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< a href = "#define-members" > Macros< / a > |
< a href = "#func-members" > Functions< / a > < / div >
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< div class = "title" > ode_semi_implicit_euler.c File Reference< / div > < / div >
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< p > Solve a multivariable first order < a href = "https://en.wikipedia.org/wiki/Ordinary_differential_equation" > ordinary differential equation (ODEs)< / a > using < a href = "https://en.wikipedia.org/wiki/Semi-implicit_Euler_method" > semi implicit Euler method< / a >
< a href = "#details" > More...< / a > < / p >
< div class = "textblock" > < code > #include < math.h> < / code > < br / >
< code > #include < stdio.h> < / code > < br / >
< code > #include < stdlib.h> < / code > < br / >
< code > #include < time.h> < / code > < br / >
< / div > < div class = "textblock" > < div class = "dynheader" >
Include dependency graph for ode_semi_implicit_euler.c:< / div >
< div class = "dyncontent" >
< div class = "center" > < iframe scrolling = "no" frameborder = "0" src = "../../d9/d85/ode__semi__implicit__euler_8c__incl.svg" width = "331" height = "127" > < p > < b > This browser is not able to show SVG: try Firefox, Chrome, Safari, or Opera instead.< / b > < / p > < / iframe >
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< / div > < table class = "memberdecls" >
< tr class = "heading" > < td colspan = "2" > < h2 class = "groupheader" > < a name = "define-members" > < / a >
Macros< / h2 > < / td > < / tr >
< tr class = "memitem:a9ceb646336224ee890a269d0b4600d09" > < td class = "memItemLeft" align = "right" valign = "top" > #define  < / td > < td class = "memItemRight" valign = "bottom" > < a class = "el" href = "../../d4/d99/ode__semi__implicit__euler_8c.html#a9ceb646336224ee890a269d0b4600d09" > order< / a >       2< / td > < / tr >
< tr class = "separator:a9ceb646336224ee890a269d0b4600d09" > < td class = "memSeparator" colspan = "2" >   < / td > < / tr >
< / table > < table class = "memberdecls" >
< tr class = "heading" > < td colspan = "2" > < h2 class = "groupheader" > < a name = "func-members" > < / a >
Functions< / h2 > < / td > < / tr >
< tr class = "memitem:a97075291390a68c262ed66e157a57eb4" > < td class = "memItemLeft" align = "right" valign = "top" > void  < / td > < td class = "memItemRight" valign = "bottom" > < a class = "el" href = "../../d4/d99/ode__semi__implicit__euler_8c.html#a97075291390a68c262ed66e157a57eb4" > problem< / a > (const double *x, double *y, double *dy)< / td > < / tr >
< tr class = "memdesc:a97075291390a68c262ed66e157a57eb4" > < td class = "mdescLeft" >   < / td > < td class = "mdescRight" > Problem statement for a system with first-order differential equations. Updates the system differential variables. < a href = "../../d4/d99/ode__semi__implicit__euler_8c.html#a97075291390a68c262ed66e157a57eb4" > More...< / a > < br / > < / td > < / tr >
< tr class = "separator:a97075291390a68c262ed66e157a57eb4" > < td class = "memSeparator" colspan = "2" >   < / td > < / tr >
< tr class = "memitem:a8caee977b26888d34040b122e0e28e3a" > < td class = "memItemLeft" align = "right" valign = "top" > void  < / td > < td class = "memItemRight" valign = "bottom" > < a class = "el" href = "../../d4/d99/ode__semi__implicit__euler_8c.html#a8caee977b26888d34040b122e0e28e3a" > exact_solution< / a > (const double *x, double *y)< / td > < / tr >
< tr class = "memdesc:a8caee977b26888d34040b122e0e28e3a" > < td class = "mdescLeft" >   < / td > < td class = "mdescRight" > Exact solution of the problem. Used for solution comparison. < a href = "../../d4/d99/ode__semi__implicit__euler_8c.html#a8caee977b26888d34040b122e0e28e3a" > More...< / a > < br / > < / td > < / tr >
< tr class = "separator:a8caee977b26888d34040b122e0e28e3a" > < td class = "memSeparator" colspan = "2" >   < / td > < / tr >
< tr class = "memitem:a720b7e995d2bbc615f94a2c7dbcf84eb" > < td class = "memItemLeft" align = "right" valign = "top" > void  < / td > < td class = "memItemRight" valign = "bottom" > < a class = "el" href = "../../d4/d99/ode__semi__implicit__euler_8c.html#a720b7e995d2bbc615f94a2c7dbcf84eb" > semi_implicit_euler_step< / a > (double dx, double *x, double *y, double *dy)< / td > < / tr >
< tr class = "memdesc:a720b7e995d2bbc615f94a2c7dbcf84eb" > < td class = "mdescLeft" >   < / td > < td class = "mdescRight" > Compute next step approximation using the semi-implicit-Euler method. < a href = "../../d4/d99/ode__semi__implicit__euler_8c.html#a720b7e995d2bbc615f94a2c7dbcf84eb" > More...< / a > < br / > < / td > < / tr >
< tr class = "separator:a720b7e995d2bbc615f94a2c7dbcf84eb" > < td class = "memSeparator" colspan = "2" >   < / td > < / tr >
