Data Analysis Mathematics Linear Algebra Statistics
New Version 6.0!

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.

```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)```