gaps in plot of piecewise function
in [Mathematica]
|
Prev: ZTransform for a non-causal unstable signal. How to make Mathematica
Next: Setting default zoom factor in notebooks
From: Sjoerd C. de Vries on 12 Mar 2010 07:11 The latter functions are equal for all x. Doesn't hold for the former two. If you increase PlotPoints to well over a couple of hundred with MaxRecursion at 15 I don't see a gap. Cheers -- Sjoerd > -----Original Message----- > From: Patrick Scheibe [mailto:pscheibe(a)trm.uni-leipzig.de] > Sent: 11 March 2010 14:23 > To: David Park; Benjamin Hell; Sjoerd C. de Vries; Peter Pein; gekko; > Matthias Hunstig > Cc: mathgroup(a)smc.vnet.net > Subject: Re: Re: gaps in plot of piecewise > function > > Hi again, > > I hope everyone saw now that Exclusions->None or using not Piecewise > but > e.g. Which will do the trick. In the documentation it sounds to me that > many functions are generally connected to Piecewise (look at Properties > and Relations in the Piecewise doc). > > My question is, why would it be wrong to connect the plot in Piecewise > when the Limits are the same? Following example: > > Manipulate[ > Plot[Piecewise[{{Exp[1] x, x < 1}}, Exp[x]], {x, 0, 3}, > MaxRecursion -> mr, MeshStyle -> {Red, PointSize[0.005]}, > Mesh -> All, PlotPoints -> pp, > ImageSize -> 500], {{pp, 5, "PlotPoints"}, 3, 30, > 1}, {{mr, 1, "MaxRecursion"}, 1, 10, 1}] > > The function has the same limit at x->1 and the same derivative. I > would > clearly expect a plot without a gap even without the Exclusions > options. > Where am I wrong? > > Is it too unpredictable to check at least numerically the limits? > But why is this working? > > Manipulate[ > Plot[Piecewise[{{Sin[x], x < 1.334}}, Cos[ x - Pi/2]], {x, 0, 3}, > MaxRecursion -> mr, MeshStyle -> {Red, PointSize[0.005]}, > Mesh -> All, PlotPoints -> pp, > ImageSize -> 500], {{pp, 5, "PlotPoints"}, 3, 30, > 1}, {{mr, 1, "MaxRecursion"}, 1, 10, 1}] > > What bothers me is that when using PiecewiseExpand you get an > equivalent > presentation of one and the same function but you get different plots > in > an, say not really predictable way. > > Cheers > Patrick > > On Thu, 2010-03-11 at 06:34 -0500, David Park wrote: > > I'm not certain of the exact underlying mechanics, but basically > because of > > the steep curve as x -> 2 from below, and the piecewise function, > > Mathematica sees a discontinuity and leaves a gap. The way to > overcome this > > is to use the Exclusions option. > > > > s[x_] := Piecewise[{{-Sqrt[2]/2*Sqrt[-x + 0.5] + 2, x < 0.5}, {2, > > x >= 0.5}}]; > > > > Plot[s[x], {x, 0, 1}, > > Exclusions -> None, > > Frame -> True, > > PlotRangePadding -> .1] > > > > > > David Park > > djmpark(a)comcast.net > > http://home.comcast.net/~djmpark/ > > > > > > From: Benjamin Hell [mailto:hell(a)exoneon.de] > > > > Hi, > > I want to plot a piecewise function, but I don't want any gaps to > appear > > at the junctures. An easy example is: > > > > s[x_] := Piecewise[{{-Sqrt[2]/2*Sqrt[-x + 0.5] + 2, x < 0.5}, {2, x > >= > > 0.5}}]; > > Plot[s[x], {x, 0, 1}] > > > > It should be clear, that the piecewise function defined above is > > continuous, even at x=0.5. So there should not be any gaps appearing > in > > the plot, but they do. Maybe it's a feature of mathematica, but > > nevertheless I want to get rid of the gaps. Any suggestions on how to > > achieve that. > > > > > > Thanks in advance. > > > > > >
