Ways to speed up user implemented RK4

So, I've implemented RK4, and I'm wondering what I can do to make it more efficient? What I've got so far is below. I wish to still record all steps. I think AppendTo is doing the most damage to the time, is there a faster alternative?

rk4[f_, variables_, valtinit_, tinit_, tfinal_, nsteps_] := 

  Module[{table, xlist, ylist, step, k1, k2, k3, k4}, 

   xlist = tinit;

   step = N[(tfinal - tinit)/(nsteps)];

   ylist = valtinit;

   table = {{xlist, ylist}};

   Table[

    k1 = step* f /. MapThread[Rule, {variables, ylist}]; (* 

    Equivalent to step* f/.Thread[Rule[variables,ylist]]*)

    k2 = step*f /. MapThread[Rule, {variables, k1/2 + ylist}];

    k3 = step*f /. MapThread[Rule, {variables, k2/2 + ylist}];

    k4 = step*f /. MapThread[Rule, {variables, k3 + ylist}];

    ylist += 1/6 (k1 + 2 (k2 + k3) + k4);

    xlist += step;

    AppendTo[table, {xlist, ylist}];

    {xlist, ylist}, nsteps];

   table

   ];

Example Input:

funclist = {-x + y, x - y};

initials = {1, 2};

variables = {x, y};

init = 0;

final = 200;

nstep = 20000;

approx = rk4[funclist, variables, initials, init, final, nstep]//AbsoluteTiming;

{3.59932,{...}}

I'd love some suggestions!

edited 11 mins ago

xzczd

27.4k573255

asked 5 hours ago

Shinaolord

908

1

$begingroup$
AppendTo is quadratic time complexity. Might be better to preallocate and set by index. Also it'll be much faster to not use Rule and instead code stuff up a little bit more explicitly. As a general rule, too, use vectorized operators. Those can be very fast. And if everything can be totally functional over "packed arrays" (look them up here) it'll be very quick too.
$endgroup$
– b3m2a1
4 hours ago

$begingroup$
I'll work on implementing it more explicity, this is what came to find first though. It'll require some changes to the inputs, I'll have to ponder this. And preallocating the list is a quick change that won't be an issue to do, I can't believe I forgot that's faster :(. Thanks though!
$endgroup$
– Shinaolord
4 hours ago

$begingroup$
Shinaoloard, using Join[ {{xlist, ylist}}, Table[ k1 = step*f /. MapThread[Rule, {variables, ylist}]; k2 = step*f /. MapThread[Rule, {variables, k1/2 + ylist}]; k3 = step*f /. MapThread[Rule, {variables, k2/2 + ylist}]; k4 = step*f /. MapThread[Rule, {variables, k3 + ylist}]; ylist += 1/6 (k1 + 2 (k2 + k3) + k4); xlist += step; {xlist, ylist}, nsteps ] ] as return value is already a first step. There is no point in appending if you use a Table anyways.
$endgroup$
– Henrik Schumacher
4 hours ago

$begingroup$
@HenrikSchumacher do you think it would be faster to Pre-allocate a list of length nsteps, and append the values, or to join the values using table? I can obviously change Table to Do to remove the time it takes to make the table list, going by b3m2a1's method, or I could use Join as you have suggested. I'm thinking your method may be faster, though. I've already removed the MapThread part, I am testing the speed increase granted by that at the moment. Just curious which path you think will be faster.
$endgroup$
– Shinaolord
4 hours ago

$begingroup$
I am currently testing the speed difference between the one in the post and rk4t2[f_, valtinit_, tinit_, tfinal_, nsteps_] := Module[{test, table, xlist, ylist, step, k1, k2, k3, k4}, xlist = tinit; step = N[(tfinal - tinit)/(nsteps)]; ylist = valtinit; table = {{xlist, ylist}}; test = Table[ k1 = step* f[ylist] ; k2 = step*f[k1/2 + ylist]; k3 = step*f[k2/2 + ylist]; k4 = step*f[k3 + ylist]; ylist += 1/6 (k1 + 2 (k2 + k3) + k4); xlist += step; {xlist, ylist}, nsteps]; Join[table, test] ];
$endgroup$
– Shinaolord
4 hours ago

|
show 1 more comment

rk4[f_, variables_, valtinit_, tinit_, tfinal_, nsteps_] := 

  Module[{table, xlist, ylist, step, k1, k2, k3, k4}, 

   xlist = tinit;

   step = N[(tfinal - tinit)/(nsteps)];

   ylist = valtinit;

   table = {{xlist, ylist}};

