Minimizing with differential evolution
4
$begingroup$
A differential evolution algorithm is given here. I would like to get this kind of animation. I thought I could use NMinimize, given
DifferentialEvolution as an option, but it turns out that does not work as I espected.
Is it possible to extract intermediate step in DifferentialEvolution, or do I have to implement algorithm myself?
f[x_, y_] :=
-20 E^(-0.2 Sqrt[0.5 (x^2 + y^2)]) - E^(0.5 (Cos[2 π x] + Cos[2 π y])) + E + 20
p1 =
Plot3D[f[x, y], {x, -5, 5}, {y, -5, 5},
PerformanceGoal -> "Quality",
ColorFunction -> "WatermelonColors",
Mesh -> None,
BoxRatios -> {1, 1, 1}];
p2 =
DensityPlot[f[x, y], {x, -5, 5}, {y, -5, 5},
ColorFunction -> "WatermelonColors",
PlotPoints -> 200,
PerformanceGoal -> "Quality",
Frame -> False,
PlotRangePadding -> None];
p3 = Plot3D[0, {x, -5, 5}, {y, -5, 5}, PlotStyle -> Texture[p2], Mesh -> None];
Show[p1, p3, PlotRange -> {0, 15}]
When I use StepMonitor to track iterations as follows, it does not work.
{fit, intermediates} =
Reap[NMinimize[{f[x, y], -5 <= x <= 5, -5 <= y <= 5}, {x, y},
MaxIterations -> 1000,
Method -> {"DifferentialEvolution", "InitialPoints" -> Tuples[Range[-5, 5], 2]},
StepMonitor :> Sow[{x, y}]]];
Table[
ListPlot[Take[intermediates[[1, i ;; i + 10]]],
Frame -> True, ImageSize -> 350, AspectRatio -> 1],
{i, 10, 1000, 100}]
EDIT
Here is the result when we used @Michael E2 solution. Cool!!
f[x_, y_] := -20 E^(-0.2 Sqrt[0.5 (x^2 + y^2)]) -
E^(0.5 (Cos[2 [Pi] x] + Cos[2 [Pi] y])) + E + 20
p1 = Plot3D[f[x, y], {x, -5, 5}, {y, -5, 5},
PerformanceGoal -> "Quality", ColorFunction -> "WatermelonColors",
Mesh -> None, BoxRatios -> {1, 1, 1}];
p2 = DensityPlot[f[x, y], {x, -5, 5}, {y, -5, 5},
ColorFunction -> "WatermelonColors", PerformanceGoal -> "Quality",
Frame -> False, PlotRangePadding -> None];
p3 = Plot3D[-0.5, {x, -5, 5}, {y, -5, 5}, PlotStyle -> Texture[p2],
Mesh -> None];
p4 = Show[p1, p3, PlotRange -> {-0.5, 15}]
Block[{f},
f[x_, y_] := -20 E^(-0.2 Sqrt[0.5 (x^2 + y^2)]) -
E^(0.5 (Cos[2 [Pi] x] + Cos[2 [Pi] y])) + E + 20;
{fit, intermediates} =
Reap[NMinimize[{f[x, y], -5 <= x <= 5, -5 <= y <= 5}, {x, y},
MaxIterations -> 30,
Method -> {"DifferentialEvolution",
"InitialPoints" -> Tuples[Range[-5, 5], 2]},
StepMonitor :>
Sow[{Optimization`NMinimizeDump`vecs,
Optimization`NMinimizeDump`vals}]]];] // Quiet
Multicolumn[
Table[Show[p4,
ListPointPlot3D[{Append[#, 0] & /@ intermediates[[1, i, 1]]},
PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, Boxed -> False,
PlotStyle -> Directive[AbsolutePointSize[3], Black]]], {i, 1, 30,
2}], 5, Appearance -> "Horizontal"]
mathematical-optimization visualization
edited 13 mins ago
J. M. is computer-less♦
97.4k10303463
asked 5 hours ago
Okkes DulgerciOkkes Dulgerci
5,2691917
$endgroup$
add a comment |
4
$begingroup$
A differential evolution algorithm is given here. I would like to get this kind of animation. I thought I could use NMinimize, given
DifferentialEvolution as an option, but it turns out that does not work as I espected.
