Optimize temperature tables for accuracy

Temperature tables are emitted by selecting arbitrary sample values
to be used for the linear lookup table.  This is fine in the range
where the thermistor produces linear output, but it is markedly wrong
near the extremes where the thermister output begins to curve.

Introduce a new sample selector which chooses samples based on the "most
incorrect estimate" and improves from there to ensure we get a cleaner
approximation across the selected range.

Traumflug: this topic is tracked here:
https://github.com/Traumflug/Teacup_Firmware/pull/208

configtool/thermistortablefile.py

@@ -99,9 +99,41 @@ def BetaTable(ofp, params, names, settings, finalTable):
 
   hiadc = thrm.setting(0)[0]
   N = int(settings.numTemps)
-  step = int(hiadc / (N - 1))
 
-  for i in range(1, int(hiadc), step):
+  # This is a variation of the Ramer-Douglas-Peucker algorithm, see
+  # https://en.wikipedia.org/wiki/Ramer%E2%80%93Douglas%E2%80%93Peucker_algorithm
+  #
+  # It works like this:
+  #
+  #   - Calculate all (1024) ideal values.
+  #   - Insert the two extremes into our sample list.
+  #   - Calculate the linear approximation of the remaining values.
+  #   - Insert the correct value for the "most-wrong" estimation into our
+  #     sample list.
+  #   - Repeat until "N" values are chosen as requested.
+
+  # Calculate actual temps for all ADC values.
+  actual = dict([(x, thrm.temp(1.0 * x)) for x in range(1, int(hiadc + 1))])
+
+  # Build a lookup table starting with the extremes.
+  lookup = dict([(x, actual[x]) for x in [1, int(hiadc)]])
+
+  A = 1
+  B = int(hiadc)
+  error = dict({})
+  while len(lookup) < N:
+    error.update(dict([(x, abs(actual[x] - LinearTableEstimate(lookup, x)))
+                       for x in range(A + 1, B)]))
+
+    # Correct the most-wrong lookup value.
+    next = max(error, key = error.get)
+    lookup[next] = actual[next]
+
+    # Prepare to update the error range.
+    A = before(lookup, next)
+    B = after(lookup, next)
+
+  for i in sorted(lookup.keys()):
     t = int(thrm.temp(i))
     if t is None:
       ofp.output("// ERROR CALCULATING THERMISTOR VALUES AT ADC %d" % i)
@@ -112,7 +144,7 @@ def BetaTable(ofp, params, names, settings, finalTable):
 
     vTherm = i * vadc / 1024
     ptherm = vTherm * vTherm / r
-    if i + step >= int(hiadc):
+    if i == max(lookup):
       c = " "
     else:
       c = ","
@@ -162,3 +194,20 @@ def SteinhartHartTable(ofp, params, names, settings, finalTable):
     ofp.output("  }")
   else:
     ofp.output("  },")
+
+def after(lookup, value):
+  return min([x for x in lookup.keys() if x > value])
+
+def before(lookup, value):
+  return max([x for x in lookup.keys() if x < value])
+
+def LinearTableEstimate(lookup, value):
+  if value in lookup:
+    return lookup[value]
+
+  # Estimate result with linear estimation algorithm.
+  x0 = before(lookup, value)
+  x1 = after(lookup, value)
+  y0 = lookup[x0]
+  y1 = lookup[x1]
+  return ((value - x0) * y1 + (x1 - value) * y0) / (x1 - x0)