   Table[

    k1 = step* f /. MapThread[Rule, {variables, ylist}]; (* 

    Equivalent to step* f/.Thread[Rule[variables,ylist]]*)

    k2 = step*f /. MapThread[Rule, {variables, k1/2 + ylist}];

    k3 = step*f /. MapThread[Rule, {variables, k2/2 + ylist}];

    k4 = step*f /. MapThread[Rule, {variables, k3 + ylist}];

    ylist += 1/6 (k1 + 2 (k2 + k3) + k4);

    xlist += step;

    AppendTo[table, {xlist, ylist}];

    {xlist, ylist}, nsteps];

   table

   ];

Example Input:

funclist = {-x + y, x - y};

initials = {1, 2};

variables = {x, y};

init = 0;

final = 200;

nstep = 20000;

approx = rk4[funclist, variables, initials, init, final, nstep]//AbsoluteTiming;

{3.59932,{...}}

I'd love some suggestions!

edited 11 mins ago

xzczd

27.4k573255

asked 5 hours ago

Shinaolord

908

1

$begingroup$
AppendTo is quadratic time complexity. Might be better to preallocate and set by index. Also it'll be much faster to not use Rule and instead code stuff up a little bit more explicitly. As a general rule, too, use vectorized operators. Those can be very fast. And if everything can be totally functional over "packed arrays" (look them up here) it'll be very quick too.
$endgroup$
– b3m2a1
4 hours ago

$begingroup$
I'll work on implementing it more explicity, this is what came to find first though. It'll require some changes to the inputs, I'll have to ponder this. And preallocating the list is a quick change that won't be an issue to do, I can't believe I forgot that's faster :(. Thanks though!
$endgroup$
– Shinaolord
4 hours ago

$begingroup$
Shinaoloard, using Join[ {{xlist, ylist}}, Table[ k1 = step*f /. MapThread[Rule, {variables, ylist}]; k2 = step*f /. MapThread[Rule, {variables, k1/2 + ylist}]; k3 = step*f /. MapThread[Rule, {variables, k2/2 + ylist}]; k4 = step*f /. MapThread[Rule, {variables, k3 + ylist}]; ylist += 1/6 (k1 + 2 (k2 + k3) + k4); xlist += step; {xlist, ylist}, nsteps ] ] as return value is already a first step. There is no point in appending if you use a Table anyways.
$endgroup$
– Henrik Schumacher
4 hours ago

$begingroup$
@HenrikSchumacher do you think it would be faster to Pre-allocate a list of length nsteps, and append the values, or to join the values using table? I can obviously change Table to Do to remove the time it takes to make the table list, going by b3m2a1's method, or I could use Join as you have suggested. I'm thinking your method may be faster, though. I've already removed the MapThread part, I am testing the speed increase granted by that at the moment. Just curious which path you think will be faster.
$endgroup$
– Shinaolord
4 hours ago

$begingroup$
I am currently testing the speed difference between the one in the post and rk4t2[f_, valtinit_, tinit_, tfinal_, nsteps_] := Module[{test, table, xlist, ylist, step, k1, k2, k3, k4}, xlist = tinit; step = N[(tfinal - tinit)/(nsteps)]; ylist = valtinit; table = {{xlist, ylist}}; test = Table[ k1 = step* f[ylist] ; k2 = step*f[k1/2 + ylist]; k3 = step*f[k2/2 + ylist]; k4 = step*f[k3 + ylist]; ylist += 1/6 (k1 + 2 (k2 + k3) + k4); xlist += step; {xlist, ylist}, nsteps]; Join[table, test] ];
$endgroup$
– Shinaolord
4 hours ago

|
show 1 more comment

rk4[f_, variables_, valtinit_, tinit_, tfinal_, nsteps_] := 

  Module[{table, xlist, ylist, step, k1, k2, k3, k4}, 

   xlist = tinit;

   step = N[(tfinal - tinit)/(nsteps)];

   ylist = valtinit;

   table = {{xlist, ylist}};