Is it possible to extract intermediate step in DifferentialEvolution, or do I have to implement algorithm myself?
f[x_, y_] :=
-20 E^(-0.2 Sqrt[0.5 (x^2 + y^2)]) - E^(0.5 (Cos[2 π x] + Cos[2 π y])) + E + 20
p1 =
Plot3D[f[x, y], {x, -5, 5}, {y, -5, 5},
PerformanceGoal -> "Quality",
ColorFunction -> "WatermelonColors",
Mesh -> None,
BoxRatios -> {1, 1, 1}];
p2 =
DensityPlot[f[x, y], {x, -5, 5}, {y, -5, 5},
ColorFunction -> "WatermelonColors",
PlotPoints -> 200,
PerformanceGoal -> "Quality",
Frame -> False,
PlotRangePadding -> None];
p3 = Plot3D[0, {x, -5, 5}, {y, -5, 5}, PlotStyle -> Texture[p2], Mesh -> None];
Show[p1, p3, PlotRange -> {0, 15}]
When I use StepMonitor to track iterations as follows, it does not work.
{fit, intermediates} =
Reap[NMinimize[{f[x, y], -5 <= x <= 5, -5 <= y <= 5}, {x, y},
MaxIterations -> 1000,
Method -> {"DifferentialEvolution", "InitialPoints" -> Tuples[Range[-5, 5], 2]},
StepMonitor :> Sow[{x, y}]]];
Table[
ListPlot[Take[intermediates[[1, i ;; i + 10]]],
Frame -> True, ImageSize -> 350, AspectRatio -> 1],
{i, 10, 1000, 100}]
EDIT
Here is the result when we used @Michael E2 solution. Cool!!
f[x_, y_] := -20 E^(-0.2 Sqrt[0.5 (x^2 + y^2)]) -
E^(0.5 (Cos[2 [Pi] x] + Cos[2 [Pi] y])) + E + 20
p1 = Plot3D[f[x, y], {x, -5, 5}, {y, -5, 5},
PerformanceGoal -> "Quality", ColorFunction -> "WatermelonColors",
Mesh -> None, BoxRatios -> {1, 1, 1}];
p2 = DensityPlot[f[x, y], {x, -5, 5}, {y, -5, 5},
ColorFunction -> "WatermelonColors", PerformanceGoal -> "Quality",
Frame -> False, PlotRangePadding -> None];
p3 = Plot3D[-0.5, {x, -5, 5}, {y, -5, 5}, PlotStyle -> Texture[p2],
Mesh -> None];
p4 = Show[p1, p3, PlotRange -> {-0.5, 15}]
Block[{f},
f[x_, y_] := -20 E^(-0.2 Sqrt[0.5 (x^2 + y^2)]) -
E^(0.5 (Cos[2 [Pi] x] + Cos[2 [Pi] y])) + E + 20;
{fit, intermediates} =
Reap[NMinimize[{f[x, y], -5 <= x <= 5, -5 <= y <= 5}, {x, y},
MaxIterations -> 30,
Method -> {"DifferentialEvolution",
"InitialPoints" -> Tuples[Range[-5, 5], 2]},
StepMonitor :>
Sow[{Optimization`NMinimizeDump`vecs,
Optimization`NMinimizeDump`vals}]]];] // Quiet
Multicolumn[
Table[Show[p4,
ListPointPlot3D[{Append[#, 0] & /@ intermediates[[1, i, 1]]},
PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, Boxed -> False,
PlotStyle -> Directive[AbsolutePointSize[3], Black]]], {i, 1, 30,
2}], 5, Appearance -> "Horizontal"]
mathematical-optimization visualization
edited 13 mins ago
J. M. is computer-less♦
97.4k10303463
asked 5 hours ago
Okkes DulgerciOkkes Dulgerci
5,2691917
$endgroup$
$begingroup$
Note that blockingf(Block[{f}, ...]) isn't necessary. It was just to preventffrom being defined, which is a habit I have with single-lettter symbols on SE, esp. ones I use likef,x, etc. -- thanks for the accept!
$endgroup$
– Michael E2
3 hours ago
add a comment |
4
4
4
2
$begingroup$
A differential evolution algorithm is given here. I would like to get this kind of animation. I thought I could use NMinimize, given
DifferentialEvolution as an option, but it turns out that does not work as I espected.
