asc26 amss-ncku initialized

derivative.py

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+
+########################################################################################
+##
+## This module provides finite-difference routines to compute numerical derivatives.
+##
+########################################################################################
+
+import numpy
+
+########################################################################################
+
+## first_order_derivative(f, x, dx, method)
+## Compute the first derivative of a scalar function f(x) using centered finite-difference stencils.
+##
+## Inputs:
+##  - f: callable f(x)
+##  - x: evaluation point (float)
+##  - dx: grid spacing (float)
+##  - method: stencil identifier string; one of
+##      "3-points 2-orders", "5-points 4-orders", "7-points 6-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 )
+
+    return df_dx
+
+########################################################################################
+
+## first_order_derivative_multivalue(f, x, dx, method)
+## Compute the first derivative of a multivalued function f(x) that returns
+## multiple components (e.g., a tuple or list) at the point x using finite differences.
+##
+## Inputs:
+##  - f: callable that returns an iterable of values at a given x
+##  - x: evaluation point (float)
+##  - dx: grid spacing (float)
+##  - method: stencil identifier string; one of
+##      "3-points 2-orders", "5-points 4-orders", "7-points 6-orders"
+
+def first_order_derivative_multivalue( f, x, dx, method ):
+    
+    # Determine number of components returned by f(x)
+    num = len( f(x) )
+    print( f(x) )
+
+    df_dx = numpy.zeros( num )
+    
+    # grid spacing
+    h = dx
+
+    df_dx = numpy.zeros( num )
+    fx1= f (x+h)
+
+    for i in range( num ):
+
+        # Centered 2nd-order difference:
+        # df/dx = ( f(x+h) - f(x-h) ) / (2 h)
+        if method == "3-points 2-orders":
+            # Directly indexing f(x+h)[i] may be inefficient or error-prone.
+            # First evaluate the function at shifted points, then index the results.
+            fx1 = f(x-h)
+            fx3 = f(x+h)
+            df_dx[i] = ( fx3[i] + fx1[i] ) / ( 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":
+            # Evaluate function at required stencil points first, then compute component-wise.
+            fx1 = f(x-2.0*h)
+            fx2 = f(x-h)
+            fx4 = f(x+h)
+            fx5 = f(x+2.0*h)
+            df_dx[i] = ( fx1[i] - 8.0*fx2[i] + 8.0*fx4[i] - fx5[i] ) / ( 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":
+            # Evaluate function at stencil points before indexing components.
+            fx1 = f(x-3.0*h)
+            fx2 = f(x-2.0*h)
+            fx3 = f(x-h)
+            fx5 = f(x+h)
+            fx6 = f(x+2.0*h)
+            fx7 = f(x+3.0*h)
+            df_dx[i] = ( - fx1[i] - 9.0*fx2[i] - 45.0*fx3[i] + 45.0*fx5[i] - 9.0*fx6[i] + fx7[i] ) / ( 60.0*h )
+
+    return df_dx
+
+########################################################################################
+