< tr class = "memitem:ad80059877222f885b549f2d0a3dc6b55" > < td class = "memItemLeft" align = "right" valign = "top" > double  < / td > < td class = "memItemRight" valign = "bottom" > < a class = "el" href = "../../d4/d99/ode__semi__implicit__euler_8c.html#ad80059877222f885b549f2d0a3dc6b55" > semi_implicit_euler< / a > (double dx, double x0, double x_max, double *y, char save_to_file)< / td > < / tr >
< tr class = "memdesc:ad80059877222f885b549f2d0a3dc6b55" > < td class = "mdescLeft" >   < / td > < td class = "mdescRight" > Compute approximation using the semi-implicit-Euler method in the given limits. < a href = "../../d4/d99/ode__semi__implicit__euler_8c.html#ad80059877222f885b549f2d0a3dc6b55" > More...< / a > < br / > < / td > < / tr >
< tr class = "separator:ad80059877222f885b549f2d0a3dc6b55" > < td class = "memSeparator" colspan = "2" >   < / td > < / tr >
< tr class = "memitem:a0ddf1224851353fc92bfbff6f499fa97" > < td class = "memItemLeft" align = "right" valign = "top" > int  < / td > < td class = "memItemRight" valign = "bottom" > < a class = "el" href = "../../d4/d99/ode__semi__implicit__euler_8c.html#a0ddf1224851353fc92bfbff6f499fa97" > main< / a > (int argc, char *argv[])< / td > < / tr >
< tr class = "separator:a0ddf1224851353fc92bfbff6f499fa97" > < td class = "memSeparator" colspan = "2" >   < / td > < / tr >
< / table >
< a name = "details" id = "details" > < / a > < h2 class = "groupheader" > Detailed Description< / h2 >
< div class = "textblock" > < p > Solve a multivariable first order < a href = "https://en.wikipedia.org/wiki/Ordinary_differential_equation" > ordinary differential equation (ODEs)< / a > using < a href = "https://en.wikipedia.org/wiki/Semi-implicit_Euler_method" > semi implicit Euler method< / a > < / p >
< dl class = "section author" > < dt > Authors< / dt > < dd > < a href = "https://github.com/kvedala" > Krishna Vedala< / a > < / dd > < / dl >
< p > The ODE being solved is: < / p > < p class = "formulaDsp" >
\begin{eqnarray*} \dot{u} & =& v\\ \dot{v} & =& -\omega^2 u\\ \omega & =& 1\\ [x_0, u_0, v_0] & =& [0,1,0]\qquad\ldots\text{(initial values)} \end{eqnarray*}
< / p >
< p > The exact solution for the above problem is: < / p > < p class = "formulaDsp" >
\begin{eqnarray*} u(x) & =& \cos(x)\\ v(x) & =& -\sin(x)\\ \end{eqnarray*}
< / p >
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< p > The computation results are stored to a text file < code > semi_implicit_euler.csv< / code > and the exact soltuion results in < code > exact.csv< / code > for comparison. < img src = "https://raw.githubusercontent.com/TheAlgorithms/C/docs/images/numerical_methods/ode_semi_implicit_euler.svg" alt = "Implementation solution" style = "pointer-events: none;" class = "inline" / > < / p >
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< p > To implement < a href = "https://en.wikipedia.org/wiki/Van_der_Pol_oscillator" > Van der Pol oscillator< / a > , change the < a class = "el" href = "../../d4/d99/ode__semi__implicit__euler_8c.html#a97075291390a68c262ed66e157a57eb4" title = "Problem statement for a system with first-order differential equations. Updates the system differenti..." > problem< / a > function to: < / p > < div class = "fragment" > < div class = "line" > < span class = "keyword" > const< / span > < span class = "keywordtype" > double< / span > mu = 2.0;< / div >
< div class = "line" > dy[0] = y[1];< / div >
< div class = "line" > dy[1] = mu * (1.f - y[0] * y[0]) * y[1] - y[0];< / div >
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< / div > <!-- fragment --> < p > < a href = "https://en.wikipedia.org/wiki/Van_der_Pol_oscillator" > < img src = "https://raw.githubusercontent.com/TheAlgorithms/C/docs/images/numerical_methods/van_der_pol_implicit_euler.svg" alt = "Van der Pol Oscillator solution" style = "pointer-events: none;" class = "inline" / > < / a > < / p >
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< dl class = "section see" > < dt > See also< / dt > < dd > < a class = "el" href = "../../d4/d07/ode__forward__euler_8c.html" title = "Solve a multivariable first order ordinary differential equation (ODEs) using forward Euler method" > ode_forward_euler.c< / a > , < a class = "el" href = "../../d1/dc2/ode__midpoint__euler_8c.html" title = "Solve a multivariable first order ordinary differential equation (ODEs) using midpoint Euler method" > ode_midpoint_euler.c< / a > < / dd > < / dl >
< / div > < h2 class = "groupheader" > Macro Definition Documentation< / h2 >
< a id = "a9ceb646336224ee890a269d0b4600d09" > < / a >
< h2 class = "memtitle" > < span class = "permalink" > < a href = "#a9ceb646336224ee890a269d0b4600d09" > ◆ < / a > < / span > order< / h2 >
< div class = "memitem" >
< div class = "memproto" >
< table class = "memname" >
< tr >
< td class = "memname" > #define order      2< / td >
< / tr >
< / table >
< / div > < div class = "memdoc" >