Is it possible to extract intermediate step in DifferentialEvolution, or do I have to implement algorithm myself?
f[x_, y_] :=
-20 E^(-0.2 Sqrt[0.5 (x^2 + y^2)]) - E^(0.5 (Cos[2 π x] + Cos[2 π y])) + E + 20
p1 =
Plot3D[f[x, y], {x, -5, 5}, {y, -5, 5},
PerformanceGoal -> "Quality",
ColorFunction -> "WatermelonColors",
Mesh -> None,
BoxRatios -> {1, 1, 1}];
p2 =
DensityPlot[f[x, y], {x, -5, 5}, {y, -5, 5},
ColorFunction -> "WatermelonColors",
PlotPoints -> 200,
PerformanceGoal -> "Quality",
Frame -> False,
PlotRangePadding -> None];
p3 = Plot3D[0, {x, -5, 5}, {y, -5, 5}, PlotStyle -> Texture[p2], Mesh -> None];
Show[p1, p3, PlotRange -> {0, 15}]
When I use StepMonitor to track iterations as follows, it does not work.
{fit, intermediates} =
Reap[NMinimize[{f[x, y], -5 <= x <= 5, -5 <= y <= 5}, {x, y},
MaxIterations -> 1000,
Method -> {"DifferentialEvolution", "InitialPoints" -> Tuples[Range[-5, 5], 2]},
StepMonitor :> Sow[{x, y}]]];
Table[
ListPlot[Take[intermediates[[1, i ;; i + 10]]],
Frame -> True, ImageSize -> 350, AspectRatio -> 1],
{i, 10, 1000, 100}]
EDIT
Here is the result when we used @Michael E2 solution. Cool!!
f[x_, y_] := -20 E^(-0.2 Sqrt[0.5 (x^2 + y^2)]) -
E^(0.5 (Cos[2 [Pi] x] + Cos[2 [Pi] y])) + E + 20
p1 = Plot3D[f[x, y], {x, -5, 5}, {y, -5, 5},
PerformanceGoal -> "Quality", ColorFunction -> "WatermelonColors",
Mesh -> None, BoxRatios -> {1, 1, 1}];
p2 = DensityPlot[f[x, y], {x, -5, 5}, {y, -5, 5},
ColorFunction -> "WatermelonColors", PerformanceGoal -> "Quality",
Frame -> False, PlotRangePadding -> None];
p3 = Plot3D[-0.5, {x, -5, 5}, {y, -5, 5}, PlotStyle -> Texture[p2],
Mesh -> None];
p4 = Show[p1, p3, PlotRange -> {-0.5, 15}]
Block[{f},
f[x_, y_] := -20 E^(-0.2 Sqrt[0.5 (x^2 + y^2)]) -
E^(0.5 (Cos[2 [Pi] x] + Cos[2 [Pi] y])) + E + 20;
{fit, intermediates} =
Reap[NMinimize[{f[x, y], -5 <= x <= 5, -5 <= y <= 5}, {x, y},
MaxIterations -> 30,
Method -> {"DifferentialEvolution",
"InitialPoints" -> Tuples[Range[-5, 5], 2]},
StepMonitor :>
Sow[{Optimization`NMinimizeDump`vecs,
Optimization`NMinimizeDump`vals}]]];] // Quiet
Multicolumn[
Table[Show[p4,
ListPointPlot3D[{Append[#, 0] & /@ intermediates[[1, i, 1]]},
PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, Boxed -> False,
PlotStyle -> Directive[AbsolutePointSize[3], Black]]], {i, 1, 30,
2}], 5, Appearance -> "Horizontal"]
mathematical-optimization visualization
edited 13 mins ago
J. M. is computer-less♦
97.4k10303463
asked 5 hours ago
Okkes DulgerciOkkes Dulgerci
5,2691917
$endgroup$
A differential evolution algorithm is given here. I would like to get this kind of animation. I thought I could use NMinimize, given
DifferentialEvolution as an option, but it turns out that does not work as I espected.
