/// <summary> /// Constructs a new Cholesky Decomposition. /// </summary> /// /// <param name="value"> /// The symmetric matrix, given in upper triangular form, to be decomposed.</param> /// <param name="robust"> /// True to perform a square-root free LDLt decomposition, false otherwise.</param> /// <param name="inPlace"> /// True to perform the decomposition in place, storing the factorization in the /// lower triangular part of the given matrix.</param> /// <param name="valueType"> /// How to interpret the matrix given to be decomposed. Using this parameter, a lower or /// upper-triangular matrix can be interpreted as a symmetric matrix by assuming both lower /// and upper parts contain the same elements. Use this parameter in conjunction with inPlace /// to save memory by storing the original matrix and its decomposition at the same memory /// location (lower part will contain the decomposition's L matrix, upper part will contains /// the original matrix).</param> /// public CholeskyDecompositionF(Single[,] value, bool robust = false, bool inPlace = false, MatrixType valueType = MatrixType.UpperTriangular) { if (value.Rows() != value.Columns()) { throw new DimensionMismatchException("value", "Matrix is not square."); } if (!inPlace) { value = value.Copy(); } this.n = value.Rows(); this.L = value.ToUpperTriangular(valueType, result: value); this.robust = robust; if (robust) { LDLt(); // Compute square-root free decomposition } else { LLt(); // Compute standard Cholesky decomposition } }
/// <summary>Least squares solution of <c>X * A = B</c></summary> /// <param name="value">Right-hand-side matrix with as many columns as <c>A</c> and any number of rows.</param> /// <returns>A matrix that minimized the two norm of <c>X * Q * R - B</c>.</returns> /// <exception cref="T:System.ArgumentException">Matrix column dimensions must be the same.</exception> /// <exception cref="T:System.InvalidOperationException">Matrix is rank deficient.</exception> public Single[,] SolveTranspose(Single[,] value) { if (value == null) { throw new ArgumentNullException("value", "Matrix cannot be null."); } if (value.Columns() != qr.Rows()) { throw new ArgumentException("Matrix row dimensions must agree."); } if (!this.FullRank) { throw new InvalidOperationException("Matrix is rank deficient."); } // Copy right hand side int count = value.Rows(); var X = value.Transpose(); // Compute Y = transpose(Q)*B for (int k = 0; k < p; k++) { for (int j = 0; j < count; j++) { Single s = 0; for (int i = k; i < n; i++) { s += qr[i, k] * X[i, j]; } s = -s / qr[k, k]; for (int i = k; i < n; i++) { X[i, j] += s * qr[i, k]; } } } // Solve R*X = Y; for (int k = m - 1; k >= 0; k--) { for (int j = 0; j < count; j++) { X[k, j] /= Rdiag[k]; } for (int i = 0; i < k; i++) { for (int j = 0; j < count; j++) { X[i, j] -= X[k, j] * qr[i, k]; } } } return(Matrix.Create(count, p, X, transpose: true)); }
/// <summary> /// Creates a new Cholesky decomposition directly from /// an already computed left triangular matrix <c>L</c>. /// </summary> /// <param name="leftTriangular">The left triangular matrix from a Cholesky decomposition.</param> /// public static CholeskyDecompositionF FromLeftTriangularMatrix(Single[,] leftTriangular) { var chol = new CholeskyDecompositionF(); chol.n = leftTriangular.Rows(); chol.L = leftTriangular; chol.positiveDefinite = true; chol.robust = false; chol.D = Vector.Ones <Single>(chol.n); return(chol); }