< p > number of dependent variables in < a class = "el" href = "../../d4/d99/ode__semi__implicit__euler_8c.html#a97075291390a68c262ed66e157a57eb4" title = "Problem statement for a system with first-order differential equations. Updates the system differenti..." > problem< / a > < / p >
< / div >
< / div >
< h2 class = "groupheader" > Function Documentation< / h2 >
< a id = "a8caee977b26888d34040b122e0e28e3a" > < / a >
< h2 class = "memtitle" > < span class = "permalink" > < a href = "#a8caee977b26888d34040b122e0e28e3a" > ◆ < / a > < / span > exact_solution()< / h2 >
< div class = "memitem" >
< div class = "memproto" >
< table class = "memname" >
< tr >
< td class = "memname" > void exact_solution < / td >
< td > (< / td >
< td class = "paramtype" > const double *  < / td >
< td class = "paramname" > < em > x< / em > , < / td >
< / tr >
< tr >
< td class = "paramkey" > < / td >
< td > < / td >
< td class = "paramtype" > double *  < / td >
< td class = "paramname" > < em > y< / em >   < / td >
< / tr >
< tr >
< td > < / td >
< td > )< / td >
< td > < / td > < td > < / td >
< / tr >
< / table >
< / div > < div class = "memdoc" >
< p > Exact solution of the problem. Used for solution comparison. < / p >
< dl class = "params" > < dt > Parameters< / dt > < dd >
< table class = "params" >
< tr > < td class = "paramdir" > [in]< / td > < td class = "paramname" > x< / td > < td > independent variable < / td > < / tr >
< tr > < td class = "paramdir" > [in,out]< / td > < td class = "paramname" > y< / td > < td > dependent variable < / td > < / tr >
< / table >
< / dd >
< / dl >
< div class = "fragment" > < div class = "line" > < a name = "l00072" > < / a > < span class = "lineno" > 72< / span >   {< / div >
< div class = "line" > < a name = "l00073" > < / a > < span class = "lineno" > 73< / span >   y[0] = cos(x[0]);< / div >
< div class = "line" > < a name = "l00074" > < / a > < span class = "lineno" > 74< / span >   y[1] = -sin(x[0]);< / div >
< div class = "line" > < a name = "l00075" > < / a > < span class = "lineno" > 75< / span >   }< / div >
< / div > <!-- fragment -->
< / div >
< / div >
< a id = "a0ddf1224851353fc92bfbff6f499fa97" > < / a >
< h2 class = "memtitle" > < span class = "permalink" > < a href = "#a0ddf1224851353fc92bfbff6f499fa97" > ◆ < / a > < / span > main()< / h2 >
< div class = "memitem" >
< div class = "memproto" >
< table class = "memname" >
< tr >
< td class = "memname" > int main < / td >
< td > (< / td >
< td class = "paramtype" > int  < / td >
< td class = "paramname" > < em > argc< / em > , < / td >
< / tr >
< tr >
< td class = "paramkey" > < / td >
< td > < / td >
< td class = "paramtype" > char *  < / td >
< td class = "paramname" > < em > argv< / em > []  < / td >
< / tr >
< tr >
< td > < / td >
< td > )< / td >
< td > < / td > < td > < / td >
< / tr >
< / table >
< / div > < div class = "memdoc" >
< p > Main Function < / p >
< div class = "fragment" > < div class = "line" > < a name = "l00148" > < / a > < span class = "lineno" > 148< / span >   {< / div >
< div class = "line" > < a name = "l00149" > < / a > < span class = "lineno" > 149< / span >   < span class = "keywordtype" > double< / span > X0 = 0.f; < span class = "comment" > /* initial value of x0 */< / span > < / div >
< div class = "line" > < a name = "l00150" > < / a > < span class = "lineno" > 150< / span >   < span class = "keywordtype" > double< / span > X_MAX = 10.F; < span class = "comment" > /* upper limit of integration */< / span > < / div >
< div class = "line" > < a name = "l00151" > < / a > < span class = "lineno" > 151< / span >   < span class = "keywordtype" > double< / span > Y0[] = {1.f, 0.f}; < span class = "comment" > /* initial value Y = y(x = x_0) */< / span > < / div >
< div class = "line" > < a name = "l00152" > < / a > < span class = "lineno" > 152< / span >   < span class = "keywordtype" > double< / span > step_size;< / div >
< div class = "line" > < a name = "l00153" > < / a > < span class = "lineno" > 153< / span >   < / div >
< div class = "line" > < a name = "l00154" > < / a > < span class = "lineno" > 154< / span >   < span class = "keywordflow" > if< / span > (argc == 1)< / div >
< div class = "line" > < a name = "l00155" > < / a > < span class = "lineno" > 155< / span >   {< / div >
< div class = "line" > < a name = "l00156" > < / a > < span class = "lineno" > 156< / span >   printf(< span class = "stringliteral" > " \nEnter the step size: " < / span > );< / div >
< div class = "line" > < a name = "l00157" > < / a > < span class = "lineno" > 157< / span >   scanf(< span class = "stringliteral" > " %lg" < / span > , & step_size);< / div >
< div class = "line" > < a name = "l00158" > < / a > < span class = "lineno" > 158< / span >   }< / div >
< div class = "line" > < a name = "l00159" > < / a > < span class = "lineno" > 159< / span >   < span class = "keywordflow" > else< / span > < / div >