Is it possible to extract intermediate step in DifferentialEvolution, or do I have to implement algorithm myself?
f[x_, y_] :=
-20 E^(-0.2 Sqrt[0.5 (x^2 + y^2)]) - E^(0.5 (Cos[2 π x] + Cos[2 π y])) + E + 20
p1 =
Plot3D[f[x, y], {x, -5, 5}, {y, -5, 5},
PerformanceGoal -> "Quality",
ColorFunction -> "WatermelonColors",
Mesh -> None,
BoxRatios -> {1, 1, 1}];
p2 =
DensityPlot[f[x, y], {x, -5, 5}, {y, -5, 5},
ColorFunction -> "WatermelonColors",
PlotPoints -> 200,
PerformanceGoal -> "Quality",
Frame -> False,
PlotRangePadding -> None];
p3 = Plot3D[0, {x, -5, 5}, {y, -5, 5}, PlotStyle -> Texture[p2], Mesh -> None];
Show[p1, p3, PlotRange -> {0, 15}]
When I use StepMonitor to track iterations as follows, it does not work.
{fit, intermediates} =
Reap[NMinimize[{f[x, y], -5 <= x <= 5, -5 <= y <= 5}, {x, y},
MaxIterations -> 1000,
Method -> {"DifferentialEvolution", "InitialPoints" -> Tuples[Range[-5, 5], 2]},
StepMonitor :> Sow[{x, y}]]];
Table[
ListPlot[Take[intermediates[[1, i ;; i + 10]]],
Frame -> True, ImageSize -> 350, AspectRatio -> 1],
{i, 10, 1000, 100}]
EDIT
Here is the result when we used @Michael E2 solution. Cool!!
f[x_, y_] := -20 E^(-0.2 Sqrt[0.5 (x^2 + y^2)]) -
E^(0.5 (Cos[2 [Pi] x] + Cos[2 [Pi] y])) + E + 20
p1 = Plot3D[f[x, y], {x, -5, 5}, {y, -5, 5},
PerformanceGoal -> "Quality", ColorFunction -> "WatermelonColors",
Mesh -> None, BoxRatios -> {1, 1, 1}];
p2 = DensityPlot[f[x, y], {x, -5, 5}, {y, -5, 5},
ColorFunction -> "WatermelonColors", PerformanceGoal -> "Quality",
Frame -> False, PlotRangePadding -> None];
p3 = Plot3D[-0.5, {x, -5, 5}, {y, -5, 5}, PlotStyle -> Texture[p2],
Mesh -> None];
p4 = Show[p1, p3, PlotRange -> {-0.5, 15}]
Block[{f},
f[x_, y_] := -20 E^(-0.2 Sqrt[0.5 (x^2 + y^2)]) -
E^(0.5 (Cos[2 [Pi] x] + Cos[2 [Pi] y])) + E + 20;
{fit, intermediates} =
Reap[NMinimize[{f[x, y], -5 <= x <= 5, -5 <= y <= 5}, {x, y},
MaxIterations -> 30,
Method -> {"DifferentialEvolution",
"InitialPoints" -> Tuples[Range[-5, 5], 2]},
StepMonitor :>
Sow[{Optimization`NMinimizeDump`vecs,
Optimization`NMinimizeDump`vals}]]];] // Quiet
Multicolumn[
Table[Show[p4,
ListPointPlot3D[{Append[#, 0] & /@ intermediates[[1, i, 1]]},
PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, Boxed -> False,
PlotStyle -> Directive[AbsolutePointSize[3], Black]]], {i, 1, 30,
2}], 5, Appearance -> "Horizontal"]
mathematical-optimization visualization
mathematical-optimization visualization
edited 13 mins ago
J. M. is computer-less♦
97.4k10303463
asked 5 hours ago
Okkes DulgerciOkkes Dulgerci
5,2691917
edited 13 mins ago
J. M. is computer-less♦
97.4k10303463
asked 5 hours ago
Okkes DulgerciOkkes Dulgerci
5,2691917
edited 13 mins ago
J. M. is computer-less♦
97.4k10303463
edited 13 mins ago
J. M. is computer-less♦
97.4k10303463
edited 13 mins ago
J. M. is computer-less♦
97.4k10303463
97.4k10303463
asked 5 hours ago
Okkes DulgerciOkkes Dulgerci
5,2691917
asked 5 hours ago
Okkes DulgerciOkkes Dulgerci
5,2691917
asked 5 hours ago
Okkes DulgerciOkkes Dulgerci
5,2691917
5,2691917
$begingroup$
Note that blockingf(Block[{f}, ...]) isn't necessary. It was just to preventffrom being defined, which is a habit I have with single-lettter symbols on SE, esp. ones I use likef,x, etc. -- thanks for the accept!
