/// <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();
}