< div class = "line" > < a name = "l00160" > < / a > < span class = "lineno" > 160< / span >   < span class = "comment" > // use commandline argument as independent variable step size< / span > < / div >
< div class = "line" > < a name = "l00161" > < / a > < span class = "lineno" > 161< / span >   step_size = atof(argv[1]);< / div >
< div class = "line" > < a name = "l00162" > < / a > < span class = "lineno" > 162< / span >   < / div >
< div class = "line" > < a name = "l00163" > < / a > < span class = "lineno" > 163< / span >   < span class = "comment" > // get approximate solution< / span > < / div >
< div class = "line" > < a name = "l00164" > < / a > < span class = "lineno" > 164< / span >   < span class = "keywordtype" > double< / span > total_time = < a class = "code" href = "../../d4/d99/ode__semi__implicit__euler_8c.html#ad80059877222f885b549f2d0a3dc6b55" > semi_implicit_euler< / a > (step_size, X0, X_MAX, Y0, 1);< / div >
< div class = "line" > < a name = "l00165" > < / a > < span class = "lineno" > 165< / span >   printf(< span class = "stringliteral" > " \tTime = %.6g ms\n" < / span > , total_time);< / div >
< div class = "line" > < a name = "l00166" > < / a > < span class = "lineno" > 166< / span >   < / div >
< div class = "line" > < a name = "l00167" > < / a > < span class = "lineno" > 167< / span >   < span class = "comment" > /* compute exact solution for comparion */< / span > < / div >
< div class = "line" > < a name = "l00168" > < / a > < span class = "lineno" > 168< / span >   FILE *fp = fopen(< span class = "stringliteral" > " exact.csv" < / span > , < span class = "stringliteral" > " w+" < / span > );< / div >
< div class = "line" > < a name = "l00169" > < / a > < span class = "lineno" > 169< / span >   < span class = "keywordflow" > if< / span > (fp == NULL)< / div >
< div class = "line" > < a name = "l00170" > < / a > < span class = "lineno" > 170< / span >   {< / div >
< div class = "line" > < a name = "l00171" > < / a > < span class = "lineno" > 171< / span >   perror(< span class = "stringliteral" > " Error! " < / span > );< / div >
< div class = "line" > < a name = "l00172" > < / a > < span class = "lineno" > 172< / span >   < span class = "keywordflow" > return< / span > -1;< / div >
< div class = "line" > < a name = "l00173" > < / a > < span class = "lineno" > 173< / span >   }< / div >
< div class = "line" > < a name = "l00174" > < / a > < span class = "lineno" > 174< / span >   < span class = "keywordtype" > double< / span > x = X0;< / div >
< div class = "line" > < a name = "l00175" > < / a > < span class = "lineno" > 175< / span >   < span class = "keywordtype" > double< / span > *y = & (Y0[0]);< / div >
< div class = "line" > < a name = "l00176" > < / a > < span class = "lineno" > 176< / span >   printf(< span class = "stringliteral" > " Finding exact solution\n" < / span > );< / div >
< div class = "line" > < a name = "l00177" > < / a > < span class = "lineno" > 177< / span >   clock_t t1 = clock();< / div >
< div class = "line" > < a name = "l00178" > < / a > < span class = "lineno" > 178< / span >   < / div >
< div class = "line" > < a name = "l00179" > < / a > < span class = "lineno" > 179< / span >   < span class = "keywordflow" > do< / span > < / div >
< div class = "line" > < a name = "l00180" > < / a > < span class = "lineno" > 180< / span >   {< / div >
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< div class = "line" > < a name = "l00181" > < / a > < span class = "lineno" > 181< / span >   fprintf(fp, < span class = "stringliteral" > " %.4g,%.4g,%.4g\n" < / span > , x, y[0], y[1]); < span class = "comment" > // write to file< / span > < / div >
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< div class = "line" > < a name = "l00182" > < / a > < span class = "lineno" > 182< / span >   < a class = "code" href = "../../d4/d99/ode__semi__implicit__euler_8c.html#a8caee977b26888d34040b122e0e28e3a" > exact_solution< / a > (& x, y);< / div >
< div class = "line" > < a name = "l00183" > < / a > < span class = "lineno" > 183< / span >   x += step_size;< / div >
< div class = "line" > < a name = "l00184" > < / a > < span class = "lineno" > 184< / span >   } < span class = "keywordflow" > while< / span > (x < = X_MAX);< / div >
< div class = "line" > < a name = "l00185" > < / a > < span class = "lineno" > 185< / span >   < / div >
< div class = "line" > < a name = "l00186" > < / a > < span class = "lineno" > 186< / span >   clock_t t2 = clock();< / div >
< div class = "line" > < a name = "l00187" > < / a > < span class = "lineno" > 187< / span >   total_time = (t2 - t1) / CLOCKS_PER_SEC;< / div >
< div class = "line" > < a name = "l00188" > < / a > < span class = "lineno" > 188< / span >   printf(< span class = "stringliteral" > " \tTime = %.6g ms\n" < / span > , total_time);< / div >
< div class = "line" > < a name = "l00189" > < / a > < span class = "lineno" > 189< / span >   fclose(fp);< / div >
