C# DenseMatrix.ToString方法代码示例

        /// <summary>
        /// Run example
        /// </summary>
        /// <seealso cref="http://en.wikipedia.org/wiki/Triangular_matrix">Triangular matrix</seealso>
        public void Run()
        {
            // Format matrix output to console
            var formatProvider = (CultureInfo)CultureInfo.InvariantCulture.Clone();
            formatProvider.TextInfo.ListSeparator = " ";

            // Create square matrix
            var matrix = new DenseMatrix(10);
            var k = 0;
            for (var i = 0; i < matrix.RowCount; i++)
            {
                for (var j = 0; j < matrix.ColumnCount; j++)
                {
                    matrix[i, j] = k++;
                }
            }

            Console.WriteLine(@"Initial square matrix");
            Console.WriteLine(matrix.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 1. Retrieve a new matrix containing the lower triangle of the matrix
            var lower = matrix.LowerTriangle();

            // Puts the lower triangle of the matrix into the result matrix.
            matrix.LowerTriangle(lower);
            Console.WriteLine(@"1. Lower triangle of the matrix");
            Console.WriteLine(lower.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 2. Retrieve a new matrix containing the upper triangle of the matrix
            var upper = matrix.UpperTriangle();

            // Puts the upper triangle of the matrix into the result matrix.
            matrix.UpperTriangle(lower);
            Console.WriteLine(@"2. Upper triangle of the matrix");
            Console.WriteLine(upper.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 3. Retrieve a new matrix containing the strictly lower triangle of the matrix
            var strictlylower = matrix.StrictlyLowerTriangle();

            // Puts the strictly lower triangle of the matrix into the result matrix.
            matrix.StrictlyLowerTriangle(strictlylower);
            Console.WriteLine(@"3. Strictly lower triangle of the matrix");
            Console.WriteLine(strictlylower.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 4. Retrieve a new matrix containing the strictly upper triangle of the matrix
            var strictlyupper = matrix.StrictlyUpperTriangle();

            // Puts the strictly upper triangle of the matrix into the result matrix.
            matrix.StrictlyUpperTriangle(strictlyupper);
            Console.WriteLine(@"4. Strictly upper triangle of the matrix");
            Console.WriteLine(strictlyupper.ToString("#0.00\t", formatProvider));
            Console.WriteLine();
        }

        /// <summary>
        /// Run example
        /// </summary>
        /// <seealso cref="http://en.wikipedia.org/wiki/Transpose">Transpose</seealso>
        /// <seealso cref="http://en.wikipedia.org/wiki/Invertible_matrix">Invertible matrix</seealso>
        public void Run()
        {
            // Format matrix output to console
            var formatProvider = (CultureInfo)CultureInfo.InvariantCulture.Clone();
            formatProvider.TextInfo.ListSeparator = " ";

            // Create random square matrix
            var matrix = new DenseMatrix(5);
            var rnd = new Random(1);
            for (var i = 0; i < matrix.RowCount; i++)
            {
                for (var j = 0; j < matrix.ColumnCount; j++)
                {
                    matrix[i, j] = rnd.NextDouble();
                }
            }

            Console.WriteLine(@"Initial matrix");
            Console.WriteLine(matrix.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 1. Get matrix inverse
            var inverse = matrix.Inverse();
            Console.WriteLine(@"1. Matrix inverse");
            Console.WriteLine(inverse.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 2. Matrix multiplied by its inverse gives identity matrix
            var identity = matrix * inverse;
            Console.WriteLine(@"2. Matrix multiplied by its inverse");
            Console.WriteLine(identity.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 3. Get matrix transpose
            var transpose = matrix.Transpose();
            Console.WriteLine(@"3. Matrix transpose");
            Console.WriteLine(transpose.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 4. Get orthogonal  matrix, i.e. do QR decomposition and get matrix Q
            var orthogonal = matrix.QR().Q;
            Console.WriteLine(@"4. Orthogonal  matrix");
            Console.WriteLine(orthogonal.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 5. Transpose and multiply orthogonal matrix by iteslf gives identity matrix
            identity = orthogonal.TransposeAndMultiply(orthogonal);
            Console.WriteLine(@"Transpose and multiply orthogonal matrix by iteslf");
            Console.WriteLine(identity.ToString("#0.00\t", formatProvider));
            Console.WriteLine();
        }