   Table[

    k1 = step* f /. MapThread[Rule, {variables, ylist}]; (* 

    Equivalent to step* f/.Thread[Rule[variables,ylist]]*)

    k2 = step*f /. MapThread[Rule, {variables, k1/2 + ylist}];

    k3 = step*f /. MapThread[Rule, {variables, k2/2 + ylist}];

    k4 = step*f /. MapThread[Rule, {variables, k3 + ylist}];

    ylist += 1/6 (k1 + 2 (k2 + k3) + k4);

    xlist += step;

    AppendTo[table, {xlist, ylist}];

    {xlist, ylist}, nsteps];

   table

   ];

Example Input:

funclist = {-x + y, x - y};

initials = {1, 2};

variables = {x, y};

init = 0;

final = 200;

nstep = 20000;

approx = rk4[funclist, variables, initials, init, final, nstep]//AbsoluteTiming;

{3.59932,{...}}

I'd love some suggestions!

edited 11 mins ago

xzczd

27.4k573255

asked 5 hours ago

Shinaolord

908

rk4[f_, variables_, valtinit_, tinit_, tfinal_, nsteps_] := 

  Module[{table, xlist, ylist, step, k1, k2, k3, k4}, 

   xlist = tinit;

   step = N[(tfinal - tinit)/(nsteps)];

   ylist = valtinit;

   table = {{xlist, ylist}};

   Table[

    k1 = step* f /. MapThread[Rule, {variables, ylist}]; (* 

    Equivalent to step* f/.Thread[Rule[variables,ylist]]*)

    k2 = step*f /. MapThread[Rule, {variables, k1/2 + ylist}];

    k3 = step*f /. MapThread[Rule, {variables, k2/2 + ylist}];

    k4 = step*f /. MapThread[Rule, {variables, k3 + ylist}];

    ylist += 1/6 (k1 + 2 (k2 + k3) + k4);

    xlist += step;

    AppendTo[table, {xlist, ylist}];

    {xlist, ylist}, nsteps];

   table

   ];

Example Input:

funclist = {-x + y, x - y};

initials = {1, 2};

variables = {x, y};

init = 0;

final = 200;

nstep = 20000;

approx = rk4[funclist, variables, initials, init, final, nstep]//AbsoluteTiming;

{3.59932,{...}}

I'd love some suggestions!

differential-equations numerical-integration performance-tuning

edited 11 mins ago

xzczd

27.4k573255

asked 5 hours ago

Shinaolord

908

edited 11 mins ago

xzczd

27.4k573255

asked 5 hours ago

Shinaolord

908

edited 11 mins ago

xzczd

27.4k573255

edited 11 mins ago

xzczd

27.4k573255

edited 11 mins ago

xzczd

27.4k573255

asked 5 hours ago

Shinaolord

908

asked 5 hours ago

Shinaolord

908

asked 5 hours ago

Shinaolord

908

1

$begingroup$
AppendTo is quadratic time complexity. Might be better to preallocate and set by index. Also it'll be much faster to not use Rule and instead code stuff up a little bit more explicitly. As a general rule, too, use vectorized operators. Those can be very fast. And if everything can be totally functional over "packed arrays" (look them up here) it'll be very quick too.
$endgroup$
– b3m2a1
4 hours ago

$begingroup$
I'll work on implementing it more explicity, this is what came to find first though. It'll require some changes to the inputs, I'll have to ponder this. And preallocating the list is a quick change that won't be an issue to do, I can't believe I forgot that's faster :(. Thanks though!
$endgroup$
– Shinaolord
4 hours ago

$begingroup$
Shinaoloard, using Join[ {{xlist, ylist}}, Table[ k1 = step*f /. MapThread[Rule, {variables, ylist}]; k2 = step*f /. MapThread[Rule, {variables, k1/2 + ylist}]; k3 = step*f /. MapThread[Rule, {variables, k2/2 + ylist}]; k4 = step*f /. MapThread[Rule, {variables, k3 + ylist}]; ylist += 1/6 (k1 + 2 (k2 + k3) + k4); xlist += step; {xlist, ylist}, nsteps ] ] as return value is already a first step. There is no point in appending if you use a Table anyways.
$endgroup$
– Henrik Schumacher
4 hours ago