From: Patrick Scheibe on 12 Mar 2010 07:13 Hi again, I hope everyone saw now that Exclusions->None or using not Piecewise but e.g. Which will do the trick. In the documentation it sounds to me that many functions are generally connected to Piecewise (look at Properties and Relations in the Piecewise doc). My question is, why would it be wrong to connect the plot in Piecewise when the Limits are the same? Following example: Manipulate[ Plot[Piecewise[{{Exp[1] x, x < 1}}, Exp[x]], {x, 0, 3}, MaxRecursion -> mr, MeshStyle -> {Red, PointSize[0.005]}, Mesh -> All, PlotPoints -> pp, ImageSize -> 500], {{pp, 5, "PlotPoints"}, 3, 30, 1}, {{mr, 1, "MaxRecursion"}, 1, 10, 1}] The function has the same limit at x->1 and the same derivative. I would clearly expect a plot without a gap even without the Exclusions options. Where am I wrong? Is it too unpredictable to check at least numerically the limits? But why is this working? Manipulate[ Plot[Piecewise[{{Sin[x], x < 1.334}}, Cos[ x - Pi/2]], {x, 0, 3}, MaxRecursion -> mr, MeshStyle -> {Red, PointSize[0.005]}, Mesh -> All, PlotPoints -> pp, ImageSize -> 500], {{pp, 5, "PlotPoints"}, 3, 30, 1}, {{mr, 1, "MaxRecursion"}, 1, 10, 1}] What bothers me is that when using PiecewiseExpand you get an equivalent presentation of one and the same function but you get different plots in an, say not really predictable way. Cheers Patrick On Thu, 2010-03-11 at 06:34 -0500, David Park wrote: > I'm not certain of the exact underlying mechanics, but basically because of > the steep curve as x -> 2 from below, and the piecewise function, > Mathematica sees a discontinuity and leaves a gap. The way to overcome this > is to use the Exclusions option. > > s[x_] := Piecewise[{{-Sqrt[2]/2*Sqrt[-x + 0.5] + 2, x < 0.5}, {2, > x >= 0.5}}]; > > Plot[s[x], {x, 0, 1}, > Exclusions -> None, > Frame -> True, > PlotRangePadding -> .1] > > > David Park > djmpark(a)comcast.net > http://home.comcast.net/~djmpark/ > > > From: Benjamin Hell [mailto:hell(a)exoneon.de] > > Hi, > I want to plot a piecewise function, but I don't want any gaps to appear > at the junctures. An easy example is: > > s[x_] := Piecewise[{{-Sqrt[2]/2*Sqrt[-x + 0.5] + 2, x < 0.5}, {2, x >= > 0.5}}]; > Plot[s[x], {x, 0, 1}] > > It should be clear, that the piecewise function defined above is > continuous, even at x=0.5. So there should not be any gaps appearing in > the plot, but they do. Maybe it's a feature of mathematica, but > nevertheless I want to get rid of the gaps. Any suggestions on how to > achieve that. > > > Thanks in advance. > > >
From: Patrick Scheibe on 13 Mar 2010 07:55
Hi, > The latter functions are equal for all x. Doesn't hold for the former two. yep, but Sin[x] == (I/2)/E^(I*x) - (I/2)*E^(I*x) // Simplify Manipulate[ Plot[Piecewise[{{Sin[x], x < 1.334}}, (I/2)/E^(I*x) - (I/2)* E^(I*x)], {x, 0, 3}, MaxRecursion -> mr, MeshStyle -> {Red, PointSize[0.005]}, Mesh -> All, PlotPoints -> pp, ImageSize -> 500], {{pp, 5, "PlotPoints"}, 3, 30, 1}, {{mr, 1, "MaxRecursion"}, 1, 10, 1}] > If you increase PlotPoints to well over a couple of hundred with > MaxRecursion at 15 I don't see a gap. "See" doesn't mean it's not there. Please set PlotPoints to 200 and MaxRecursion to 15 and check the zoomed result Manipulate[ Plot[Piecewise[{{Exp[1] x, x < 1}}, Exp[x]], {x, 0, 3}, MaxRecursion -> mr, MeshStyle -> {Red, PointSize[0.005]}, Mesh -> All, PlotPoints -> pp, ImageSize -> 500, PlotRange -> {{1 - zoom, 1 + zoom}, Automatic}], {{pp, 5, "PlotPoints"}, 3, 200, 1}, {{mr, 1, "MaxRecursion"}, 1, 15, 1}, {{zoom, 1}, 1, 0}] Cheers Patrick > Cheers -- Sjoerd > > > -----Original Message----- > > From: Patrick Scheibe [mailto:pscheibe(a)trm.uni-leipzig.de] > > Sent: 11 March 2010 14:23 > > To: David Park; Benjamin Hell; Sjoerd C. de Vries; Peter Pein; gekko; > > Matthias Hunstig > > Cc: mathgroup(a)smc.vnet.net > > Subject: Re: Re: gaps in plot of piecewise > > function > > > > Hi again, > > > > I hope everyone saw now that Exclusions->None or using not Piecewise > > but > > e.g. Which will do the trick. In the documentation it sounds to me that > > many functions are generally connected to Piecewise (look at Properties > > and Relations in the Piecewise doc). > > > > My question is, why would it be wrong to connect the plot in Piecewise > > when the Limits are the same? Following example: > > > > Manipulate[ > > Plot[Piecewise[{{Exp[1] x, x < 1}}, Exp[x]], {x, 0, 3}, > > MaxRecursion -> mr, MeshStyle -> {Red, PointSize[0.005]}, > > Mesh -> All, PlotPoints -> pp, > > ImageSize -> 500], {{pp, 5, "PlotPoints"}, 3, 30, > > 1}, {{mr, 1, "MaxRecursion"}, 1, 10, 1}] > > > > The function has the same limit at x->1 and the same derivative. I > > would > > clearly expect a plot without a gap even without the Exclusions > > options. > > Where am I wrong? > > > > Is it too unpredictable to check at least numerically the limits? > > But why is this working? > > > > Manipulate[ > > Plot[Piecewise[{{Sin[x], x < 1.334}}, Cos[ x - Pi/2]], {x, 0, 3}, > > MaxRecursion -> mr, MeshStyle -> {Red, PointSize[0.005]}, > > Mesh -> All, PlotPoints -> pp, > > ImageSize -> 500], {{pp, 5, "PlotPoints"}, 3, 30, > > 1}, {{mr, 1, "MaxRecursion"}, 1, 10, 1}] > > > > What bothers me is that when using PiecewiseExpand you get an > > equivalent > > presentation of one and the same function but you get different plots > > in > > an, say not really predictable way. > > > > Cheers > > Patrick > > > > On Thu, 2010-03-11 at 06:34 -0500, David Park wrote: > > > I'm not certain of the exact underlying mechanics, but basically > > because of > > > the steep curve as x -> 2 from below, and the piecewise function, > > > Mathematica sees a discontinuity and leaves a gap. The way to > > overcome this > > > is to use the Exclusions option. > > > > > > s[x_] := Piecewise[{{-Sqrt[2]/2*Sqrt[-x + 0.5] + 2, x < 0.5}, {2, > > > x >= 0.5}}]; > > > > > > Plot[s[x], {x, 0, 1}, > > > Exclusions -> None, > > > Frame -> True, > > > PlotRangePadding -> .1] > > > > > > > > > David Park > > > djmpark(a)comcast.net > > > http://home.comcast.net/~djmpark/ > > > > > > > > > From: Benjamin Hell [mailto:hell(a)exoneon.de] > > > > > > Hi, > > > I want to plot a piecewise function, but I don't want any gaps to > > appear > > > at the junctures. An easy example is: > > > > > > s[x_] := Piecewise[{{-Sqrt[2]/2*Sqrt[-x + 0.5] + 2, x < 0.5}, {2, x > > >= > > > 0.5}}]; > > > Plot[s[x], {x, 0, 1}] > > > > > > It should be clear, that the piecewise function defined above is > > > continuous, even at x=0.5. So there should not be any gaps appearing > > in > > > the plot, but they do. Maybe it's a feature of mathematica, but > > > nevertheless I want to get rid of the gaps. Any suggestions on how to > > > achieve that. > > > > > > > > > Thanks in advance. > > > > > > > > > > |