        /// <summary>
        /// Run example
        /// </summary>
        /// <seealso cref="http://en.wikipedia.org/wiki/Cholesky_decomposition">Cholesky decomposition</seealso>
        public void Run()
        {
            // Format matrix output to console
            var formatProvider = (CultureInfo)CultureInfo.InvariantCulture.Clone();
            formatProvider.TextInfo.ListSeparator = " ";

            // Create square, symmetric, positive definite matrix
            var matrix = new DenseMatrix(new[,] { { 2.0, 1.0 }, { 1.0, 2.0 } });
            Console.WriteLine(@"Initial square, symmetric, positive definite matrix");
            Console.WriteLine(matrix.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // Perform Cholesky decomposition
            var cholesky = matrix.Cholesky();
            Console.WriteLine(@"Perform Cholesky decomposition");

            // 1. Lower triangular form of the Cholesky matrix
            Console.WriteLine(@"1. Lower triangular form of the Cholesky matrix");
            Console.WriteLine(cholesky.Factor.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 2. Reconstruct initial matrix: A = L * LT
            var reconstruct = cholesky.Factor * cholesky.Factor.Transpose();
            Console.WriteLine(@"2. Reconstruct initial matrix: A = L*LT");
            Console.WriteLine(reconstruct.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 3. Get determinant of the matrix
            Console.WriteLine(@"3. Determinant of the matrix");
            Console.WriteLine(cholesky.Determinant);
            Console.WriteLine();

            // 4. Get log determinant of the matrix
            Console.WriteLine(@"4. Log determinant of the matrix");
            Console.WriteLine(cholesky.DeterminantLn);
            Console.WriteLine();
        }

        /// <summary>
        /// Run example
        /// </summary>
        public void Run()
        {
            // 1. Initialize a new instance of the matrix from a 2D array. This constructor will allocate a completely new memory block for storing the dense matrix.
            var matrix1 = new DenseMatrix(new[,] { { 1.0, 2.0, 3.0 }, { 4.0, 5.0, 6.0 } });

            // 2. Initialize a new instance of the empty square matrix with a given order.
            var matrix2 = new DenseMatrix(3);

            // 3. Initialize a new instance of the empty matrix with a given size.
            var matrix3 = new DenseMatrix(2, 3);

            // 4. Initialize a new instance of the matrix with all entries set to a particular value.
            var matrix4 = new DenseMatrix(2, 3, 3.0);

            // 4. Initialize a new instance of the matrix from a one dimensional array. This array should store the matrix in column-major order. 
            var matrix5 = new DenseMatrix(2, 3, new[] { 1.0, 4.0, 2.0, 5.0, 3.0, 6.0 });

            // 5. Initialize a square matrix with all zero's except for ones on the diagonal. Identity matrix (http://en.wikipedia.org/wiki/Identity_matrix).
            var matrixI = DenseMatrix.Identity(5);

            // Format matrix output to console
            var formatProvider = (CultureInfo)CultureInfo.InvariantCulture.Clone();
            formatProvider.TextInfo.ListSeparator = " ";

            Console.WriteLine(@"Matrix 1");
            Console.WriteLine(matrix1.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            Console.WriteLine(@"Matrix 2");
            Console.WriteLine(matrix2.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            Console.WriteLine(@"Matrix 3");
            Console.WriteLine(matrix3.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            Console.WriteLine(@"Matrix 4");
            Console.WriteLine(matrix4.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            Console.WriteLine(@"Matrix 5");
            Console.WriteLine(matrix5.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            Console.WriteLine(@"Identity matrix");
            Console.WriteLine(matrixI.ToString("#0.00\t", formatProvider));
            Console.WriteLine();
        }

        /// <summary>
        /// Run example
        /// </summary>
        public void Run()
        {
            // Format vector output to console
            var formatProvider = (CultureInfo)CultureInfo.InvariantCulture.Clone();
            formatProvider.TextInfo.ListSeparator = " ";