< div class = "line" > < a name = "l00190" > < / a > < span class = "lineno" > 190< / span >   < / div >
< div class = "line" > < a name = "l00191" > < / a > < span class = "lineno" > 191< / span >   < span class = "keywordflow" > return< / span > 0;< / div >
< div class = "line" > < a name = "l00192" > < / a > < span class = "lineno" > 192< / span >   }< / div >
< / div > <!-- fragment --> < div class = "dynheader" >
Here is the call graph for this function:< / div >
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< div class = "center" > < iframe scrolling = "no" frameborder = "0" src = "../../d4/d99/ode__semi__implicit__euler_8c_a0ddf1224851353fc92bfbff6f499fa97_cgraph.svg" width = "572" height = "88" > < p > < b > This browser is not able to show SVG: try Firefox, Chrome, Safari, or Opera instead.< / b > < / p > < / iframe >
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< a id = "a97075291390a68c262ed66e157a57eb4" > < / a >
< h2 class = "memtitle" > < span class = "permalink" > < a href = "#a97075291390a68c262ed66e157a57eb4" > ◆ < / a > < / span > problem()< / h2 >
< div class = "memitem" >
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< table class = "memname" >
< tr >
< td class = "memname" > void problem < / td >
< td > (< / td >
< td class = "paramtype" > const double *  < / td >
< td class = "paramname" > < em > x< / em > , < / td >
< / tr >
< tr >
< td class = "paramkey" > < / td >
< td > < / td >
< td class = "paramtype" > double *  < / td >
< td class = "paramname" > < em > y< / em > , < / td >
< / tr >
< tr >
< td class = "paramkey" > < / td >
< td > < / td >
< td class = "paramtype" > double *  < / td >
< td class = "paramname" > < em > dy< / em >   < / td >
< / tr >
< tr >
< td > < / td >
< td > )< / td >
< td > < / td > < td > < / td >
< / tr >
< / table >
< / div > < div class = "memdoc" >
< p > Problem statement for a system with first-order differential equations. Updates the system differential variables. < / p >
< dl class = "section note" > < dt > Note< / dt > < dd > This function can be updated to and ode of any order.< / dd > < / dl >
< dl class = "params" > < dt > Parameters< / dt > < dd >
< table class = "params" >
< tr > < td class = "paramdir" > [in]< / td > < td class = "paramname" > x< / td > < td > independent variable(s) < / td > < / tr >
< tr > < td class = "paramdir" > [in,out]< / td > < td class = "paramname" > y< / td > < td > dependent variable(s) < / td > < / tr >
< tr > < td class = "paramdir" > [in,out]< / td > < td class = "paramname" > dy< / td > < td > first-derivative of dependent variable(s) < / td > < / tr >
< / table >
< / dd >
< / dl >
< div class = "fragment" > < div class = "line" > < a name = "l00059" > < / a > < span class = "lineno" > 59< / span >   {< / div >
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< div class = "line" > < a name = "l00060" > < / a > < span class = "lineno" > 60< / span >   < span class = "keyword" > const< / span > < span class = "keywordtype" > double< / span > omega = 1.F; < span class = "comment" > // some const for the problem< / span > < / div >
< div class = "line" > < a name = "l00061" > < / a > < span class = "lineno" > 61< / span >   dy[0] = y[1]; < span class = "comment" > // x dot< / span > < / div >
< div class = "line" > < a name = "l00062" > < / a > < span class = "lineno" > 62< / span >   dy[1] = -omega * omega * y[0]; < span class = "comment" > // y dot< / span > < / div >
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< div class = "line" > < a name = "l00063" > < / a > < span class = "lineno" > 63< / span >   }< / div >
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< a id = "ad80059877222f885b549f2d0a3dc6b55" > < / a >
< h2 class = "memtitle" > < span class = "permalink" > < a href = "#ad80059877222f885b549f2d0a3dc6b55" > ◆ < / a > < / span > semi_implicit_euler()< / h2 >
< div class = "memitem" >
< div class = "memproto" >
< table class = "memname" >
< tr >
< td class = "memname" > double semi_implicit_euler < / td >
< td > (< / td >
< td class = "paramtype" > double  < / td >
< td class = "paramname" > < em > dx< / em > , < / td >
< / tr >
< tr >
< td class = "paramkey" > < / td >
< td > < / td >
< td class = "paramtype" > double  < / td >
< td class = "paramname" > < em > x0< / em > , < / td >
< / tr >
< tr >
< td class = "paramkey" > < / td >
< td > < / td >
< td class = "paramtype" > double  < / td >
< td class = "paramname" > < em > x_max< / em > , < / td >
< / tr >
< tr >
< td class = "paramkey" > < / td >
< td > < / td >
< td class = "paramtype" > double *  < / td >
< td class = "paramname" > < em > y< / em > , < / td >
< / tr >
< tr >
< td class = "paramkey" > < / td >
< td > < / td >
< td class = "paramtype" > char  < / td >
< td class = "paramname" > < em > save_to_file< / em >   < / td >
< / tr >
< tr >
< td > < / td >
