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Diffstat (limited to 'tutorials/module_3/1_numerical_differentiation.md')
| -rw-r--r-- | tutorials/module_3/1_numerical_differentiation.md | 41 |
1 files changed, 29 insertions, 12 deletions
diff --git a/tutorials/module_3/1_numerical_differentiation.md b/tutorials/module_3/1_numerical_differentiation.md index aa45c3a..68e0953 100644 --- a/tutorials/module_3/1_numerical_differentiation.md +++ b/tutorials/module_3/1_numerical_differentiation.md @@ -11,7 +11,8 @@ s = 34 * np.exp(3 * t) Then we can use the following methods to approximate the definitive derivatives as follows. -## Forward Difference +## Finite Difference +### Forward Difference Uses the point at which we want to find the derivative and a point forwards in the array. $$ f'(x_i) = \frac{f(x_{i+1})-f(x_i)}{x_{i+1}-x_i} @@ -19,14 +20,21 @@ $$ *Note: If we apply this to an array, consider what happens at the last point.* ```python +for s in t: + ds_dt = f(x+) +``` + + + +```python # Forward difference using python arrays -dsdt = (y[1:] - y[:-1]) / (x[1:] - x[:-1]) +dsdt = (s[1:] - s[:-1]) / (t[1:] - t[:-1]) import matplotlib.pyplot as plt # Plot the function -plt.plot(x, s, label=r'$y(x)$') -plt.plot(x, dsdt, label=b'$/frac{ds}{dt}$') +plt.plot(t, s, label=r'$s(t)$') +plt.plot(t, dsdt, label=b'$/frac{ds}{dt}$') plt.xlabel('Time (t)') plt.ylabel('Displacement (s)') plt.title('Plot of $34e^{3t}$') @@ -36,7 +44,7 @@ plt.show() ``` -## Backwards Difference +### Backwards Difference Uses the point at which we want to find and the previous point in the array. $$ f'(x_i) = \frac{f(x_{i})-f(x_{i-1})}{x_i - x_{i-1}} @@ -44,23 +52,32 @@ $$ ```python -dsdt = (y[1:] - y[:-1]) / (x[1:] - x[:-1]) +dsdt = (s[1:] - s[:-1]) / (t[1:] - t[:-1]) # Plot the function -plt.plot(x, y, label=r'$y(x)$') -plt.plot(x, dydx, label=b'$/frac{ds}{dt}$') -plt.xlabel('x') -plt.ylabel('y') -plt.title('Plot of $34e^{3x}$') +plt.plot(t, s, label=r'$s(t)$') +plt.plot(t, dydx, label=b'$/frac{ds}{dt}$') + +plt.xlabel('Time (t)') +plt.ylabel('Displacement (s)') +plt.title('Plot of $34e^{3t}$') plt.grid(True) plt.legend() plt.show() ``` -## Central Difference +Try plotting both forward and backwards + +### Central Difference $$ f'(x_i) = \frac{f(x_{i+1})-f(x_{i-1})}{x_{i+1}-x_{i-1}} $$ +--- +# Advanced Derivatives +## High-Accuracy Differentiation Formulas +## Richardson Extrapolation +## Derivative of Unequally Spaced Data +## Partial Derivatives |