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I'd like to plot 

Abs[s]/λ - 1 + Log[λ/s]

as a function of $s > 0$, for some fixed value for $\lambda$, say 0.1. This function behaves essentially linearly for $s > \lambda$, and the log-linear plot is linear for $s < \lambda$. It would therefore be nice to see the plot from $s=e^{-8}$ to $s=8$. There should be roughly the same number of ticks on both sides, if I'm not mistaken.

I'd like to depict this by having a plot that is log linear to the left of $\lambda$, and linear to the right of $\lambda$. Any advice?

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  • 1
    $\begingroup$ Your function assumes complex values for $\lambda=0.1$ and $s<0$. Could you check your expressions? $\endgroup$ – MarcoB May 25 '18 at 14:37
  • $\begingroup$ s must be nonnegative. I've fixed the associated typo. $\endgroup$ – Wannabe Mathematician May 26 '18 at 1:04
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Using the scaling functions {f[λ], if[λ]} from Brett's excellent answer and adding a few embellishments:

Quiet[With[{λ = 0.3}, 
 {lscale, rscale} = Thread[{0,  NMaxValue[{Abs[s]/λ - 1 + Log[λ/s] , #}, s] & /@ 
   {10^(-4) <= s <= λ , λ <= s <= 8}}];
 plot = Plot[If[s < λ, Abs[s]/λ - 1 + Log[λ/s], 
       Rescale[Abs[s]/λ - 1 + Log[λ/s], rscale, lscale ]], {s, 10^-4, 8}, 
    ScalingFunctions -> {{f[λ], if[λ]}, None}, 
    Frame -> True, 
    FrameTicks -> {{Automatic,  Charting`FindTicks[lscale, rscale]}, {All, All}}, 
    Mesh -> {{λ}}, MeshShading -> {Red, Blue}, 
    FrameStyle -> {{Red, Blue}, {Automatic, Automatic}}];
 ticks = FrameTicks /. Options[plot, FrameTicks];
 ticks[[2, 1]] = Join[ticks[[2, 1]], {λ, λ}, {#, #} & /@ Range[8]]; 
 ticks[[2, 2]] = ticks[[2, 1]];
 Show[plot, FrameTicks -> ticks, ImageSize -> 500]]]

enter image description here

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Here's my attempt, using ScalingFunctions.

The first step is to define a function (and it's inverse) that is linear for inputs greater than $\lambda$, and is log-scaled for input below, with a shift so that we preserve continuity:

f[λ_][a_] := If[a < λ, Log[a] + λ - Log[λ], a]
if[λ_][a_] := If[a < λ, E^(-λ + a) λ, a]

Just for fun, I've included λ as a parameter to the scaling functions.

Now we just plot as usual:

With[{λ = 0.1}, 
     Plot[Abs[s]/λ - 1 + Log[λ/s], {s, 10^-4, 8}, 
          ScalingFunctions -> {{f[λ], if[λ]}, None}
     ]
] // VisualQuiet

enter image description here

The ticks aren't perhaps what you'd want, since in this case Plot decided to go for powers of 10 across the domain; that could be changed with a custom Ticks specification.

Also, at least in V11.3, the graphic pinks because of some apparently benign recursion error I haven't debugged yet, so I'm throwing in a utility I have for telling the frontend to not turn the graphic pink if there are errors:

VisualQuiet[expr_] := Style[expr, AutoStyleOptions -> {
    "FormattingErrorTooltips" -> False, 
    "HighlightFormattingErrors" -> False
}]
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It is hard to see how to improve on side-by-side plots:

With[{λ = 0.1}, GraphicsRow@{
   LogLinearPlot[Abs[s]/λ - 1 + Log[λ/s], {s, Exp[-8], λ}],
   Plot[Abs[s]/λ - 1 + Log[λ/s], {s, λ, 8}]
   }]
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  • $\begingroup$ I'm now figuring out how to make the two graphs overlap so that I actually realize my goal. $\endgroup$ – Wannabe Mathematician May 26 '18 at 1:06
  • $\begingroup$ The solution of @kglr gives you that (and elegantly), but we usually plot for communication, and I suspect communication will be much more rapid with side-by-side separate plots. (E.g., ask why color is needed to make the combined plot comprehensible.) $\endgroup$ – Alan May 26 '18 at 13:22

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