$endgroup$
– Michael E2
3 hours ago
add a comment |
$begingroup$
Note that blockingf(Block[{f}, ...]) isn't necessary. It was just to preventffrom being defined, which is a habit I have with single-lettter symbols on SE, esp. ones I use likef,x, etc. -- thanks for the accept!
$endgroup$
– Michael E2
3 hours ago
$begingroup$
Note that blocking
f (Block[{f}, ...]) isn't necessary. It was just to prevent f from being defined, which is a habit I have with single-lettter symbols on SE, esp. ones I use like f, x, etc. -- thanks for the accept!$endgroup$
– Michael E2
3 hours ago
$begingroup$
Note that blocking
f (Block[{f}, ...]) isn't necessary. It was just to prevent f from being defined, which is a habit I have with single-lettter symbols on SE, esp. ones I use like f, x, etc. -- thanks for the accept!$endgroup$
– Michael E2
3 hours ago
add a comment |
1 Answer
1
active
oldest
votes
5
$begingroup$
Here's a way:
Block[{f},
f[x_, y_] := -20 E^(-0.2 Sqrt[0.5 (x^2 + y^2)]) -
E^(0.5 (Cos[2 [Pi] x] + Cos[2 [Pi] y])) + E + 20;
{fit, intermediates} =
Reap[NMinimize[{f[x, y], -5 <= x <= 5, -5 <= y <= 5}, {x, y},
MaxIterations -> 30,
Method -> {"DifferentialEvolution",
"InitialPoints" -> Tuples[Range[-5, 5], 2]},
StepMonitor :>
Sow[{Optimization`NMinimizeDump`vecs,
Optimization`NMinimizeDump`vals}]]];
]
Manipulate[
Graphics[{
PointSize[Medium],
Point[intermediates[[1, n, 1]],
VertexColors ->
ColorData["Rainbow"] /@
Rescale[intermediates[[1, n, 2]],
MinMax[intermediates[[1, All, 2]]]]]
},
PlotRange -> 5, Frame -> True],
{n, 1, Length@intermediates[[1]], 1}
]
You can find out about things like Optimization`NMinimizeDump`vecs by inspecting the code for Optimization`NMinimizeDump`CoreDE.
answered 4 hours ago
Michael E2Michael E2
148k12198478
$endgroup$
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
5
$begingroup$
Here's a way:
Block[{f},
f[x_, y_] := -20 E^(-0.2 Sqrt[0.5 (x^2 + y^2)]) -
E^(0.5 (Cos[2 [Pi] x] + Cos[2 [Pi] y])) + E + 20;
{fit, intermediates} =
Reap[NMinimize[{f[x, y], -5 <= x <= 5, -5 <= y <= 5}, {x, y},
MaxIterations -> 30,
Method -> {"DifferentialEvolution",
"InitialPoints" -> Tuples[Range[-5, 5], 2]},
StepMonitor :>
Sow[{Optimization`NMinimizeDump`vecs,
Optimization`NMinimizeDump`vals}]]];
]
Manipulate[
Graphics[{
PointSize[Medium],
Point[intermediates[[1, n, 1]],
VertexColors ->
ColorData["Rainbow"] /@
Rescale[intermediates[[1, n, 2]],
MinMax[intermediates[[1, All, 2]]]]]
},
PlotRange -> 5, Frame -> True],
{n, 1, Length@intermediates[[1]], 1}
]
You can find out about things like Optimization`NMinimizeDump`vecs by inspecting the code for Optimization`NMinimizeDump`CoreDE.