< td > )< / td >
< td > < / td > < td > < / td >
< / tr >
< / table >
< / div > < div class = "memdoc" >
< p > Compute approximation using the semi-implicit-Euler method in the given limits. < / p >
< dl class = "params" > < dt > Parameters< / dt > < dd >
< table class = "params" >
< tr > < td class = "paramdir" > [in]< / td > < td class = "paramname" > dx< / td > < td > step size < / td > < / tr >
< tr > < td class = "paramdir" > [in]< / td > < td class = "paramname" > x0< / td > < td > initial value of independent variable < / td > < / tr >
< tr > < td class = "paramdir" > [in]< / td > < td class = "paramname" > x_max< / td > < td > final value of independent variable < / td > < / tr >
< tr > < td class = "paramdir" > [in,out]< / td > < td class = "paramname" > y< / td > < td > take \(y_n\) and compute \(y_{n+1}\) < / td > < / tr >
< tr > < td class = "paramdir" > [in]< / td > < td class = "paramname" > save_to_file< / td > < td > flag to save results to a CSV file (1) or not (0) < / td > < / tr >
< / table >
< / dd >
< / dl >
< dl class = "section return" > < dt > Returns< / dt > < dd > time taken for computation in seconds < / dd > < / dl >
< div class = "fragment" > < div class = "line" > < a name = "l00111" > < / a > < span class = "lineno" > 111< / span >   {< / div >
< div class = "line" > < a name = "l00112" > < / a > < span class = "lineno" > 112< / span >   < span class = "keywordtype" > double< / span > dy[< a class = "code" href = "../../d4/d99/ode__semi__implicit__euler_8c.html#a9ceb646336224ee890a269d0b4600d09" > order< / a > ];< / div >
< div class = "line" > < a name = "l00113" > < / a > < span class = "lineno" > 113< / span >   < / div >
< div class = "line" > < a name = "l00114" > < / a > < span class = "lineno" > 114< / span >   FILE *fp = NULL;< / div >
< div class = "line" > < a name = "l00115" > < / a > < span class = "lineno" > 115< / span >   < span class = "keywordflow" > if< / span > (save_to_file)< / div >
< div class = "line" > < a name = "l00116" > < / a > < span class = "lineno" > 116< / span >   {< / div >
< div class = "line" > < a name = "l00117" > < / a > < span class = "lineno" > 117< / span >   fp = fopen(< span class = "stringliteral" > " semi_implicit_euler.csv" < / span > , < span class = "stringliteral" > " w+" < / span > );< / div >
< div class = "line" > < a name = "l00118" > < / a > < span class = "lineno" > 118< / span >   < span class = "keywordflow" > if< / span > (fp == NULL)< / div >
< div class = "line" > < a name = "l00119" > < / a > < span class = "lineno" > 119< / span >   {< / div >
< div class = "line" > < a name = "l00120" > < / a > < span class = "lineno" > 120< / span >   perror(< span class = "stringliteral" > " Error! " < / span > );< / div >
< div class = "line" > < a name = "l00121" > < / a > < span class = "lineno" > 121< / span >   < span class = "keywordflow" > return< / span > -1;< / div >
< div class = "line" > < a name = "l00122" > < / a > < span class = "lineno" > 122< / span >   }< / div >
< div class = "line" > < a name = "l00123" > < / a > < span class = "lineno" > 123< / span >   }< / div >
< div class = "line" > < a name = "l00124" > < / a > < span class = "lineno" > 124< / span >   < / div >
< div class = "line" > < a name = "l00125" > < / a > < span class = "lineno" > 125< / span >   < span class = "comment" > /* start integration */< / span > < / div >
< div class = "line" > < a name = "l00126" > < / a > < span class = "lineno" > 126< / span >   clock_t t1 = clock();< / div >
< div class = "line" > < a name = "l00127" > < / a > < span class = "lineno" > 127< / span >   < span class = "keywordtype" > double< / span > x = x0;< / div >
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< div class = "line" > < a name = "l00128" > < / a > < span class = "lineno" > 128< / span >   < span class = "keywordflow" > do< / span > < span class = "comment" > // iterate for each step of independent variable< / span > < / div >
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< div class = "line" > < a name = "l00129" > < / a > < span class = "lineno" > 129< / span >   {< / div >
< div class = "line" > < a name = "l00130" > < / a > < span class = "lineno" > 130< / span >   < span class = "keywordflow" > if< / span > (save_to_file & & fp)< / div >
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< div class = "line" > < a name = "l00131" > < / a > < span class = "lineno" > 131< / span >   fprintf(fp, < span class = "stringliteral" > " %.4g,%.4g,%.4g\n" < / span > , x, y[0], y[1]); < span class = "comment" > // write to file< / span > < / div >
< div class = "line" > < a name = "l00132" > < / a > < span class = "lineno" > 132< / span >   < a class = "code" href = "../../d4/d99/ode__semi__implicit__euler_8c.html#a720b7e995d2bbc615f94a2c7dbcf84eb" > semi_implicit_euler_step< / a > (dx, & x, y, dy); < span class = "comment" > // perform integration< / span > < / div >
