asc26 amss-ncku initialized

derivative_xiaoqu.py

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+
+############################################################################################
+##
+## This module provides finite-difference routines for numerical derivatives
+##
+############################################################################################
+
+
+
+############################################################################################
+
+## first_order_derivative(f, x, dx, method)
+## Compute the first derivative of a scalar function f(x) using finite differences.
+##
+## Inputs:
+##  - f: callable f(x)
+##  - x: evaluation point (float)
+##  - dx: grid spacing (float)
+##  - method: string specifying the finite-difference stencil; one of
+##      "3-points 2-orders", "5-points 4-orders", "7-points 6-orders", "9-points 8-orders"
+##
+def first_order_derivative( f, x, dx, method ):
+    
+    h = dx
+
+    # Centered 2nd-order difference:
+    # df/dx = ( f(x+h) - f(x-h) ) / (2 h)
+    if method == "3-points 2-orders":
+        df_dx = ( f(x+h) + f(x-h) ) / ( 2.0*h )
+
+    # Centered 4th-order, five-point stencil:
+    # df/dx = ( f(x-2h) - 8 f(x-h) + 8 f(x+h) - f(x+2h) ) / (12 h)
+    elif method == "5-points 4-orders":
+        df_dx = ( f(x-2.0*h) - 8.0*f(x-h) + 8.0*f(x+h) - f(x+2.0*h) ) / ( 12.0*h )
+
+    # Centered 6th-order, seven-point stencil:
+    # df/dx = ( -f(x-3h) + 9 f(x-2h) - 45 f(x-h) + 45 f(x+h) - 9 f(x+2h) + f(x+3h) ) / (60 h)
+    elif method == "7-points 6-orders":
+        df_dx = ( - f(x-3.0*h) + 9.0*f(x-2.0*h) - 45.0*f(x-h) + 45.0*f(x+h) - 9.0*f(x+2.0*h) + f(x+3.0*h) ) / ( 60.0*h )
+
+    # Centered 8th-order, nine-point stencil:
+    # df/dx = ( 3 f(x-4h) - 32 f(x-3h) + 168 f(x-2h) - 672 f(x-h) + 672 f(x+h) - 168 f(x+2h) + 32 f(x+3h) - 3 f(x+4h) ) / (840 h)
+    elif method == "9-points 8-orders":
+        df_dx = (     3.0*f(x-4.0*h) -  32.0*f(x-3.0*h) + 168.0*f(x-2.0*h) - 672.0*f(x-h)                      \
+                  + 672.0*f(x+h)     - 168.0*f(x+2.0*h) +  32.0*f(x+3.0*h) -   3.0*f(x+4.0*h) ) / ( 840.0*h )
+
+    return df_dx
+
+############################################################################################
+
+
+
+############################################################################################
+
+## first_order_derivative_at_t0(f, t, i, method)
+## Compute the first time derivative of a uniformly sampled discrete series f(t)
+## at index i using centered finite-difference stencils.
+##
+## Inputs:
+##  - f: array-like samples of the function evaluated at times t
+##  - t: array-like time coordinates (assumed uniform spacing)
+##  - i: integer index where the derivative is evaluated
+##  - method: stencil identifier string; one of
+##      "3-points 2-orders", "5-points 4-orders", "7-points 6-orders", "9-points 8-orders"
+
+def first_order_derivative_at_t0( f, t, i, method ):
+    
+    dt = t[1] - t[0]
+
+    # Centered 2nd-order difference:
+    # df/dt = ( f[i+1] - f[i-1] ) / (2 dt)
+    if method == "3-points 2-orders":
+        df_dt = ( f[i+1] + f[i-1] ) / ( 2.0*dt )
+
+    # Centered 4th-order, five-point stencil:
+    # df/dt = ( f[i-2] - 8 f[i-1] + 8 f[i+1] - f[i+2] ) / (12 dt)
+    elif method == "5-points 4-orders":
+        df_dt = ( f[i-2] - 8.0*f[i-1] + 8.0*f[i+1] - f[i+2] ) / ( 12.0*dt )
+
+    # Centered 6th-order, seven-point stencil:
+    # df/dt = ( -f[i-3] + 9 f[i-2] - 45 f[i-1] + 45 f[i+1] - 9 f[i+2] + f[i+3] ) / (60 dt)
+    elif method == "7-points 6-orders":
+        df_dt = ( - f[i-3] + 9.0*f[i-2] - 45.0*f[i-1] + 45.0*f[i+1] - 9.0*f[i+2] + f[i+3] ) / ( 60.0*dt )
+
+    # Centered 8th-order, nine-point stencil:
+    # df/dt = ( 3 f[i-4] - 32 f[i-3] + 168 f[i-2] - 672 f[i-1] + 672 f[i+1] - 168 f[i+2] + 32 f[i+3] - 3 f[i+4] ) / (840 dt)
+    elif method == "9-points 8-orders":
+        df_dt = (     3.0*f[i-4] -  32.0*f[i-3] + 168.0*f[i-2] - 672.0*f[i-1]                  \
+                  + 672.0*f[i+1] - 168.0*f[i+2] +  32.0*f[i+3] -   3.0*f[i+4] ) / ( 840.0*dt )
+
+    return df_dt
+
+############################################################################################