Structured Linear Equations in IronPython QuickStart Sample

Illustrates how to solve systems of simultaneous linear equations that have special structure in IronPython.

This sample is also available in: C#, Visual Basic, F#.

Overview

This QuickStart sample demonstrates how to solve systems of linear equations when the coefficient matrix has special structure - specifically triangular or symmetric matrices. These special structures allow for more efficient solution methods compared to general matrices.

The sample shows:

• Creating and working with triangular matrices
• Solving triangular systems of equations using the Solve method
• Creating and working with symmetric matrices
• Solving symmetric systems of equations
• Using the Cholesky decomposition for positive definite symmetric matrices
• Handling multiple right-hand sides
• Computing matrix properties like determinants, inverses and condition numbers
• Error handling for indefinite matrices
• Performance considerations when solving multiple systems

Code examples demonstrate proper usage of the relevant Numerics.NET classes and methods, including error checking and handling special cases. The sample includes examples of both single right-hand side vectors and multiple right-hand side matrices.

The code

import numerics

from System import Array

from Extreme.Mathematics import *
# The structured matrix classes reside in the 
# Extreme.Mathematics.LinearAlgebra namespace.
from Extreme.Mathematics.LinearAlgebra import *

# Illustrates solving symmetrical and triangular systems 
# of simultaneous linear equations using classes 
# in the Extreme.Mathematics.LinearAlgebra namespace of the Extreme 
# Optimization Mathematics Library for .NET.

# To learn more about solving general systems of
# simultaneous linear equations, see the
# LinearEquations QuickStart Sample.
#
# The methods and classes available for solving
# structured systems of equations are similar
# to those for general equations.

#
# Triangular systems and matrices
#

print "Triangular matrices:"
# For the basics of working with triangular 
# matrices, see the TriangularMatrices QuickStart
# Sample.
#
# Let's start with a triangular matrix. Remember
# that elements are stored in column-major order
# by default.
t = Matrix.CreateUpperTriangular(4, 4, Array[float]([ \
	1, 0, 0, 0, \
    1, 2, 0, 0, \
    1, 4, 1, 0, \
    1, 3, 1, 2 ]), \
	MatrixElementOrder.ColumnMajor)
b1 = Vector([ 1, 3, 6, 3 ])
b2 = Matrix([[1,2], [3,3], [6,5], [3,8]])
print "t = {0:.4f}".format(t)

#
# The Solve method
#

# The following solves m x = b1. The second 
# parameter specifies whether to overwrite the
# right-hand side with the result.
x1 = t.Solve(b1, False)
print "x1 = {0:.4f}".format(x1)
# If the overwrite parameter is omitted, the
# right-hand-side is overwritten with the solution:
t.Solve(b1)
print "b1 = {0:.4f}".format(b1)
# You can solve for multiple right hand side 
# vectors by passing them in a DenseMatrix:
x2 = t.Solve(b2, False)
print "x2 = {0:.4f}".format(x2)

#
# Related Methods
#

# You can verify whether a matrix is singular
# using the IsSingular method:
print "IsSingular(t) =", t.IsSingular()
# The inverse matrix is returned by the Inverse
# method:
print "Inverse(t) = {0:.4f}".format(t.GetInverse())
# The determinant is also available:
print "Det(t) = {0:.4f}".format(t.GetDeterminant())
# The condition number is an estimate for the
# loss of precision in solving the equations
print "Cond(t) = {0:.4f}".format(t.EstimateConditionNumber())
print 

#
# Symmetric systems and matrices
#

print "Symmetric matrices:"
# For the basics of working with symmetric 
# matrices, see the SymmetricMatrices QuickStart
# Sample.
#
# Let's start with a symmetric matrix. Remember
# that elements are stored in column-major order
# by default.
s = Matrix.CreateSymmetric(4, Array[float]([ \
	1, 0, 0, 0, \
    1, 2, 0, 0, \
    1, 1, 2, 0, \
    1, 0, 1, 4 ]), MatrixTriangle.Upper)
b1 = Vector.Create(1, 3, 6, 3)
print "s = {0:.4f}".format(s)

#
# The Solve method
#

# The following solves m x = b1. The second 
# parameter specifies whether to overwrite the
# right-hand side with the result.
x1 = s.Solve(b1, False)
print "x1 = {0:.4f}".format(x1)
# If the overwrite parameter is omitted, the
# right-hand-side is overwritten with the solution:
s.Solve(b1)
print "b1 = {0:.4f}".format(b1)
# You can solve for multiple right hand side 
# vectors by passing them in a DenseMatrix:
x2 = s.Solve(b2, False)
print "x2 = {0:.4f}".format(x2)

#
# Related Methods
#

# You can verify whether a matrix is singular
# using the IsSingular method:
print "IsSingular(s) =", s.IsSingular()
# The inverse matrix is returned by the Inverse
# method:
print "Inverse(s) = {0:.4f}".format(s.GetInverse())
# The determinant is also available:
print "Det(s) = {0:.4f}".format(s.GetDeterminant())
# The condition number is an estimate for the
# loss of precision in solving the equations
print "Cond(s) = {0:.4f}".format(s.EstimateConditionNumber())
print 

#
# The CholeskyDecomposition class
#

# If the symmetric matrix is positive definite, # you can use the CholeskyDecomposition class
# to optimize performance if multiple operations 
# need to be performed. This class does the
# bulk of the calculations only once. This
# decomposes the matrix into G x transpose(G)
# where G is a lower triangular matrix.
#
# If the matrix is indefinite, you need to use
# the LUDecomposition class instead. See the
# LinearEquations QuickStart Sample for details.
print "Using Cholesky Decomposition:"
# The constructor takes an optional second argument
# indicating whether to overwrite the original
# matrix with its decomposition:
cf = s.GetCholeskyDecomposition(False)
# The Factorize method performs the actual
# factorization. It is called automatically
# if needed.
cf.Decompose()
# All methods mentioned earlier are still available:
x2 = cf.Solve(b2, False)
print "x2 = {0:.4f}".format(x2)
print "IsSingular(m) =", cf.IsSingular()
print "Inverse(m) = {0:.4f}".format(cf.GetInverse())
print "Det(m) = {0:.4f}".format(cf.GetDeterminant())
print "Cond(m) = {0:.4f}".format(cf.EstimateConditionNumber())
# In addition, you have access to the
# triangular matrix, G, of the composition.
print "  G = {0:.4f}".format(cf.LowerTriangularFactor)

# Note that if the matrix is indefinite, 
# the factorization will fail and throw a
# MatrixNotPositiveDefiniteException.
s[0, 0] = -99
cf = s.GetCholeskyDecomposition()
try:
    cf.Decompose()
except MatrixNotPositiveDefiniteException as e:
    print e.Message

# To avoid this, you can use the TryDecompose method, 
# which returns True if the decomposition succeeds, 
# and false otherwise:
if cf.TryDecompose():
    print "Decomposition succeeded!"
else:
    print "Decomposition failed!"