< div class = "line" > < a name = "l00133" > < / a > < span class = "lineno" > 133< / span >   x += dx; < span class = "comment" > // update step< / span > < / div >
< div class = "line" > < a name = "l00134" > < / a > < span class = "lineno" > 134< / span >   } < span class = "keywordflow" > while< / span > (x < = x_max); < span class = "comment" > // till upper limit of independent variable< / span > < / div >
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< div class = "line" > < a name = "l00135" > < / a > < span class = "lineno" > 135< / span >   < span class = "comment" > /* end of integration */< / span > < / div >
< div class = "line" > < a name = "l00136" > < / a > < span class = "lineno" > 136< / span >   clock_t t2 = clock();< / div >
< div class = "line" > < a name = "l00137" > < / a > < span class = "lineno" > 137< / span >   < / div >
< div class = "line" > < a name = "l00138" > < / a > < span class = "lineno" > 138< / span >   < span class = "keywordflow" > if< / span > (save_to_file & & fp)< / div >
< div class = "line" > < a name = "l00139" > < / a > < span class = "lineno" > 139< / span >   fclose(fp);< / div >
< div class = "line" > < a name = "l00140" > < / a > < span class = "lineno" > 140< / span >   < / div >
< div class = "line" > < a name = "l00141" > < / a > < span class = "lineno" > 141< / span >   < span class = "keywordflow" > return< / span > (< span class = "keywordtype" > double< / span > )(t2 - t1) / CLOCKS_PER_SEC;< / div >
< div class = "line" > < a name = "l00142" > < / a > < span class = "lineno" > 142< / span >   }< / div >
< / div > <!-- fragment --> < div class = "dynheader" >
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< div class = "center" > < iframe scrolling = "no" frameborder = "0" src = "../../d4/d99/ode__semi__implicit__euler_8c_ad80059877222f885b549f2d0a3dc6b55_cgraph.svg" width = "474" height = "38" > < p > < b > This browser is not able to show SVG: try Firefox, Chrome, Safari, or Opera instead.< / b > < / p > < / iframe >
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< a id = "a720b7e995d2bbc615f94a2c7dbcf84eb" > < / a >
< h2 class = "memtitle" > < span class = "permalink" > < a href = "#a720b7e995d2bbc615f94a2c7dbcf84eb" > ◆ < / a > < / span > semi_implicit_euler_step()< / h2 >
< div class = "memitem" >
< div class = "memproto" >
< table class = "memname" >
< tr >
< td class = "memname" > void semi_implicit_euler_step < / td >
< td > (< / td >
< td class = "paramtype" > double  < / td >
< td class = "paramname" > < em > dx< / em > , < / td >
< / tr >
< tr >
< td class = "paramkey" > < / td >
< td > < / td >
< td class = "paramtype" > double *  < / td >
< td class = "paramname" > < em > x< / em > , < / td >
< / tr >
< tr >
< td class = "paramkey" > < / td >
< td > < / td >
< td class = "paramtype" > double *  < / td >
< td class = "paramname" > < em > y< / em > , < / td >
< / tr >
< tr >
< td class = "paramkey" > < / td >
< td > < / td >
< td class = "paramtype" > double *  < / td >
< td class = "paramname" > < em > dy< / em >   < / td >
< / tr >
< tr >
< td > < / td >
< td > )< / td >
< td > < / td > < td > < / td >
< / tr >
< / table >
< / div > < div class = "memdoc" >
< p > Compute next step approximation using the semi-implicit-Euler method. < / p >
< dl class = "params" > < dt > Parameters< / dt > < dd >
< table class = "params" >
< tr > < td class = "paramdir" > [in]< / td > < td class = "paramname" > dx< / td > < td > step size < / td > < / tr >
< tr > < td class = "paramdir" > [in,out]< / td > < td class = "paramname" > x< / td > < td > take \(x_n\) and compute \(x_{n+1}\) < / td > < / tr >
< tr > < td class = "paramdir" > [in,out]< / td > < td class = "paramname" > y< / td > < td > take \(y_n\) and compute \(y_{n+1}\) < / td > < / tr >
< tr > < td class = "paramdir" > [in,out]< / td > < td class = "paramname" > dy< / td > < td > compute \(y_n+\frac{1}{2}dx\,f\left(x_n,y_n\right)\) < / td > < / tr >
< / table >
< / dd >
< / dl >
< div class = "fragment" > < div class = "line" > < a name = "l00086" > < / a > < span class = "lineno" > 86< / span >   {< / div >
< div class = "line" > < a name = "l00087" > < / a > < span class = "lineno" > 87< / span >   < span class = "keywordtype" > int< / span > o;< / div >
< div class = "line" > < a name = "l00088" > < / a > < span class = "lineno" > 88< / span >   < / div >
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< div class = "line" > < a name = "l00089" > < / a > < span class = "lineno" > 89< / span >   < a class = "code" href = "../../d4/d99/ode__semi__implicit__euler_8c.html#a97075291390a68c262ed66e157a57eb4" > problem< / a > (x, y, dy); < span class = "comment" > // update dy once< / span > < / div >