            // Create new empty square matrix
            var matrix = new DenseMatrix(10);
            Console.WriteLine(@"Empty matrix");
            Console.WriteLine(matrix.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 1. Fill matrix by data using indexer []
            var k = 0;
            for (var i = 0; i < matrix.RowCount; i++)
            {
                for (var j = 0; j < matrix.ColumnCount; j++)
                {
                    matrix[i, j] = k++;
                }
            }

            Console.WriteLine(@"1. Fill matrix by data using indexer []");
            Console.WriteLine(matrix.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 2. Fill matrix by data using At. The element is set without range checking.
            for (var i = 0; i < matrix.RowCount; i++)
            {
                for (var j = 0; j < matrix.ColumnCount; j++)
                {
                    matrix.At(i, j, k--);
                }
            }

            Console.WriteLine(@"2. Fill matrix by data using At");
            Console.WriteLine(matrix.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 3. Clone matrix
            var clone = matrix.Clone();
            Console.WriteLine(@"3. Clone matrix");
            Console.WriteLine(clone.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 4. Clear matrix
            clone.Clear();
            Console.WriteLine(@"4. Clear matrix");
            Console.WriteLine(clone.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 5. Copy matrix into another matrix
            matrix.CopyTo(clone);
            Console.WriteLine(@"5. Copy matrix into another matrix");
            Console.WriteLine(clone.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 6. Get submatrix into another matrix
            var submatrix = matrix.SubMatrix(2, 2, 3, 3);
            Console.WriteLine(@"6. Copy submatrix into another matrix");
            Console.WriteLine(submatrix.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 7. Get part of the row as vector. In this example: get 4 elements from row 5 starting from column 3
            var row = matrix.Row(5, 3, 4);
            Console.WriteLine(@"7. Get part of the row as vector");
            Console.WriteLine(row.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 8. Get part of the column as vector. In this example: get 3 elements from column 2 starting from row 6
            var column = matrix.Column(2, 6, 3);
            Console.WriteLine(@"8. Get part of the column as vector");
            Console.WriteLine(column.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 9. Get columns using column enumerator. If you need all columns you may use ColumnEnumerator without parameters
            Console.WriteLine(@"9. Get columns using column enumerator");
            foreach (var keyValuePair in matrix.ColumnEnumerator(2, 4))
            {
                Console.WriteLine(@"Column {0}: {1}", keyValuePair.Item1, keyValuePair.Item2.ToString("#0.00\t", formatProvider));
            }

            Console.WriteLine();

            // 10. Get rows using row enumerator. If you need all rows you may use RowEnumerator without parameters
            Console.WriteLine(@"10. Get rows using row enumerator");
            foreach (var keyValuePair in matrix.RowEnumerator(4, 3))
            {
                Console.WriteLine(@"Row {0}: {1}", keyValuePair.Item1, keyValuePair.Item2.ToString("#0.00\t", formatProvider));
            }

            Console.WriteLine();

            // 11. Convert matrix into multidimensional array
            var data = matrix.ToArray();
            Console.WriteLine(@"11. Convert matrix into multidimensional array");
            for (var i = 0; i < data.GetLongLength(0); i++)
//.........这里部分代码省略.........

        /// <summary>
        /// Run example
        /// </summary>
        /// <seealso cref="http://en.wikipedia.org/wiki/Determinant">Determinant</seealso>
        /// <seealso cref="http://en.wikipedia.org/wiki/Rank_%28linear_algebra%29">Rank (linear algebra)</seealso>
        /// <seealso cref="http://en.wikipedia.org/wiki/Trace_%28linear_algebra%29">Trace (linear algebra)</seealso>
        /// <seealso cref="http://en.wikipedia.org/wiki/Condition_number">Condition number</seealso>
        public void Run()
        {
            // Format matrix output to console
            var formatProvider = (CultureInfo)CultureInfo.InvariantCulture.Clone();
            formatProvider.TextInfo.ListSeparator = " ";

            // Create random square matrix
            var matrix = new DenseMatrix(5);
            var rnd = new Random(1);
            for (var i = 0; i < matrix.RowCount; i++)
            {
                for (var j = 0; j < matrix.ColumnCount; j++)
                {
                    matrix[i, j] = rnd.NextDouble();
                }
            }