answered 4 hours ago
Michael E2Michael E2
148k12198478
$endgroup$
add a comment |
5
$begingroup$
Here's a way:
Block[{f},
f[x_, y_] := -20 E^(-0.2 Sqrt[0.5 (x^2 + y^2)]) -
E^(0.5 (Cos[2 [Pi] x] + Cos[2 [Pi] y])) + E + 20;
{fit, intermediates} =
Reap[NMinimize[{f[x, y], -5 <= x <= 5, -5 <= y <= 5}, {x, y},
MaxIterations -> 30,
Method -> {"DifferentialEvolution",
"InitialPoints" -> Tuples[Range[-5, 5], 2]},
StepMonitor :>
Sow[{Optimization`NMinimizeDump`vecs,
Optimization`NMinimizeDump`vals}]]];
]
Manipulate[
Graphics[{
PointSize[Medium],
Point[intermediates[[1, n, 1]],
VertexColors ->
ColorData["Rainbow"] /@
Rescale[intermediates[[1, n, 2]],
MinMax[intermediates[[1, All, 2]]]]]
},
PlotRange -> 5, Frame -> True],
{n, 1, Length@intermediates[[1]], 1}
]
You can find out about things like Optimization`NMinimizeDump`vecs by inspecting the code for Optimization`NMinimizeDump`CoreDE.
answered 4 hours ago
Michael E2Michael E2
148k12198478
$endgroup$
add a comment |
5
5
5
$begingroup$
Here's a way:
Block[{f},
f[x_, y_] := -20 E^(-0.2 Sqrt[0.5 (x^2 + y^2)]) -
E^(0.5 (Cos[2 [Pi] x] + Cos[2 [Pi] y])) + E + 20;
{fit, intermediates} =
Reap[NMinimize[{f[x, y], -5 <= x <= 5, -5 <= y <= 5}, {x, y},
MaxIterations -> 30,
Method -> {"DifferentialEvolution",
"InitialPoints" -> Tuples[Range[-5, 5], 2]},
StepMonitor :>
Sow[{Optimization`NMinimizeDump`vecs,
Optimization`NMinimizeDump`vals}]]];
]
Manipulate[
Graphics[{
PointSize[Medium],
Point[intermediates[[1, n, 1]],
VertexColors ->
ColorData["Rainbow"] /@
Rescale[intermediates[[1, n, 2]],
MinMax[intermediates[[1, All, 2]]]]]
},
PlotRange -> 5, Frame -> True],
{n, 1, Length@intermediates[[1]], 1}
]
You can find out about things like Optimization`NMinimizeDump`vecs by inspecting the code for Optimization`NMinimizeDump`CoreDE.
answered 4 hours ago
Michael E2Michael E2
148k12198478
$endgroup$
Here's a way:
Block[{f},
f[x_, y_] := -20 E^(-0.2 Sqrt[0.5 (x^2 + y^2)]) -
E^(0.5 (Cos[2 [Pi] x] + Cos[2 [Pi] y])) + E + 20;
{fit, intermediates} =
Reap[NMinimize[{f[x, y], -5 <= x <= 5, -5 <= y <= 5}, {x, y},
MaxIterations -> 30,
Method -> {"DifferentialEvolution",
"InitialPoints" -> Tuples[Range[-5, 5], 2]},
StepMonitor :>
Sow[{Optimization`NMinimizeDump`vecs,
Optimization`NMinimizeDump`vals}]]];
]
Manipulate[
Graphics[{
PointSize[Medium],
Point[intermediates[[1, n, 1]],
VertexColors ->
ColorData["Rainbow"] /@
Rescale[intermediates[[1, n, 2]],
MinMax[intermediates[[1, All, 2]]]]]
},
PlotRange -> 5, Frame -> True],
{n, 1, Length@intermediates[[1]], 1}
]
You can find out about things like Optimization`NMinimizeDump`vecs by inspecting the code for Optimization`NMinimizeDump`CoreDE.
answered 4 hours ago
Michael E2Michael E2
148k12198478
answered 4 hours ago
Michael E2Michael E2
148k12198478
answered 4 hours ago
Michael E2Michael E2
148k12198478
answered 4 hours ago
Michael E2Michael E2
148k12198478
148k12198478
add a comment |
add a comment |
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$begingroup$
Note that blocking
f(Block[{f}, ...]) isn't necessary. It was just to preventffrom being defined, which is a habit I have with single-lettter symbols on SE, esp. ones I use likef,x, etc. -- thanks for the accept!$endgroup$
– Michael E2
3 hours ago