C# DenseMatrix.Svd方法代码示例

        public void CanCheckRankOfSquareSingular(int order)
        {
            var matrixA = new DenseMatrix(order, order);
            matrixA[0, 0] = 1;
            matrixA[order - 1, order - 1] = 1;
            for (var i = 1; i < order - 1; i++)
            {
                matrixA[i, i - 1] = 1;
                matrixA[i, i + 1] = 1;
                matrixA[i - 1, i] = 1;
                matrixA[i + 1, i] = 1;
            }

            var factorSvd = matrixA.Svd();

            Assert.AreEqual(factorSvd.Determinant, 0);
            Assert.AreEqual(factorSvd.Rank, order - 1);
        }

        /// <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();
        }