$begingroup$
@HenrikSchumacher do you think it would be faster to Pre-allocate a list of length nsteps, and append the values, or to join the values using table? I can obviously change Table to Do to remove the time it takes to make the table list, going by b3m2a1's method, or I could use Join as you have suggested. I'm thinking your method may be faster, though. I've already removed the MapThread part, I am testing the speed increase granted by that at the moment. Just curious which path you think will be faster.
$endgroup$
– Shinaolord
4 hours ago

$begingroup$
I am currently testing the speed difference between the one in the post and rk4t2[f_, valtinit_, tinit_, tfinal_, nsteps_] := Module[{test, table, xlist, ylist, step, k1, k2, k3, k4}, xlist = tinit; step = N[(tfinal - tinit)/(nsteps)]; ylist = valtinit; table = {{xlist, ylist}}; test = Table[ k1 = step* f[ylist] ; k2 = step*f[k1/2 + ylist]; k3 = step*f[k2/2 + ylist]; k4 = step*f[k3 + ylist]; ylist += 1/6 (k1 + 2 (k2 + k3) + k4); xlist += step; {xlist, ylist}, nsteps]; Join[table, test] ];
$endgroup$
– Shinaolord
4 hours ago

|
show 1 more comment

1

$begingroup$
AppendTo is quadratic time complexity. Might be better to preallocate and set by index. Also it'll be much faster to not use Rule and instead code stuff up a little bit more explicitly. As a general rule, too, use vectorized operators. Those can be very fast. And if everything can be totally functional over "packed arrays" (look them up here) it'll be very quick too.
$endgroup$
– b3m2a1
4 hours ago

$begingroup$
I'll work on implementing it more explicity, this is what came to find first though. It'll require some changes to the inputs, I'll have to ponder this. And preallocating the list is a quick change that won't be an issue to do, I can't believe I forgot that's faster :(. Thanks though!
$endgroup$
– Shinaolord
4 hours ago

$begingroup$
Shinaoloard, using Join[ {{xlist, ylist}}, Table[ k1 = step*f /. MapThread[Rule, {variables, ylist}]; k2 = step*f /. MapThread[Rule, {variables, k1/2 + ylist}]; k3 = step*f /. MapThread[Rule, {variables, k2/2 + ylist}]; k4 = step*f /. MapThread[Rule, {variables, k3 + ylist}]; ylist += 1/6 (k1 + 2 (k2 + k3) + k4); xlist += step; {xlist, ylist}, nsteps ] ] as return value is already a first step. There is no point in appending if you use a Table anyways.
$endgroup$
– Henrik Schumacher
4 hours ago

$begingroup$
@HenrikSchumacher do you think it would be faster to Pre-allocate a list of length nsteps, and append the values, or to join the values using table? I can obviously change Table to Do to remove the time it takes to make the table list, going by b3m2a1's method, or I could use Join as you have suggested. I'm thinking your method may be faster, though. I've already removed the MapThread part, I am testing the speed increase granted by that at the moment. Just curious which path you think will be faster.
$endgroup$
– Shinaolord
4 hours ago

$begingroup$
I am currently testing the speed difference between the one in the post and rk4t2[f_, valtinit_, tinit_, tfinal_, nsteps_] := Module[{test, table, xlist, ylist, step, k1, k2, k3, k4}, xlist = tinit; step = N[(tfinal - tinit)/(nsteps)]; ylist = valtinit; table = {{xlist, ylist}}; test = Table[ k1 = step* f[ylist] ; k2 = step*f[k1/2 + ylist]; k3 = step*f[k2/2 + ylist]; k4 = step*f[k3 + ylist]; ylist += 1/6 (k1 + 2 (k2 + k3) + k4); xlist += step; {xlist, ylist}, nsteps]; Join[table, test] ];
$endgroup$
– Shinaolord
4 hours ago

AppendTo is quadratic time complexity. Might be better to preallocate and set by index. Also it'll be much faster to not use Rule and instead code stuff up a little bit more explicitly. As a general rule, too, use vectorized operators. Those can be very fast. And if everything can be totally functional over "packed arrays" (look them up here) it'll be very quick too.

