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QuickStart Samples

Cubic Splines QuickStart Sample (IronPython)

Illustrates using natural and clamped cubic splines for interpolation using classes in the Extreme.Mathematics.LinearAlgebra namespace in IronPython.

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import numerics

from Extreme.Mathematics import *
# The CubicSpline class resides in the
# Extreme.Mathematics.Curves namespace.
from Extreme.Mathematics.Curves import *

#/ Illustrates creating natural and clamped cubic splines using 
#/ the CubicSpline class in the Extreme.Mathematics.Curves 
#/ namespace of the Extreme Optimization Mathematics 
#/ Library for .NET.

# A cubic spline is a piecewise curve that is made up
# of pieces of cubic polynomials. Its value as well as its first
# derivative are continuous, giving it a smooth appearance.
# 
# Cubic splines are implemented by the CubicSpline class, # which inherits from PiecewiseCurve.
#
# For an example of piecewise constant and piecewise 
# linear curves, see the PiecewiseCurves QuickStart
# Sample.
#

# 
# Creating Cubic Splines
#

# In order to define a spline curve completely, two extra
# conditions must be imposed.
			
# 'Natural' splines have zero second derivatives. This is
# the default.

# The data points are specified as double arrays containing
# the x and y values:
xValues = Vector([ 1, 2, 3, 4, 5, 6 ])
yValues = Vector([ 1, 3, 4, 3, 4, 2 ])
naturalSpline = CubicSpline(xValues, yValues)

# 'Clamped' splines have a fixed slope or first derivative at the 
# leftmost and rightmost points. The slopes are specified as
# two extra parameters in the constructor:
clampedSpline = CubicSpline(xValues, yValues, -1, 1)

#
# Curve Parameters
#

# The shape of any curve is determined by a set of parameters.
# These parameters can be retrieved and set through the
# Parameters collection. The number of parameters for a curve 
# is given by this collection's Count property.
#
# Cubic splines have 2n+2 parameters, where n is the number of
# data points. The first n parameters are the x-values. The next
# n parameters are the y-values. The last two parameters are
# the values of the derivative at the first and last point. For natural
# splines, these parameters are unused.

print "naturalSpline.Parameters.Count =", naturalSpline.Parameters.Count
# Parameters can easily be retrieved:
print "naturalSpline.Parameters[0] =", naturalSpline.Parameters[0]
# Parameters can also be set:
naturalSpline.Parameters[0] = 1

#
# Piecewise curve methods and properties
#

# The NumberOfIntervals property returns the number of subintervals
# on which the curve has unique definitions.
print "Number of intervals:", naturalSpline.NumberOfIntervals

# The IndexOf method returns the index of the interval
# that contains a specific value.
print "naturalSpline.IndexOf(1.4) =", naturalSpline.IndexOf(1.4)
# The method returns -1 when the value is smaller than the lower bound
# of the first interval, and NumberOfIntervals if the value is equal to or larger than
# the upper bound of the last interval.
			

#
# Curve Methods
#

# The ValueAt method returns the y value of the
# curve at the specified x value:
print "naturalSpline.ValueAt(2.4) =", naturalSpline.ValueAt(2.4)

# The SlopeAt method returns the slope of the curve
# a the specified x value:
print "naturalSpline.SlopeAt(2) =", naturalSpline.SlopeAt(2)
# You can verify that the clamped spline has the correct slope at the end points:
print "clampedSpline.SlopeAt(1) =", clampedSpline.SlopeAt(1)
print "clampedSpline.SlopeAt(6) =", clampedSpline.SlopeAt(6)

# Cubic splines do not have a defined derivative. The GetDerivative method
# returns a GeneralCurve:
derivative = naturalSpline.GetDerivative()
print "Type of derivative:", derivative.GetType().ToString()
print "derivative(2) =", derivative.ValueAt(2)

# You can get a Line that is the tangent to a curve
# at a specified x value using the TangentAt method:
tangent = clampedSpline.TangentAt(2)
print "Slope of tangent line at 2 =", tangent.Slope

# The integral of a spline curve can be calculated exactly. This technique is
# often used to approximate the integral of a tabulated function:
print "Integral of naturalSpline between 1.4 and 4.6 =", naturalSpline.Integral(1.4, 4.6)