            Console.WriteLine(@"Initial matrix");
            Console.WriteLine(matrix.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 1. Determinant
            Console.WriteLine(@"1. Determinant");
            Console.WriteLine(matrix.Determinant());
            Console.WriteLine();

            // 2. Rank
            Console.WriteLine(@"2. Rank");
            Console.WriteLine(matrix.Rank());
            Console.WriteLine();

            // 3. Condition number
            Console.WriteLine(@"2. Condition number");
            Console.WriteLine(matrix.ConditionNumber());
            Console.WriteLine();

            // 4. Trace
            Console.WriteLine(@"4. Trace");
            Console.WriteLine(matrix.Trace());
            Console.WriteLine();
        }

        /// <summary>
        /// Run example
        /// </summary>
        /// <seealso cref="http://en.wikipedia.org/wiki/Singular_value_decomposition">SVD decomposition</seealso>
        public void Run()
        {
            // Format matrix output to console
            var formatProvider = (CultureInfo)CultureInfo.InvariantCulture.Clone();
            formatProvider.TextInfo.ListSeparator = " ";

            // Create square matrix
            var matrix = new DenseMatrix(new[,] { { 4.0, 1.0 }, { 3.0, 2.0 } });
            Console.WriteLine(@"Initial square matrix");
            Console.WriteLine(matrix.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // Perform full SVD decomposition
            var svd = matrix.Svd(true);
            Console.WriteLine(@"Perform full SVD decomposition");

            // 1. Left singular vectors
            Console.WriteLine(@"1. Left singular vectors");
            Console.WriteLine(svd.U().ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 2. Singular values as vector
            Console.WriteLine(@"2. Singular values as vector");
            Console.WriteLine(svd.S().ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 3. Singular values as diagonal matrix
            Console.WriteLine(@"3. Singular values as diagonal matrix");
            Console.WriteLine(svd.W().ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 4. Right singular vectors
            Console.WriteLine(@"4. Right singular vectors");
            Console.WriteLine(svd.VT().ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 5. Multiply U matrix by its transpose
            var identinty = svd.U() * svd.U().Transpose();
            Console.WriteLine(@"5. Multiply U matrix by its transpose");
            Console.WriteLine(identinty.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 6. Multiply V matrix by its transpose
            identinty = svd.VT().TransposeAndMultiply(svd.VT());
            Console.WriteLine(@"6. Multiply V matrix by its transpose");
            Console.WriteLine(identinty.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 7. Reconstruct initial matrix: A = U*Σ*VT
            var reconstruct = svd.U() * svd.W() * svd.VT();
            Console.WriteLine(@"7. Reconstruct initial matrix: A = U*S*VT");
            Console.WriteLine(reconstruct.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 8. Condition Number of the matrix
            Console.WriteLine(@"8. Condition Number of the matrix");
            Console.WriteLine(svd.ConditionNumber);
            Console.WriteLine();

            // 9. Determinant of the matrix
            Console.WriteLine(@"9. Determinant of the matrix");
            Console.WriteLine(svd.Determinant);
            Console.WriteLine();

            // 10. 2-norm of the matrix
            Console.WriteLine(@"10. 2-norm of the matrix");
            Console.WriteLine(svd.Norm2);
            Console.WriteLine();

            // 11. Rank of the matrix
            Console.WriteLine(@"11. Rank of the matrix");
            Console.WriteLine(svd.Rank);
            Console.WriteLine();

            // Perform partial SVD decomposition, without computing the singular U and VT vectors
            svd = matrix.Svd(false);
            Console.WriteLine(@"Perform partial SVD decomposition, without computing the singular U and VT vectors");

            // 12. Singular values as vector
            Console.WriteLine(@"12. Singular values as vector");
            Console.WriteLine(svd.S().ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 13. Singular values as diagonal matrix
            Console.WriteLine(@"13. Singular values as diagonal matrix");
            Console.WriteLine(svd.W().ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 14. Access to left singular vectors when partial SVD decomposition was performed
            try
            {
                Console.WriteLine(@"14. Access to left singular vectors when partial SVD decomposition was performed");
                Console.WriteLine(svd.U().ToString("#0.00\t", formatProvider));
            }
            catch (Exception ex)
            {
//.........这里部分代码省略.........