– b3m2a1
4 hours ago

I'll work on implementing it more explicity, this is what came to find first though. It'll require some changes to the inputs, I'll have to ponder this. And preallocating the list is a quick change that won't be an issue to do, I can't believe I forgot that's faster :(. Thanks though!

– Shinaolord
4 hours ago

Shinaoloard, using

Join[  {{xlist, ylist}},  Table[   k1 = step*f /. MapThread[Rule, {variables, ylist}];   k2 = step*f /. MapThread[Rule, {variables, k1/2 + ylist}];   k3 = step*f /. MapThread[Rule, {variables, k2/2 + ylist}];   k4 = step*f /. MapThread[Rule, {variables, k3 + ylist}];   ylist += 1/6 (k1 + 2 (k2 + k3) + k4);   xlist += step;   {xlist, ylist},   nsteps   ]  ]

as return value is already a first step. There is no point in appending if you use a Table anyways.

– Henrik Schumacher
4 hours ago

Shinaoloard, using

Join[  {{xlist, ylist}},  Table[   k1 = step*f /. MapThread[Rule, {variables, ylist}];   k2 = step*f /. MapThread[Rule, {variables, k1/2 + ylist}];   k3 = step*f /. MapThread[Rule, {variables, k2/2 + ylist}];   k4 = step*f /. MapThread[Rule, {variables, k3 + ylist}];   ylist += 1/6 (k1 + 2 (k2 + k3) + k4);   xlist += step;   {xlist, ylist},   nsteps   ]  ]

as return value is already a first step. There is no point in appending if you use a Table anyways.

– Henrik Schumacher
4 hours ago

@HenrikSchumacher do you think it would be faster to Pre-allocate a list of length nsteps, and append the values, or to join the values using table? I can obviously change Table to Do to remove the time it takes to make the table list, going by b3m2a1's method, or I could use Join as you have suggested. I'm thinking your method may be faster, though. I've already removed the MapThread part, I am testing the speed increase granted by that at the moment. Just curious which path you think will be faster.

– Shinaolord
4 hours ago

I am currently testing the speed difference between the one in the post and

rk4t2[f_, valtinit_, tinit_, tfinal_, nsteps_] :=    Module[{test, table, xlist, ylist, step, k1, k2, k3, k4},     xlist = tinit;    step = N[(tfinal - tinit)/(nsteps)];    ylist = valtinit;    table = {{xlist, ylist}};    test = Table[      k1 = step* f[ylist] ;      k2 = step*f[k1/2 + ylist];      k3 = step*f[k2/2 + ylist];      k4 = step*f[k3 + ylist];      ylist += 1/6 (k1 + 2 (k2 + k3) + k4);      xlist += step;      {xlist, ylist}, nsteps];    Join[table, test]        ];

– Shinaolord
4 hours ago

I am currently testing the speed difference between the one in the post and

rk4t2[f_, valtinit_, tinit_, tfinal_, nsteps_] :=    Module[{test, table, xlist, ylist, step, k1, k2, k3, k4},     xlist = tinit;    step = N[(tfinal - tinit)/(nsteps)];    ylist = valtinit;    table = {{xlist, ylist}};    test = Table[      k1 = step* f[ylist] ;      k2 = step*f[k1/2 + ylist];      k3 = step*f[k2/2 + ylist];      k4 = step*f[k3 + ylist];      ylist += 1/6 (k1 + 2 (k2 + k3) + k4);      xlist += step;      {xlist, ylist}, nsteps];    Join[table, test]        ];

– Shinaolord
4 hours ago

|
show 1 more comment

1 Answer
1

active

oldest

votes

Just to give you an impression how fast things may get when you use the right tools.

