added an example that explains how to compute the Taylor series expan…

…sion of the elements of a Jacobian matrix

documentation/sphinx/examples/taylor_series_of_jacobian.py

@@ -0,0 +1,56 @@
+import numpy
+import algopy
+from algopy import CGraph, UTPM, Function
+
+def eval_g(x, y):
+    """ some vector-valued function """
+    retval = algopy.zeros(3, dtype=x)
+    retval[0] = algopy.sin(x**2 + y)
+    retval[1] = algopy.cos(x+y) - x
+    retval[2] = algopy.sin(x)**2 + algopy.cos(x)**2
+    return retval
+
+# trace the function evaluation
+# and store the computational graph in cg
+cg = CGraph()
+ax = 3.
+ay = 5.
+fx = Function(ax)
+fy = Function(ay)
+fz = eval_g(fx, fy)
+cg.independentFunctionList = [fx, fy]
+cg.dependentFunctionList = [fz]
+
+# compute Taylor series
+#
+#  Jx( 1. + 2.*t + 3.*t**2 + 4.*t**3 + 5.*t**5)
+#  Jy( 1. + 2.*t + 3.*t**2 + 4.*t**3 + 5.*t**5)
+#
+# where
+#
+# Jx = dg/dx
+# Jy = dg/dy
+
+
+# setup input Taylor polynomials
+D,P = 5, 3  # order D=5, number of directions P
+ax = UTPM(numpy.zeros((D, P)))
+ay = UTPM(numpy.zeros((D, P)))
+ax.data[:, :] = numpy.array([1., 2. ,3. ,4. ,5.]).reshape((5,1))  # input Taylor polynomial
+ay.data[:, :] = numpy.array([1., 2. ,3. ,4. ,5.]).reshape((5,1))  # input Taylor polynomial
+
+# forward sweep
+cg.pushforward([ax, ay])
+
+azbar = UTPM(numpy.zeros((D, P, 3)))
+azbar.data[0, ...] = numpy.eye(3)
+
+# reverse sweep
+cg.pullback([azbar])
+
+# get results
+Jx = cg.independentFunctionList[0].xbar
+Jy = cg.independentFunctionList[1].xbar
+
+print 'Taylor series of Jx =\n', Jx
+print 'Taylor series of Jy =\n', Jy