feat: add quadratic example

examples/caputo-derivative-quadratic.py

@@ -0,0 +1,66 @@
+# SPDX-FileCopyrightText: 2023 Alexandru Fikl <alexfikl@gmail.com>
+# SPDX-License-Identifier: MIT
+
+import numpy as np
+
+from pycaputo.derivatives import CaputoDerivative, Side
+from pycaputo.differentiation import CaputoSpectralMethod, diff
+from pycaputo.grid import make_jacobi_gauss_lobatto_points
+from pycaputo.utils import Array
+
+try:
+    import matplotlib  # noqa: F401
+except ImportError as exc:
+    raise SystemExit(0) from exc
+else:
+    from pycaputo.utils import figure, set_recommended_matplotlib
+
+    set_recommended_matplotlib()
+
+
+def f(x: Array) -> Array:
+    return x**2
+
+
+with figure("caputo-derivative-quadratic.pdf") as fig:
+    ax = fig.gca()
+
+    alphas = [
+        0.1,
+        0.2,
+        0.3,
+        0.4,
+        0.5,
+        0.6,
+        0.7,
+        0.8,
+        0.9,
+        1.1,
+        1.2,
+        1.3,
+        1.4,
+        1.5,
+        1.6,
+        1.7,
+        1.8,
+        1.9,
+    ]
+
+    p = make_jacobi_gauss_lobatto_points(32, a=0.0, b=2.0)
+
+    for alpha in alphas:
+        d = CaputoDerivative(order=alpha, side=Side.Left)
+        method = CaputoSpectralMethod(d)
+
+        df_num = diff(method, f, p)
+        ax.plot(p.x, df_num, color="k", alpha=0.2)
+
+    ax.plot(p.x, f(p.x), label=r"$\alpha = 0$")
+    ax.plot(p.x, 2.0 * p.x, label=r"$\alpha = 1$")
+    ax.plot(p.x, np.full(p.x.shape, 2.0), label=r"$\alpha = 2$")
+
+    ax.set_xlabel("$x$")
+    ax.set_ylabel(r"$D^\alpha_C[f](x)$")
+    ax.legend()
+
+# }}}

pycaputo/grid.py

@@ -59,7 +59,7 @@ def make_stretched_points(
     where :math:`A` is the *strength* and :math:`x_c = 1/2` for the midpoint.
 
     :arg n: number of points in :math:`[a, b]`.
-    :arg strength: a positive number that constrols the clustering of points at
+    :arg strength: a positive number that controls the clustering of points at
         the midpoint, i.e. a larger number denotes more points.
     """
     x = np.linspace(0, 1, n)

pycaputo/interpolation.py

@@ -77,7 +77,7 @@ def determine_truncation_error(offsets: Array, x: float, h: float = 1.0) -> floa
         f(x) - \sum_{k \in \text{offsets}} \ell_k(x) f_{m + k} =
         c \frac{\mathrm{d}^n f}{\mathrm{d} x^n}(\xi),
 
-    where the constant :math:`c` is approxiated as the truncation error. The
+    where the constant :math:`c` is approximated as the truncation error. The
     Error itself also depends on the derivatives of :math:`f` at a undetermined
     point :math:`\xi` in the interpolation interval.