/// <summary>Constructs a QR decomposition.</summary> /// <param name="value">The matrix A to be decomposed.</param> /// <param name="transpose">True if the decomposition should be performed on /// the transpose of A rather than A itself, false otherwise. Default is false.</param> /// <param name="inPlace">True if the decomposition should be done in place, /// overriding the given matrix <paramref name="value"/>. Default is false.</param> /// <param name="economy">True to perform the economy decomposition, where only ///.the information needed to solve linear systems is computed. If set to false, /// the full QR decomposition will be computed.</param> public QrDecompositionF(Single[,] value, bool transpose = false, bool economy = true, bool inPlace = false) { if (value == null) { throw new ArgumentNullException("value", "Matrix cannot be null."); } if ((!transpose && value.Rows() < value.Columns()) || (transpose && value.Columns() < value.Rows())) { throw new ArgumentException("Matrix has more columns than rows.", "value"); } // https://www.inf.ethz.ch/personal/gander/papers/qrneu.pdf if (transpose) { this.p = value.Rows(); if (economy) { // Compute the faster, economy-sized QR decomposition this.qr = value.Transpose(inPlace: inPlace); } else { // Create room to store the full decomposition this.qr = Matrix.Create(value.Columns(), value.Columns(), value, transpose: true); } } else { this.p = value.Columns(); if (economy) { // Compute the faster, economy-sized QR decomposition this.qr = inPlace ? value : value.Copy(); } else { // Create room to store the full decomposition this.qr = Matrix.Create(value.Rows(), value.Rows(), value, transpose: false); } } this.economy = economy; this.n = qr.Rows(); this.m = qr.Columns(); this.Rdiag = new Single[m]; for (int k = 0; k < m; k++) { // Compute 2-norm of k-th column without under/overflow. Single nrm = 0; for (int i = k; i < n; i++) { nrm = Tools.Hypotenuse(nrm, qr[i, k]); } if (nrm != 0) { // Form k-th Householder vector. if (qr[k, k] < 0) { nrm = -nrm; } for (int i = k; i < n; i++) { qr[i, k] /= nrm; } qr[k, k] += 1; // Apply transformation to remaining columns. for (int j = k + 1; j < m; j++) { Single s = 0; for (int i = k; i < n; i++) { s += qr[i, k] * qr[i, j]; } s = -s / qr[k, k]; for (int i = k; i < n; i++) { qr[i, j] += s * qr[i, k]; } } } this.Rdiag[k] = -nrm; } }
/// <summary> /// Solves a set of equation systems of type <c>A * X = B</c>. /// </summary> /// /// <param name="value">Right hand side matrix with as many rows as <c>A</c> and any number of columns.</param> /// <returns>Matrix <c>X</c> so that <c>L * L' * X = B</c>.</returns> /// <exception cref="T:System.ArgumentException">Matrix dimensions do not match.</exception> /// <exception cref="T:System.NonSymmetricMatrixException">Matrix is not symmetric.</exception> /// <exception cref="T:System.NonPositiveDefiniteMatrixException">Matrix is not positive-definite.</exception> /// <param name="inPlace">True to compute the solving in place, false otherwise.</param> /// public Single[,] Solve(Single[,] value, bool inPlace) { if (value == null) { throw new ArgumentNullException("value"); } if (value.Rows() != n) { throw new ArgumentException("Argument matrix should have the same number of rows as the decomposed matrix.", "value"); } if (!robust && !positiveDefinite) { throw new NonPositiveDefiniteMatrixException("Decomposed matrix is not positive definite."); } if (undefined) { throw new InvalidOperationException("The decomposition is undefined (zero in diagonal)."); } if (destroyed) { throw new InvalidOperationException("The decomposition has been destroyed."); } // int count = value.Columns(); var B = inPlace ? value : value.MemberwiseClone(); int m = B.Columns(); // Solve L*Y = B; for (int k = 0; k < n; k++) { for (int j = 0; j < m; j++) { for (int i = 0; i < k; i++) { B[k, j] -= B[i, j] * L[k, i]; } B[k, j] /= L[k, k]; } } if (robust) { for (int k = 0; k < D.Length; k++) { for (int j = 0; j < m; j++) { B[k, j] /= D[k]; } } } // Solve L'*X = Y; for (int k = n - 1; k >= 0; k--) { for (int j = 0; j < m; j++) { for (int i = k + 1; i < n; i++) { B[k, j] -= B[i, j] * L[i, k]; } B[k, j] /= L[k, k]; } } return(B); }