< div class = "line" > < a name = "l00090" > < / a > < span class = "lineno" > 90< / span >   y[0] += dx * dy[0]; < span class = "comment" > // update y0< / span > < / div >
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< div class = "line" > < a name = "l00091" > < / a > < span class = "lineno" > 91< / span >   < / div >
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< div class = "line" > < a name = "l00092" > < / a > < span class = "lineno" > 92< / span >   < a class = "code" href = "../../d4/d99/ode__semi__implicit__euler_8c.html#a97075291390a68c262ed66e157a57eb4" > problem< / a > (x, y, dy); < span class = "comment" > // update dy once more< / span > < / div >
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< div class = "line" > < a name = "l00093" > < / a > < span class = "lineno" > 93< / span >   < / div >
< div class = "line" > < a name = "l00094" > < / a > < span class = "lineno" > 94< / span >   < span class = "keywordflow" > for< / span > (o = 1; o < < a class = "code" href = "../../d4/d99/ode__semi__implicit__euler_8c.html#a9ceb646336224ee890a269d0b4600d09" > order< / a > ; o++)< / div >
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< div class = "line" > < a name = "l00095" > < / a > < span class = "lineno" > 95< / span >   y[o] += dx * dy[o]; < span class = "comment" > // update remaining using new dy< / span > < / div >
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< div class = "line" > < a name = "l00096" > < / a > < span class = "lineno" > 96< / span >   *x += dx;< / div >
< div class = "line" > < a name = "l00097" > < / a > < span class = "lineno" > 97< / span >   }< / div >
< / div > <!-- fragment --> < div class = "dynheader" >
Here is the call graph for this function:< / div >
< div class = "dyncontent" >
< div class = "center" > < iframe scrolling = "no" frameborder = "0" src = "../../d4/d99/ode__semi__implicit__euler_8c_a720b7e995d2bbc615f94a2c7dbcf84eb_cgraph.svg" width = "292" height = "38" > < p > < b > This browser is not able to show SVG: try Firefox, Chrome, Safari, or Opera instead.< / b > < / p > < / iframe >
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< / div > <!-- contents -->
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< div class = "ttc" id = "aode__semi__implicit__euler_8c_html_ad80059877222f885b549f2d0a3dc6b55" > < div class = "ttname" > < a href = "../../d4/d99/ode__semi__implicit__euler_8c.html#ad80059877222f885b549f2d0a3dc6b55" > semi_implicit_euler< / a > < / div > < div class = "ttdeci" > double semi_implicit_euler(double dx, double x0, double x_max, double *y, char save_to_file)< / div > < div class = "ttdoc" > Compute approximation using the semi-implicit-Euler method in the given limits.< / div > < div class = "ttdef" > < b > Definition:< / b > ode_semi_implicit_euler.c:109< / div > < / div >
< div class = "ttc" id = "aode__semi__implicit__euler_8c_html_a8caee977b26888d34040b122e0e28e3a" > < div class = "ttname" > < a href = "../../d4/d99/ode__semi__implicit__euler_8c.html#a8caee977b26888d34040b122e0e28e3a" > exact_solution< / a > < / div > < div class = "ttdeci" > void exact_solution(const double *x, double *y)< / div > < div class = "ttdoc" > Exact solution of the problem. Used for solution comparison.< / div > < div class = "ttdef" > < b > Definition:< / b > ode_semi_implicit_euler.c:71< / div > < / div >
< div class = "ttc" id = "aode__semi__implicit__euler_8c_html_a97075291390a68c262ed66e157a57eb4" > < div class = "ttname" > < a href = "../../d4/d99/ode__semi__implicit__euler_8c.html#a97075291390a68c262ed66e157a57eb4" > problem< / a > < / div > < div class = "ttdeci" > void problem(const double *x, double *y, double *dy)< / div > < div class = "ttdoc" > Problem statement for a system with first-order differential equations. Updates the system differenti...< / div > < div class = "ttdef" > < b > Definition:< / b > ode_semi_implicit_euler.c:58< / div > < / div >
< div class = "ttc" id = "aode__semi__implicit__euler_8c_html_a9ceb646336224ee890a269d0b4600d09" > < div class = "ttname" > < a href = "../../d4/d99/ode__semi__implicit__euler_8c.html#a9ceb646336224ee890a269d0b4600d09" > order< / a > < / div > < div class = "ttdeci" > #define order< / div > < div class = "ttdef" > < b > Definition:< / b > ode_semi_implicit_euler.c:47< / div > < / div >
< div class = "ttc" id = "aode__semi__implicit__euler_8c_html_a720b7e995d2bbc615f94a2c7dbcf84eb" > < div class = "ttname" > < a href = "../../d4/d99/ode__semi__implicit__euler_8c.html#a720b7e995d2bbc615f94a2c7dbcf84eb" > semi_implicit_euler_step< / a > < / div > < div class = "ttdeci" > void semi_implicit_euler_step(double dx, double *x, double *y, double *dy)< / div > < div class = "ttdoc" > Compute next step approximation using the semi-implicit-Euler method.< / div > < div class = "ttdef" > < b > Definition:< / b > ode_semi_implicit_euler.c:85< / div > < / div >
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