For given stepsize τ and given vectorfield F, this creates a CompiledFunction cStep that computes a single Runge-Kutta step

F = X [Function] {-Indexed[X, 2], Indexed[X, 1]};



τ = 0.01;

Block[{YY, Y, k1, k2, k3, k4},



  YY = Table[Compile`GetElement[Y, i], {i, 1, 2}];

  k1 = τ F[YY];

  k2 = τ F[0.5 k1 + YY];

  k3 = τ F[0.5 k2 + YY];

  k4 = τ F[k3 + YY];



  cStep = With[{code = YY + (k1 + 2. (k2 + k3) + k4)/6. },

    Compile[{{Y, _Real, 1}},

     code,

     CompilationTarget -> "C",

     RuntimeOptions -> "Speed"

     ]

    ]

  ];

Now we can apply it 2 million times with NestList and still need only 2 seconds.

nsteps = 20000000;

xlist = Range[0., step nsteps, step];

Ylist = NestList[cStep, initials, nsteps]; // AbsoluteTiming // First

2.08678

answered 4 hours ago

Henrik Schumacher

57.9k579159

$begingroup$
Damn, you definitely know how to use Mathematica A LOT more efficiently than I do. Thanks!
$endgroup$
– Shinaolord
4 hours ago

$begingroup$
You're welcome.
$endgroup$
– Henrik Schumacher
4 hours ago

$begingroup$
I'll have to play around with Compile, it definitely seems like a massive speed up if used correctly.
$endgroup$
– Shinaolord
4 hours ago

add a comment |

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1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

Just to give you an impression how fast things may get when you use the right tools.

For given stepsize τ and given vectorfield F, this creates a CompiledFunction cStep that computes a single Runge-Kutta step

F = X [Function] {-Indexed[X, 2], Indexed[X, 1]};



τ = 0.01;

Block[{YY, Y, k1, k2, k3, k4},



  YY = Table[Compile`GetElement[Y, i], {i, 1, 2}];

  k1 = τ F[YY];

  k2 = τ F[0.5 k1 + YY];

  k3 = τ F[0.5 k2 + YY];

  k4 = τ F[k3 + YY];



  cStep = With[{code = YY + (k1 + 2. (k2 + k3) + k4)/6. },

    Compile[{{Y, _Real, 1}},

     code,

     CompilationTarget -> "C",

     RuntimeOptions -> "Speed"

     ]

    ]

  ];

Now we can apply it 2 million times with NestList and still need only 2 seconds.

nsteps = 20000000;

xlist = Range[0., step nsteps, step];

Ylist = NestList[cStep, initials, nsteps]; // AbsoluteTiming // First

2.08678

answered 4 hours ago

Henrik Schumacher

57.9k579159

$begingroup$
Damn, you definitely know how to use Mathematica A LOT more efficiently than I do. Thanks!
$endgroup$
– Shinaolord
4 hours ago

$begingroup$
You're welcome.
$endgroup$
– Henrik Schumacher
4 hours ago

$begingroup$
I'll have to play around with Compile, it definitely seems like a massive speed up if used correctly.
$endgroup$
– Shinaolord
4 hours ago

add a comment |

Just to give you an impression how fast things may get when you use the right tools.

For given stepsize τ and given vectorfield F, this creates a CompiledFunction cStep that computes a single Runge-Kutta step

F = X [Function] {-Indexed[X, 2], Indexed[X, 1]};



τ = 0.01;

Block[{YY, Y, k1, k2, k3, k4},



  YY = Table[Compile`GetElement[Y, i], {i, 1, 2}];

  k1 = τ F[YY];

  k2 = τ F[0.5 k1 + YY];

  k3 = τ F[0.5 k2 + YY];

  k4 = τ F[k3 + YY];



  cStep = With[{code = YY + (k1 + 2. (k2 + k3) + k4)/6. },

    Compile[{{Y, _Real, 1}},

     code,

     CompilationTarget -> "C",

     RuntimeOptions -> "Speed"

     ]

    ]

  ];

Now we can apply it 2 million times with NestList and still need only 2 seconds.

nsteps = 20000000;

xlist = Range[0., step nsteps, step];

Ylist = NestList[cStep, initials, nsteps]; // AbsoluteTiming // First

2.08678

answered 4 hours ago

Henrik Schumacher

57.9k579159

$begingroup$
Damn, you definitely know how to use Mathematica A LOT more efficiently than I do. Thanks!
$endgroup$
– Shinaolord
4 hours ago

$begingroup$
You're welcome.
$endgroup$
– Henrik Schumacher
4 hours ago

$begingroup$
I'll have to play around with Compile, it definitely seems like a massive speed up if used correctly.
$endgroup$
– Shinaolord
4 hours ago

add a comment |

Just to give you an impression how fast things may get when you use the right tools.

