LogPlot axes labels destroyed when working in high precision

Question

Bug introduced in 7.0 or earlier and fixed in 11.0.0

---

(I'm using Mathematica 8.)

I have a Taylor series:

poly = Normal[Series[E^x, {x, 0, 10}]]

I want to produce a log-linear plot of the error. This is easy enough with the following code:

LogPlot[Abs[E^x - poly], {x, -1, 1}]

This produces

Now, I want to plot even smaller values of the error (in particular I want the plot to be sensible near zero), so I tell LogPlot to use high precision as follows:

LogPlot[Abs[E^x - poly], {x, -1, 1}, WorkingPrecision -> 30]

However this destroys the labeling on the y-axis:

Does anyone know what has gone wrong here? How do I fix it?

Answer 1

This is not simply a mislabeling of the axes. More than that is going on: the plot produced is not even logarithmic. Let's try to use the default (non-log-transformed tick marks):

First, with MachinePrecision (correct result):

Show[
 LogPlot[Abs[E^x - poly], {x, -1, 1}, WorkingPrecision -> MachinePrecision],
 Ticks -> Automatic
]

Then with higher precision (incorrect result):

Show[
 LogPlot[Abs[E^x - poly], {x, -1, 1}, WorkingPrecision -> 30],
 Ticks -> Automatic
]

I don't think it's worth digging into how LogPlot works, as at this point this clearly seems to be a bug.

---

You can work around it by using Plot instead of LogPlot:

Plot[Log@Abs[E^x - poly], {x, -1, 1}, WorkingPrecision -> 30]

But then you have to do re-label the axes yourself (CustomTicks / LevelScheme are helpful packages). If you don't mind losing adaptive plotting, you can generate the points to be shown yourself and us ListLogPlot:

ListLogPlot[Table[Evaluate@Abs[E^x - poly], {x, -1, 1, 0.01`30}]]

(You'd probably want Joined -> True here, but seeing where the points are helps you tune the plot, so I didn't include it now.)

Answer 2

Although Szabolcs warned "I don't think it's worth digging into how LogPlot works, as at this point this clearly seems to be a bug" I thought I would give a go. I found that:

System`LogPlot calls Graphics`LogPlotDump`scaledPlot which calls System`Plot which calls System`ProtoPlotDump`iPlot which calls Visualization`Core`Plot which is not readable.

We can demonstrate that the problem exists in this innermost function by making direct use of "MappingFunctions" which is what LogPlot ultimately becomes (among a few other settings):

poly = Normal[Series[E^x, {x, 0, 10}]];

Table[
 Visualization`Core`Plot[Abs[E^x - poly], {x, -1, 1}, WorkingPrecision -> wp, 
  Method -> {"MappingFunctions" -> {{#1, Log[#2]} &, {#1, #2} &}}],
 {wp, {MachinePrecision, 30}}
]

This bug therefore appears to be beyond our reach to fix. :-(

Answer 3

neat little work-around,

poly[x_] = Normal[Series[E^x, {x, 0, 10}]];
LogPlot[(Abs[E^# - poly[#]] &@N[Rationalize[x, 0], 20]), {x, -1, 1}]

Answer 4

EDIT: Problem persists with Mathematica version "10.3.1 for Mac OS X x86 (64-bit) (December 9, 2015)"

THIS IS NOT AN ANSWER but rather an extended comment providing an update for Mma v10.1.

In looking at the issue referenced in LogPlot does not show all points, I noticed that use of WorkingPrecision causes really odd results on my system (Mac OS 10.10.4) with Mathematica 10.1. A related issue with LogLogPlot and version 10 is documented in WorkingPrecision in LogLogPlot creates wrong plots.

This problem appears to have gotten worse since the frame ticks and labels are not displayed at all--although they appear in a saved file using Save Selection As...

$Version

"10.1.0 for Mac OS X x86 (64-bit) (March 24, 2015)"

LogPlot[1/x, {x, 10^-12, 1}, PlotRange -> {{-.1, 1}, All}, Frame -> True, 
 WorkingPrecision -> $MachinePrecision]

AS DISPLAYED:

AS SAVED using Save Selection As...: