QuickStart Samples

# Triangular Matrices QuickStart Sample (IronPython)

Illustrates how to work efficiently with upper or lower triangular or trapezoidal matrices in IronPython.

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from System import Array import numerics # The TriangularMatrix class resides in the Extreme.Mathematics.LinearAlgebra # namespace. from Extreme.Mathematics import * from Extreme.Mathematics.LinearAlgebra import * #/ Illustrates the use of the TriangularMatrix class in the #/ Extreme.Mathematics.LinearAlgebra namespace of the Extreme Optimization #/ Mathematics Library for .NET. # Triangular matrices are matrices whose elements # above or below the diagonal are all zero. The # former is called lower triangular, the latter # lower triangular. In addition, triangular matrices # can have all 1's on the diagonal. # # Constructing triangular matrices # # Constructing triangular matrices is similar to # constructing general matrices. See the # BasicMatrices QuickStart samples for a more # complete discussion. # # All constructors take a MatrixTriangle # value as their first parameter. This indicates # whether an upper or lower triangular matrix # should be created. The following creates a # 5x5 lower triangular matrix: t1 = Matrix.CreateLowerTriangular(5, 5) # You can also specify whether the diagonal # consists of all 1's using a unitDiagonal parameter: t2 = Matrix.CreateLowerTriangular(5, 5, MatrixDiagonal.UnitDiagonal) # Triangular matrices access and modify only the # elements that are non-zero. If the diagonal # mode is UnitDiagonal, the diagonal elements # are not used, since they are all equal to 1. components = Array[float]([ \ 11, 12, 13, 14, 15, \ 21, 22, 23, 24, 25, \ 31, 32, 33, 34, 35, \ 41, 42, 43, 44, 45, \ 51, 52, 53, 54, 55 ]) # The following creates a matrix using the # upper triangular part of the above. t3 = Matrix.CreateUpperTriangular(5, 5, components, MatrixElementOrder.RowMajor) print "t3 =", t3 # Same as above, but unit diagonal: t4 = Matrix.CreateUpperTriangular(5, 5, components, MatrixDiagonal.UnitDiagonal, MatrixElementOrder.RowMajor, True) print "t4 =", t4 # # Extracting triangular matrices # # You may want to use part of a dense matrix # as a triangular matrix. The static # ExtractUpperTriangle and ExtractLowerTriangle # methods perform this task. m = Matrix.Create(5, 5, components, MatrixElementOrder.ColumnMajor) print "m =", m # Both methods are overloaded. The simplest # returns a triangular matrix of the same dimension: t5 = TriangularMatrix.ExtractLowerTriangle(m) print "t5 =", t5 # You can also specify if the matrix is unit diagonal: t6 = TriangularMatrix.ExtractUpperTriangle(m, MatrixDiagonal.UnitDiagonal) print "t6 =", t6 # Or the dimensions of the matrix if they don't # match the original: t7 = TriangularMatrix.ExtractUpperTriangle(m, 3, 3, MatrixDiagonal.UnitDiagonal) print "t7 =", t7 print # # TriangularMatrix properties # # The IsLowerTriangular and IsUpperTriangular return # a boolean value: print "t4 is lower triangular? -", t4.IsLowerTriangular print "t4 is upper triangular? -", t4.IsUpperTriangular # The IsUnitDiagonal property indicates whether the # matrix has all 1's on its diagonal: print "t3 is unit diagonal? -", t3.IsUnitDiagonal print "t4 is unit diagonal? -", t4.IsUnitDiagonal print # You can get and set matrix elements: t3[1, 3] = 55 print "t3[1, 3] =", t3[1, 3] # But trying to set an element that is zero or # is on the diagonal for a unit diagonal matrix # causes an exception to be thrown: try: t3[3, 1] = 100 except ComponentReadOnlyException as e: print "Error accessing element:", e.Message # # Rows and columns # # The GetRow and GetColumn methods are # available. row = t3.GetRow(1) row = t3[1,:] print "row 2 of t3 =", row column = t4.GetColumn(1, 0, 2) column = t4[0:3, 1] print "2nd column of t4 from row 1 to 3 =", column

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