For given stepsize τ and given vectorfield F, this creates a CompiledFunction cStep that computes a single Runge-Kutta step

F = X [Function] {-Indexed[X, 2], Indexed[X, 1]};



τ = 0.01;

Block[{YY, Y, k1, k2, k3, k4},



  YY = Table[Compile`GetElement[Y, i], {i, 1, 2}];

  k1 = τ F[YY];

  k2 = τ F[0.5 k1 + YY];

  k3 = τ F[0.5 k2 + YY];

  k4 = τ F[k3 + YY];



  cStep = With[{code = YY + (k1 + 2. (k2 + k3) + k4)/6. },

    Compile[{{Y, _Real, 1}},

     code,

     CompilationTarget -> "C",

     RuntimeOptions -> "Speed"

     ]

    ]

  ];

Now we can apply it 2 million times with NestList and still need only 2 seconds.

nsteps = 20000000;

xlist = Range[0., step nsteps, step];

Ylist = NestList[cStep, initials, nsteps]; // AbsoluteTiming // First

2.08678

answered 4 hours ago

Henrik Schumacher

57.9k579159

Just to give you an impression how fast things may get when you use the right tools.

For given stepsize τ and given vectorfield F, this creates a CompiledFunction cStep that computes a single Runge-Kutta step

F = X [Function] {-Indexed[X, 2], Indexed[X, 1]};



τ = 0.01;

Block[{YY, Y, k1, k2, k3, k4},



  YY = Table[Compile`GetElement[Y, i], {i, 1, 2}];

  k1 = τ F[YY];

  k2 = τ F[0.5 k1 + YY];

  k3 = τ F[0.5 k2 + YY];

  k4 = τ F[k3 + YY];



  cStep = With[{code = YY + (k1 + 2. (k2 + k3) + k4)/6. },

    Compile[{{Y, _Real, 1}},

     code,

     CompilationTarget -> "C",

     RuntimeOptions -> "Speed"

     ]

    ]

  ];

Now we can apply it 2 million times with NestList and still need only 2 seconds.

nsteps = 20000000;

xlist = Range[0., step nsteps, step];

Ylist = NestList[cStep, initials, nsteps]; // AbsoluteTiming // First

2.08678

answered 4 hours ago

Henrik Schumacher

57.9k579159

answered 4 hours ago

Henrik Schumacher

57.9k579159

answered 4 hours ago

Henrik Schumacher

57.9k579159

answered 4 hours ago

Henrik Schumacher

57.9k579159

$begingroup$
Damn, you definitely know how to use Mathematica A LOT more efficiently than I do. Thanks!
$endgroup$
– Shinaolord
4 hours ago

$begingroup$
You're welcome.
$endgroup$
– Henrik Schumacher
4 hours ago

$begingroup$
I'll have to play around with Compile, it definitely seems like a massive speed up if used correctly.
$endgroup$
– Shinaolord
4 hours ago

add a comment |

$begingroup$
Damn, you definitely know how to use Mathematica A LOT more efficiently than I do. Thanks!
$endgroup$
– Shinaolord
4 hours ago

$begingroup$
You're welcome.
$endgroup$
– Henrik Schumacher
4 hours ago

$begingroup$
I'll have to play around with Compile, it definitely seems like a massive speed up if used correctly.
$endgroup$
– Shinaolord
4 hours ago

Damn, you definitely know how to use Mathematica A LOT more efficiently than I do. Thanks!

– Shinaolord
4 hours ago

You're welcome.

– Henrik Schumacher
4 hours ago

I'll have to play around with Compile, it definitely seems like a massive speed up if used correctly.

– Shinaolord
4 hours ago

add a comment |

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