QuickStart Samples

# Chebyshev Series QuickStart Sample (IronPython)

Illustrates the basic use of the ChebyshevSeries class in IronPython.

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import numerics from math import * from System import Array # The ChebyshevSeries class resides in the Extreme.Mathematics.Curves # namespace. from Extreme.Mathematics.Curves import * # The Func<double, double> delegate resides in the # Extreme.Mathematics namespace. from Extreme.Mathematics import * #/ Illustrates the use of the ChebyshevSeries class #/ in the Extreme.Mathematics.Curve namespace of the Extreme Optimization #/ Numerical Libraries for .NET. # Chebyshev polynomials form an alternative basis # for polynomials. A Chebyshev expansion is a # polynomial expressed as a sum of Chebyshev # polynomials. # # Using the ChebyshevSeries class instead of # Polynomial can have two major advantages: # 1. They are numerically more stable. Higher # accuracy is maintained even for large problems. # 2. When approximating other functions with # polynomials, the coefficients in the # Chebyshev expansion will tend to decrease # in size, where those of the normal polynomial # approximation will tend to oscillate wildly. # # Constructing Chebyshev expansions # # Chebyshev expansions are defined over an interval. # The first constructor requires you to specify the # boundaries of the interval, and the coefficients # of the expansion. coefficients = Array[float]([ 1, 0.5, -0.3, 0.1 ]) chebyshev1 = ChebyshevSeries(0, 2, coefficients) # If you omit the boundaries, they are assumed to be # -1 and +1: chebyshev2 = ChebyshevSeries(coefficients) # # Chebyshev approximations # # A third constructor creates a Chebyshev # approximation to an arbitrary function. For more # about the Func<double, double> delegate, see the # FunctionDelegates QuickStart Sample. # # Chebyshev expansions allow us to obtain an # excellent approximation at minimal cost. # # The following creates a Chebyshev approximation # of degree 7 to Cos(x) over the interval [0, 2]: approximation1 = ChebyshevSeries.GetInterpolatingPolynomial(cos, 0, 2, 7) # The coefficients of the expansion are available through # the indexer property of the ChebyshevSeries object: print "Chebyshev approximation of cos(x):" for index in range(0, 8): print " c{0} = {1}".format(index, approximation1[index]) # The largest errors are approximately at the # zeroes of the Chebyshev polynomial of degree 8: for index in range(0,9): zero = 1 + cos(index * Constants.Pi / 8) error = approximation1.ValueAt(zero) - cos(zero) print " Error {0} = {1}".format(index, error) # # Least squares approximations # # We will now calculate the least squares polynomial # of degree 7 through 33 points. # First, calculate the points: xValues = Vector([ 1 + cos(index * Constants.Pi / 32) for index in range(0,33) ]) yValues = Vector([ cos(x) for x in xValues ]) # Next, define a ChebyshevBasis object for the # approximation we want: interval [0,2] and degree # is 7. basis = ChebyshevBasis(0, 2, 7) # Now we can calculate the least squares fit: approximation2 = basis.LeastSquaresFit(xValues, yValues) # We can see it is close to the original # approximation we found earlier: for index in range(0,7): print " c{0} = {1}".format(index, approximation2[index])

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