        /// <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>
        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();

            // Create stop criteriums to monitor an iterative calculation. There are next available stop criteriums:
            // - DivergenceStopCriterium: monitors an iterative calculation for signs of divergence;
            // - FailureStopCriterium: monitors residuals for NaN's;
            // - IterationCountStopCriterium: monitors the numbers of iteration steps;
            // - ResidualStopCriterium: monitors residuals if calculation is considered converged;

            // Stop calculation if 1000 iterations reached during calculation
            var iterationCountStopCriterium = new IterationCountStopCriterium(1000);

            // Stop calculation if residuals are below 1E-10 --> the calculation is considered converged
            var residualStopCriterium = new ResidualStopCriterium(1e-10);

            // Create monitor with defined stop criteriums
            var monitor = new Iterator(new IIterationStopCriterium[] { iterationCountStopCriterium, residualStopCriterium });

            // Create Multiple-Lanczos Bi-Conjugate Gradient Stabilized solver
            var solver = new MlkBiCgStab(monitor);

            // 1. Solve the matrix equation
            var resultX = solver.Solve(matrixA, vectorB);
            Console.WriteLine(@"1. Solve the matrix equation");
            Console.WriteLine();

            // 2. Check solver status of the iterations.
            // Solver has property IterationResult which contains the status of the iteration once the calculation is finished.
            // Possible values are:
            // - CalculationCancelled: calculation was cancelled by the user;
            // - CalculationConverged: calculation has converged to the desired convergence levels;
            // - CalculationDiverged: calculation diverged;
            // - CalculationFailure: calculation has failed for some reason;
            // - CalculationIndetermined: calculation is indetermined, not started or stopped;
            // - CalculationRunning: calculation is running and no results are yet known;
            // - CalculationStoppedWithoutConvergence: calculation has been stopped due to reaching the stopping limits, but that convergence was not achieved;
            Console.WriteLine(@"2. Solver status of the iterations");
            Console.WriteLine(solver.IterationResult);
            Console.WriteLine();

            // 3. Solution result vector of the matrix equation
            Console.WriteLine(@"3. Solution result vector of the matrix equation");
            Console.WriteLine(resultX.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 4. Verify result. Multiply coefficient matrix "A" by result vector "x"
            var reconstructVecorB = matrixA * resultX;
            Console.WriteLine(@"4. Multiply coefficient matrix 'A' by result vector 'x'");
            Console.WriteLine(reconstructVecorB.ToString("#0.00\t", formatProvider));
            Console.WriteLine();
        }

 public string ToString(DenseMatrix m)
 {
     return m.ToString("",formatProvider);
 }

        /// <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();
        }

        /// <summary>
        /// Run example
        /// </summary>
        /// <seealso cref="http://en.wikipedia.org/wiki/Matrix_norm">Matrix norm</seealso>
        public void Run()
        {
            // Format matrix output to console
            var formatProvider = (CultureInfo)CultureInfo.InvariantCulture.Clone();
            formatProvider.TextInfo.ListSeparator = " ";

            // Create square matrix
            var matrix = new DenseMatrix(new[,] { { 1.0, 2.0, 3.0 }, { 6.0, 5.0, 4.0 }, { 8.0, 9.0, 7.0 } });
            Console.WriteLine(@"Initial square matrix");
            Console.WriteLine(matrix.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 1. 1-norm of the matrix
            Console.WriteLine(@"1. 1-norm of the matrix");
            Console.WriteLine(matrix.L1Norm());
            Console.WriteLine();

            // 2. 2-norm of the matrix
            Console.WriteLine(@"2. 2-norm of the matrix");
            Console.WriteLine(matrix.L2Norm());
            Console.WriteLine();

            // 3. Frobenius norm of the matrix
            Console.WriteLine(@"3. Frobenius norm of the matrix");
            Console.WriteLine(matrix.FrobeniusNorm());
            Console.WriteLine();

            // 4. Infinity norm of the matrix
            Console.WriteLine(@"4. Infinity norm of the matrix");
            Console.WriteLine(matrix.InfinityNorm());
            Console.WriteLine();

