/// <summary> /// Run example /// </summary> public void Run() { // Format matrix output to console var formatProvider = (CultureInfo)CultureInfo.InvariantCulture.Clone(); formatProvider.TextInfo.ListSeparator = " "; // Solve next system of linear equations (Ax=b): // 5*x + 2*y - 4*z = -7 // 3*x - 7*y + 6*z = 38 // 4*x + 1*y + 5*z = 43 // Create matrix "A" with coefficients var matrixA = new DenseMatrix(new[,] { { 5.00, 2.00, -4.00 }, { 3.00, -7.00, 6.00 }, { 4.00, 1.00, 5.00 } }); Console.WriteLine(@"Matrix 'A' with coefficients"); Console.WriteLine(matrixA.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // Create vector "b" with the constant terms. var vectorB = new DenseVector(new[] { -7.0, 38.0, 43.0 }); Console.WriteLine(@"Vector 'b' with the constant terms"); Console.WriteLine(vectorB.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // 1. Solve linear equations using LU decomposition var resultX = matrixA.LU().Solve(vectorB); Console.WriteLine(@"1. Solution using LU decomposition"); Console.WriteLine(resultX.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // 2. Solve linear equations using QR decomposition resultX = matrixA.QR().Solve(vectorB); Console.WriteLine(@"2. Solution using QR decomposition"); Console.WriteLine(resultX.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // 3. Solve linear equations using SVD decomposition matrixA.Svd(true).Solve(vectorB, resultX); Console.WriteLine(@"3. Solution using SVD decomposition"); Console.WriteLine(resultX.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // 4. Solve linear equations using Gram-Shmidt decomposition matrixA.GramSchmidt().Solve(vectorB, resultX); Console.WriteLine(@"4. Solution using Gram-Shmidt decomposition"); Console.WriteLine(resultX.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // 5. Verify result. Multiply coefficient matrix "A" by result vector "x" var reconstructVecorB = matrixA * resultX; Console.WriteLine(@"5. Multiply coefficient matrix 'A' by result vector 'x'"); Console.WriteLine(reconstructVecorB.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // To use Cholesky or Eigenvalue decomposition coefficient matrix must be // symmetric (for Evd and Cholesky) and positive definite (for Cholesky) // Multipy matrix "A" by its transpose - the result will be symmetric and positive definite matrix var newMatrixA = matrixA.TransposeAndMultiply(matrixA); Console.WriteLine(@"Symmetric positive definite matrix"); Console.WriteLine(newMatrixA.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // 6. Solve linear equations using Cholesky decomposition newMatrixA.Cholesky().Solve(vectorB, resultX); Console.WriteLine(@"6. Solution using Cholesky decomposition"); Console.WriteLine(resultX.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // 7. Solve linear equations using eigen value decomposition newMatrixA.Evd().Solve(vectorB, resultX); Console.WriteLine(@"7. Solution using eigen value decomposition"); Console.WriteLine(resultX.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // 8. Verify result. Multiply new coefficient matrix "A" by result vector "x" reconstructVecorB = newMatrixA * resultX; Console.WriteLine(@"8. Multiply new coefficient matrix 'A' by result vector 'x'"); Console.WriteLine(reconstructVecorB.ToString("#0.00\t", formatProvider)); Console.WriteLine(); }
/// <summary> /// Run example /// </summary> /// <seealso cref="http://en.wikipedia.org/wiki/QR_decomposition">QR decomposition</seealso> public void Run() { // Format matrix output to console var formatProvider = (CultureInfo)CultureInfo.InvariantCulture.Clone(); formatProvider.TextInfo.ListSeparator = " "; // Create 3 x 2 matrix var matrix = new DenseMatrix(new[,] { { 1.0, 2.0 }, { 3.0, 4.0 }, { 5.0, 6.0 } }); Console.WriteLine(@"Initial 3x2 matrix"); Console.WriteLine(matrix.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // Perform QR decomposition (Householder transformations) var qr = matrix.QR(); Console.WriteLine(@"QR decomposition (Householder transformations)"); // 1. Orthogonal Q matrix Console.WriteLine(@"1. Orthogonal Q matrix"); Console.WriteLine(qr.Q.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // 2. Multiply Q matrix by its transpose gives identity matrix Console.WriteLine(@"2. Multiply Q matrix by its transpose gives identity matrix"); Console.WriteLine(qr.Q.TransposeAndMultiply(qr.Q).ToString("#0.00\t", formatProvider)); Console.WriteLine(); // 3. Upper triangular factor R Console.WriteLine(@"3. Upper triangular factor R"); Console.WriteLine(qr.R.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // 4. Reconstruct initial matrix: A = Q * R var reconstruct = qr.Q * qr.R; Console.WriteLine(@"4. Reconstruct initial matrix: A = Q*R"); Console.WriteLine(reconstruct.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // Perform QR decomposition (Gram–Schmidt process) var gramSchmidt = matrix.GramSchmidt(); Console.WriteLine(@"QR decomposition (Gram–Schmidt process)"); // 5. Orthogonal Q matrix Console.WriteLine(@"5. Orthogonal Q matrix"); Console.WriteLine(gramSchmidt.Q.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // 6. Multiply Q matrix by its transpose gives identity matrix Console.WriteLine(@"6. Multiply Q matrix by its transpose gives identity matrix"); Console.WriteLine((gramSchmidt.Q.Transpose() * gramSchmidt.Q).ToString("#0.00\t", formatProvider)); Console.WriteLine(); // 7. Upper triangular factor R Console.WriteLine(@"7. Upper triangular factor R"); Console.WriteLine(gramSchmidt.R.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // 8. Reconstruct initial matrix: A = Q * R reconstruct = gramSchmidt.Q * gramSchmidt.R; Console.WriteLine(@"8. Reconstruct initial matrix: A = Q*R"); Console.WriteLine(reconstruct.ToString("#0.00\t", formatProvider)); Console.WriteLine(); }