq, t, y)% Evaluate second derivatives. r = sqrt(y(1)^2+y(2)^2)^3; f = zeros(2,1); ydp(1) = -y(1)/r; ydp(2) = -y(2)/r;function display_plot(ykeep, nhalf)% Formatting for title and axis labels. titleFmt = {'FontName', 'Helvetica', 'FontWeight', 'Bold', 'FontSize', 14}; labFmt = {'FontName', 'Helvetica', 'FontWeight', 'Bold', 'FontSize', 13}; set(gca, 'FontSize', 13); % for legend, axis tick labels, etc. % Find the end points of the three orbits. norb1 = 2*(nhalf-1); norb2 = 4*(nhalf-1); norb3 = 6*(nhalf-1); % Plot the orbits, including displacements. plot(ykeep(1:norb1,1), ykeep(1:norb1,2), '-+', ... ykeep(norb1+1:norb2,1), ykeep(norb1+1:norb2,2)+0.1, '--x', ... ykeep(norb2+1:norb3,1), ykeep(norb2+1:norb3,2)+0.2, ':*'); % Add title. title({'Second-order ODE Solution using Runge-Kutta-Nystrom.', ... 'The Two-body Problem (using shifts to distinguish orbits)'}, ... titleFmt{:}); % Label the axes. xlabel('x', labFmt{:}); ylabel('y', labFmt{:}); % Add a legend. legend('1st orbit', '2nd orbit+(0,0.1)', '3rd orbit+(0,0.2)', ... 'Location', 'Best');

d02la example program results Calculation with tol = 1.0e-04 T Y(1) Y(2) 0.0 0.50000 0.00000 2.0 -1.20573 0.61357 4.0 -1.33476 -0.47685 6.0 0.35748 -0.44558 8.0 -1.03762 0.73022 10.0 -1.42617 -0.32658 12.0 0.05515 -0.72032 14.0 -0.82880 0.81788 16.0 -1.48103 -0.16788 18.0 -0.26719 -0.84223 20.0 -0.57803 0.86339 Number of successful steps = 108 Number of failed steps = 16 Calculation with tol = 1.0e-05 T Y(1) Y(2) 0.0 0.50000 0.00000 2.0 -1.20573 0.61357 4.0 -1.33476 -0.47685 6.0 0.35748 -0.44558 8.0 -1.03762 0.73022 10.0 -1.42617 -0.32658 12.0 0.05516 -0.72031 14.0 -0.82880 0.81787 16.0 -1.48103 -0.16789 18.0 -0.26718 -0.84223 20.0 -0.57804 0.86338 Number of successful steps = 169 Number of failed steps = 15

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