            // 5. Normalize matrix columns
            Console.WriteLine(@"5. Normalize matrix columns: before normalize");
            foreach (var keyValuePair in matrix.ColumnEnumerator())
            {
                Console.WriteLine(@"Column {0} 2-nd norm is: {1}", keyValuePair.Item1, keyValuePair.Item2.Norm(2));
            }

            Console.WriteLine();
            var normalized = matrix.NormalizeColumns(2);
            Console.WriteLine(@"5. Normalize matrix columns: after normalize");
            foreach (var keyValuePair in normalized.ColumnEnumerator())
            {
                Console.WriteLine(@"Column {0} 2-nd norm is: {1}", keyValuePair.Item1, keyValuePair.Item2.Norm(2));
            }

            Console.WriteLine();

            // 6. Normalize matrix columns
            Console.WriteLine(@"6. Normalize matrix rows: before normalize");
            foreach (var keyValuePair in matrix.RowEnumerator())
            {
                Console.WriteLine(@"Row {0} 2-nd norm is: {1}", keyValuePair.Item1, keyValuePair.Item2.Norm(2));
            }

            Console.WriteLine();
            normalized = matrix.NormalizeRows(2);
            Console.WriteLine(@"6. Normalize matrix rows: after normalize");
            foreach (var keyValuePair in normalized.RowEnumerator())
            {
                Console.WriteLine(@"Row {0} 2-nd norm is: {1}", keyValuePair.Item1, keyValuePair.Item2.Norm(2));
            }
        }

        /// <summary>
        /// Run example
        /// </summary>
        /// <seealso cref="http://en.wikipedia.org/wiki/LU_decomposition">LU decomposition</seealso>
        /// <seealso cref="http://en.wikipedia.org/wiki/Invertible_matrix">Invertible matrix</seealso>
        public void Run()
        {
            // Format matrix output to console
            var formatProvider = (CultureInfo)CultureInfo.InvariantCulture.Clone();
            formatProvider.TextInfo.ListSeparator = " ";

            // Create square matrix
            var matrix = new DenseMatrix(new[,] { { 1.0, 2.0 }, { 3.0, 4.0 } });
            Console.WriteLine(@"Initial square matrix");
            Console.WriteLine(matrix.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // Perform LU decomposition
            var lu = matrix.LU();
            Console.WriteLine(@"Perform LU decomposition");

            // 1. Lower triangular factor
            Console.WriteLine(@"1. Lower triangular factor");
            Console.WriteLine(lu.L.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 2. Upper triangular factor
            Console.WriteLine(@"2. Upper triangular factor");
            Console.WriteLine(lu.U.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 3. Permutations applied to LU factorization
            Console.WriteLine(@"3. Permutations applied to LU factorization");
            for (var i = 0; i < lu.P.Dimension; i++)
            {
                if (lu.P[i] > i)
                {
                    Console.WriteLine(@"Row {0} permuted with row {1}", lu.P[i], i);
                }
            }

            Console.WriteLine();

            // 4. Reconstruct initial matrix: PA = L * U
            var reconstruct = lu.L * lu.U;

            // The rows of the reconstructed matrix should be permuted to get the initial matrix
            reconstruct.PermuteRows(lu.P.Inverse());
            Console.WriteLine(@"4. Reconstruct initial matrix: PA = L*U");
            Console.WriteLine(reconstruct.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 5. Get the determinant of the matrix
            Console.WriteLine(@"5. Determinant of the matrix");
            Console.WriteLine(lu.Determinant);
            Console.WriteLine();

            // 6. Get the inverse of the matrix
            var matrixInverse = lu.Inverse();
            Console.WriteLine(@"6. Inverse of the matrix");
            Console.WriteLine(matrixInverse.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 7. Matrix multiplied by its inverse
            var identity = matrix * matrixInverse;
            Console.WriteLine(@"7. Matrix multiplied by its inverse ");
            Console.WriteLine(identity.ToString("#0.00\t", formatProvider));
            Console.WriteLine();
        }

        /// <summary>
        /// Run example
        /// </summary>
        /// <seealso cref="http://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_and_eigenspace">EVD decomposition</seealso>
        public void Run()
        {
            // Format matrix output to console
            var formatProvider = (CultureInfo)CultureInfo.InvariantCulture.Clone();
            formatProvider.TextInfo.ListSeparator = " ";

