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QuickStart Samples

# Band Matrices QuickStart Sample (IronPython)

Illustrates how to work with the BandMatrix class in IronPython.

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import numerics # The BandMatrix class resides in the Extreme.Mathematics.LinearAlgebra # namespace. from Extreme.Mathematics import * from Extreme.Mathematics.LinearAlgebra import * #/ Illustrates the use of the BandMatrix class in the #/ Extreme.Mathematics.LinearAlgebra namespace of the Extreme Optimization #/ Numerical Libraries for .NET. # Band matrices are matrices whose elements # are nonzero only in a diagonal band around # the main diagonal. # # General band matrices, upper and lower band # matrices, and symmetric band matrices are all # represented by a single class: BandMatrix. # # Constructing band matrices # # Constructing band matrices is similar to # constructing general matrices. It is done by # calling a factory method on the Matrix class. # See theBasicMatrices QuickStart samples # for a more complete discussion. # The following creates a 7x5 band matrix with # upper bandwidth 1 and lower bandwidth 2: b1 = Matrix.CreateBanded(7, 5, 2, 1) # Once the upper and lower bandwidth are set, # it cannot be changed. Elements that are outside # the band cannot be set. # A second factory method lets you create upper # or lower band matrices. The following constructs # an 11x11 upper band matrix with unit diagonal # and three non-zero upper diagonals. b2 = Matrix.CreateUpperBanded(11, 11, 3, MatrixDiagonal.UnitDiagonal) # To create a symmetric band matrix, you only need # the size and the bandwith. The following creates # a 6x6 symmetric tri-diagonal matrix: b3 = Matrix.CreateSymmetricBanded(7, 1) # We can assign values to the components by using # the GetDiagonal method. b3.GetDiagonal(0).SetValue(2) b3.GetDiagonal(1).SetValue(-1) # Extracting band matrices # Another way to construct a band matrix is by # extracting them from an existing matrix. m = Matrix([[1,3,5], [2,4,4], [3,3,5], [2,4,7]]) # To get the lower band part of m with bandwidth 2: b4 = BandMatrix.Extract(m, 2, 0) # # BandMatrix properties # # A number of properties are available to determine # whether a BandMatrix has a special structure: print "b2 is upper?", b2.IsUpperTriangular print "b2 is lower?", b2.IsUpperTriangular print "b2 is unit diagonal?", b2.IsUnitDiagonal print "b2 is symmetrical?", b2.IsSymmetrical # # BandMatrix methods # # You can get and set matrix elements: b3[2, 3] = 55 print "b3[2, 3] =", b3[2, 3] # And the change will automatically be reflected # in the symmetric element: print "b3[3, 2] =", b3[3, 2] # # Row and column views # # The GetRow and GetColumn methods are # available. row = b2.GetRow(1) row = b2[1,:] print "row 1 of b2 =", row column = b2.GetColumn(2, 3, 4) column = b2[3:5,2] print "column 3 of b2 from row 4 to row 5 =", column # # Band matrix decompositions # # Specialized classes exist to represent the # LU decomposition of a general band matrix # and the Cholesky decomposition of a # symmetric band matrix. # Because of pivoting, the upper band matrix of # the LU decomposition has larger bandwidth. # You need to allocate extra space to be able to # overwrite a matrix with its LU decomposition. # The following creates a 7x5 band matrix with # upper bandwidth 1 and lower bandwidth 2. b5 = Matrix.CreateBanded(7, 7, 2, 1, True) b5.GetDiagonal(0).SetValue(2.0) b5.GetDiagonal(-2).SetValue(-1.0) b5.GetDiagonal(1).SetValue(-1.0) # Other than that, the API is the same as # other decomposition classes. blu = b5.GetLUDecomposition(True) solution = blu.Solve(Vector.CreateConstant(b5.ColumnCount, 1.0)) print "solution of b5*x = ones: {0:.4f}".format(solution)

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