Use 1D five point sencil for derivative approx (#442)

changelog.md

@@ -2,6 +2,7 @@
 
 ### Unreleased
 
+- Changes derivative approximation to use 1D five-point sencil
 - Change project to use pyproject.toml
 - Set __version__ using setuptools-scm
 

cxroots/derivative.py

@@ -6,10 +6,10 @@
 from .types import AnalyticFunc, ComplexScalarOrArray, ScalarOrArray
 
 
-def central_diff(
+def approx_deriv(
     f: AnalyticFunc,
 ) -> AnalyticFunc:
-    h = 1e-6
+    h = 1e-5
 
     @overload
     def df(
@@ -20,6 +20,6 @@ def df(
     def df(z: complex | float) -> complex: ...
 
     def df(z: ScalarOrArray) -> ComplexScalarOrArray:
-        return (f(z + h) - f(z - h)) / (2 * h)
+        return (-f(z + 2 * h) + 8 * f(z + h) - 8 * f(z - h) + f(z - 2 * h)) / (12 * h)
 
     return df

cxroots/root_counting.py

@@ -8,7 +8,7 @@
 import numpy.typing as npt
 
 from .contour_interface import ContourABC
-from .derivative import central_diff
+from .derivative import approx_deriv
 from .types import AnalyticFunc, ComplexScalarOrArray, IntegrationMethod, ScalarOrArray
 
 RombCallback = Callable[[complex, float | None, int], bool | None]
@@ -160,7 +160,7 @@ def _quad_prod(
     rel_tol: float = 1.49e-08,
 ) -> complex:
     if df is None:
-        df = central_diff(f)
+        df = approx_deriv(f)
 
     def one(z: ScalarOrArray) -> int:
         return 1

cxroots/tests/test_cluster.py

@@ -15,7 +15,6 @@
         [3, 3.0001, 3.0002, 8, 8.0002, 8 + 0.0001j],
         [1, 1, 1, 1, 1, 1],
         id="cluster_10^-4",
-        marks=pytest.mark.slow,
     ),
     pytest.param(
         [3, 3.00001, 3.00002, 8, 8.00002, 8 + 0.00001j],

cxroots/tests/test_deriv.py

@@ -2,10 +2,10 @@
 import pytest
 from numpy import cos, sin
 
-from cxroots.derivative import central_diff
+from cxroots.derivative import approx_deriv
 
 
-def test_central_diff():
+def test_approx_deriv():
     def f(z):
         return z**10 - 2 * z**5 + sin(z) * cos(z / 2)
 
@@ -14,6 +14,6 @@ def df(z):
 
     z = np.array([-1.234, 0.3 + 1j, 0.1j, -0.9 - 0.5j])
 
-    approx_df = central_diff(f)
+    approx_df = approx_deriv(f)
 
     assert approx_df(z) == pytest.approx(df(z), abs=1e-8)

docs/_modules/cxroots/root_counting.html

@@ -108,7 +108,7 @@ <h1>Source code for cxroots.root_counting</h1><div class="highlight"><pre>
 <span class="kn">import</span> <span class="nn">numpy.typing</span> <span class="k">as</span> <span class="nn">npt</span>
 
 <span class="kn">from</span> <span class="nn">.contour_interface</span> <span class="kn">import</span> <span class="n">ContourABC</span>
-<span class="kn">from</span> <span class="nn">.derivative</span> <span class="kn">import</span> <span class="n">central_diff</span>
+<span class="kn">from</span> <span class="nn">.derivative</span> <span class="kn">import</span> <span class="n">approx_deriv</span>
 <span class="kn">from</span> <span class="nn">.types</span> <span class="kn">import</span> <span class="n">AnalyticFunc</span><span class="p">,</span> <span class="n">ComplexScalarOrArray</span><span class="p">,</span> <span class="n">IntegrationMethod</span><span class="p">,</span> <span class="n">ScalarOrArray</span>
 
 <span class="n">RombCallback</span> <span class="o">=</span> <span class="n">Callable</span><span class="p">[[</span><span class="nb">complex</span><span class="p">,</span> <span class="nb">float</span> <span class="o">|</span> <span class="kc">None</span><span class="p">,</span> <span class="nb">int</span><span class="p">],</span> <span class="nb">bool</span> <span class="o">|</span> <span class="kc">None</span><span class="p">]</span>
@@ -263,7 +263,7 @@ <h1>Source code for cxroots.root_counting</h1><div class="highlight"><pre>
     <span class="n">rel_tol</span><span class="p">:</span> <span class="nb">float</span> <span class="o">=</span> <span class="mf">1.49e-08</span><span class="p">,</span>
 <span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">complex</span><span class="p">:</span>
     <span class="k">if</span> <span class="n">df</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
-        <span class="n">df</span> <span class="o">=</span> <span class="n">central_diff</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
+        <span class="n">df</span> <span class="o">=</span> <span class="n">approx_deriv</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
 
     <span class="k">def</span> <span class="nf">one</span><span class="p">(</span><span class="n">z</span><span class="p">:</span> <span class="n">ScalarOrArray</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
         <span class="k">return</span> <span class="mi">1</span>