            // Create square symmetric matrix
            var matrix = new DenseMatrix(new[,] { { 1.0, 2.0, 3.0 }, { 2.0, 1.0, 4.0 }, { 3.0, 4.0, 1.0 } });
            Console.WriteLine(@"Initial square symmetric matrix");
            Console.WriteLine(matrix.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // Perform eigenvalue decomposition of symmetric matrix
            var evd = matrix.Evd();
            Console.WriteLine(@"Perform eigenvalue decomposition of symmetric matrix");

            // 1. Eigen vectors
            Console.WriteLine(@"1. Eigen vectors");
            Console.WriteLine(evd.EigenVectors().ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 2. Eigen values as a complex vector
            Console.WriteLine(@"2. Eigen values as a complex vector");
            Console.WriteLine(evd.EigenValues().ToString("N", formatProvider));
            Console.WriteLine();

            // 3. Eigen values as the block diagonal matrix
            Console.WriteLine(@"3. Eigen values as the block diagonal matrix");
            Console.WriteLine(evd.D().ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 4. Multiply V by its transpose VT
            var identity = evd.EigenVectors().TransposeAndMultiply(evd.EigenVectors());
            Console.WriteLine(@"4. Multiply V by its transpose VT: V*VT = I");
            Console.WriteLine(identity.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 5. Reconstruct initial matrix: A = V*D*V'
            var reconstruct = evd.EigenVectors() * evd.D() * evd.EigenVectors().Transpose();
            Console.WriteLine(@"5. Reconstruct initial matrix: A = V*D*V'");
            Console.WriteLine(reconstruct.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 6. Determinant of the matrix
            Console.WriteLine(@"6. Determinant of the matrix");
            Console.WriteLine(evd.Determinant);
            Console.WriteLine();

            // 7. Rank of the matrix
            Console.WriteLine(@"7. Rank of the matrix");
            Console.WriteLine(evd.Rank);
            Console.WriteLine();

            // Fill matrix by random values
            var rnd = new Random(1);
            for (var i = 0; i < matrix.RowCount; i++)
            {
                for (var j = 0; j < matrix.ColumnCount; j++)
                {
                    matrix[i, j] = rnd.NextDouble();
                }
            }

            Console.WriteLine(@"Fill matrix by random values");
            Console.WriteLine(matrix.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // Perform eigenvalue decomposition of non-symmetric matrix
            evd = matrix.Evd();
            Console.WriteLine(@"Perform eigenvalue decomposition of non-symmetric matrix");

            // 8. Eigen vectors
            Console.WriteLine(@"8. Eigen vectors");
            Console.WriteLine(evd.EigenVectors().ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 9. Eigen values as a complex vector
            Console.WriteLine(@"9. Eigen values as a complex vector");
            Console.WriteLine(evd.EigenValues().ToString("N", formatProvider));
            Console.WriteLine();

            // 10. Eigen values as the block diagonal matrix
            Console.WriteLine(@"10. Eigen values as the block diagonal matrix");
            Console.WriteLine(evd.D().ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 11. Multiply A * V
            var av = matrix * evd.EigenVectors();
            Console.WriteLine(@"11. Multiply A * V");
            Console.WriteLine(av.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 12. Multiply V * D
            var vd = evd.EigenVectors() * evd.D();
            Console.WriteLine(@"12. Multiply V * D");
            Console.WriteLine(vd.ToString("#0.00\t", formatProvider));
//.........这里部分代码省略.........

        /// <summary>
        /// Run example
        /// </summary>
        /// <seealso cref="http://en.wikipedia.org/wiki/Matrix_multiplication#Scalar_multiplication">Multiply matrix by scalar</seealso>
        /// <seealso cref="http://reference.wolfram.com/mathematica/tutorial/MultiplyingVectorsAndMatrices.html">Multiply matrix by vector</seealso>
        /// <seealso cref="http://en.wikipedia.org/wiki/Matrix_multiplication#Matrix_product">Multiply matrix by matrix</seealso>
        /// <seealso cref="http://en.wikipedia.org/wiki/Matrix_multiplication#Hadamard_product">Pointwise multiplies matrix with another matrix</seealso>
        /// <seealso cref="http://en.wikipedia.org/wiki/Matrix_%28mathematics%29#Basic_operations">Addition and subtraction</seealso>
        public void Run()
        {
            // Initialize IFormatProvider to print matrix/vector data
            var formatProvider = (CultureInfo)CultureInfo.InvariantCulture.Clone();
            formatProvider.TextInfo.ListSeparator = " ";

            // Create matrix "A"
            var matrixA = new DenseMatrix(new[,] { { 1.0, 2.0, 3.0 }, { 4.0, 5.0, 6.0 }, { 7.0, 8.0, 9.0 } });
            Console.WriteLine(@"Matrix A");
            Console.WriteLine(matrixA.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // Create matrix "B"
            var matrixB = new DenseMatrix(new[,] { { 1.0, 3.0, 5.0 }, { 2.0, 4.0, 6.0 }, { 3.0, 5.0, 7.0 } });
            Console.WriteLine(@"Matrix B");
            Console.WriteLine(matrixB.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // Multiply matrix by scalar
            // 1. Using operator "*"
            var resultM = 3.0 * matrixA;
            Console.WriteLine(@"Multiply matrix by scalar using operator *. (result = 3.0 * A)");
            Console.WriteLine(resultM.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 2. Using Multiply method and getting result into different matrix instance
            resultM = (DenseMatrix)matrixA.Multiply(3.0);
            Console.WriteLine(@"Multiply matrix by scalar using method Multiply. (result = A.Multiply(3.0))");
            Console.WriteLine(resultM.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 3. Using Multiply method and updating matrix itself
            matrixA.Multiply(3.0, matrixA);
            Console.WriteLine(@"Multiply matrix by scalar using method Multiply. (A.Multiply(3.0, A))");
            Console.WriteLine(matrixA.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // Multiply matrix by vector (right-multiply)
            var vector = new DenseVector(new[] { 1.0, 2.0, 3.0 });
            Console.WriteLine(@"Vector");
            Console.WriteLine(vector.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 1. Using operator "*"
            var resultV = matrixA * vector;
            Console.WriteLine(@"Multiply matrix by vector using operator *. (result = A * vec)");
            Console.WriteLine(resultV.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 2. Using Multiply method and getting result into different vector instance
            resultV = (DenseVector)matrixA.Multiply(vector);
            Console.WriteLine(@"Multiply matrix by vector using method Multiply. (result = A.Multiply(vec))");
            Console.WriteLine(resultV.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 3. Using Multiply method and updating vector itself
            matrixA.Multiply(vector, vector);
            Console.WriteLine(@"Multiply matrix by vector using method Multiply. (A.Multiply(vec, vec))");
            Console.WriteLine(vector.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // Multiply vector by matrix (left-multiply)
            // 1. Using operator "*"
            resultV = vector * matrixA;
            Console.WriteLine(@"Multiply vector by matrix using operator *. (result = vec * A)");
            Console.WriteLine(resultV.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 2. Using LeftMultiply method and getting result into different vector instance
            resultV = (DenseVector)matrixA.LeftMultiply(vector);
            Console.WriteLine(@"Multiply vector by matrix using method LeftMultiply. (result = A.LeftMultiply(vec))");
            Console.WriteLine(resultV.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 3. Using LeftMultiply method and updating vector itself
            matrixA.LeftMultiply(vector, vector);
            Console.WriteLine(@"Multiply vector by matrix using method LeftMultiply. (A.LeftMultiply(vec, vec))");
            Console.WriteLine(vector.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // Multiply matrix by matrix
            // 1. Using operator "*"
            resultM = matrixA * matrixB;
            Console.WriteLine(@"Multiply matrix by matrix using operator *. (result = A * B)");
            Console.WriteLine(resultM.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 2. Using Multiply method and getting result into different matrix instance
            resultM = (DenseMatrix)matrixA.Multiply(matrixB);
            Console.WriteLine(@"Multiply matrix by matrix using method Multiply. (result = A.Multiply(B))");
            Console.WriteLine(resultM.ToString("#0.00\t", formatProvider));
            Console.WriteLine();
//.........这里部分代码省略.........