Two-sided Jacobi SVD decomposition of a rectangular matrix. More...
#include <JacobiSVD.h>
Two-sided Jacobi SVD decomposition of a rectangular matrix.
MatrixType | the type of the matrix of which we are computing the SVD decomposition |
QRPreconditioner | this optional parameter allows to specify the type of QR decomposition that will be used internally for the R-SVD step for non-square matrices. See discussion of possible values below. |
SVD decomposition consists in decomposing any n-by-p matrix A as a product
where U is a n-by-n unitary, V is a p-by-p unitary, and S is a n-by-p real positive matrix which is zero outside of its main diagonal; the diagonal entries of S are known as the singular values of A and the columns of U and V are known as the left and right singular vectors of A respectively.
Singular values are always sorted in decreasing order.
This JacobiSVD decomposition computes only the singular values by default. If you want U or V, you need to ask for them explicitly.
You can ask for only thin U or V to be computed, meaning the following. In case of a rectangular n-by-p matrix, letting m be the smaller value among n and p, there are only m singular vectors; the remaining columns of U and V do not correspond to actual singular vectors. Asking for thin U or V means asking for only their m first columns to be formed. So U is then a n-by-m matrix, and V is then a p-by-m matrix. Notice that thin U and V are all you need for (least squares) solving.
Here's an example demonstrating basic usage:
MatrixXf m = MatrixXf::Random(3,2); cout << "Here is the matrix m:" << endl << m << endl; JacobiSVD<MatrixXf> svd(m, ComputeThinU | ComputeThinV); cout << "Its singular values are:" << endl << svd.singularValues() << endl; cout << "Its left singular vectors are the columns of the thin U matrix:" << endl << svd.matrixU() << endl; cout << "Its right singular vectors are the columns of the thin V matrix:" << endl << svd.matrixV() << endl; Vector3f rhs(1, 0, 0); cout << "Now consider this rhs vector:" << endl << rhs << endl; cout << "A least-squares solution of m*x = rhs is:" << endl << svd.solve(rhs) << endl;
Output:
Here is the matrix m: 0.68 0.597 -0.211 0.823 0.566 -0.605 Its singular values are: 1.19 0.899 Its left singular vectors are the columns of the thin U matrix: 0.388 0.866 0.712 -0.0634 -0.586 0.496 Its right singular vectors are the columns of the thin V matrix: -0.183 0.983 0.983 0.183 Now consider this rhs vector: 1 0 0 A least-squares solution of m*x = rhs is: 0.888 0.496
This JacobiSVD class is a two-sided Jacobi R-SVD decomposition, ensuring optimal reliability and accuracy. The downside is that it's slower than bidiagonalizing SVD algorithms for large square matrices; however its complexity is still where n is the smaller dimension and p is the greater dimension, meaning that it is still of the same order of complexity as the faster bidiagonalizing R-SVD algorithms. In particular, like any R-SVD, it takes advantage of non-squareness in that its complexity is only linear in the greater dimension.
If the input matrix has inf or nan coefficients, the result of the computation is undefined, but the computation is guaranteed to terminate in finite (and reasonable) time.
The possible values for QRPreconditioner are:
typedef internal::plain_col_type<MatrixType>::type ColType |
typedef MatrixType::Index Index |
typedef _MatrixType MatrixType |
typedef Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime, MatrixOptions, MaxRowsAtCompileTime, MaxRowsAtCompileTime> MatrixUType |
typedef Matrix<Scalar, ColsAtCompileTime, ColsAtCompileTime, MatrixOptions, MaxColsAtCompileTime, MaxColsAtCompileTime> MatrixVType |
typedef NumTraits<typename MatrixType::Scalar>::Real RealScalar |
typedef internal::plain_row_type<MatrixType>::type RowType |
typedef MatrixType::Scalar Scalar |
typedef internal::plain_diag_type<MatrixType, RealScalar>::type SingularValuesType |
typedef Matrix<Scalar, DiagSizeAtCompileTime, DiagSizeAtCompileTime, MatrixOptions, MaxDiagSizeAtCompileTime, MaxDiagSizeAtCompileTime> WorkMatrixType |
anonymous enum |
JacobiSVD | ( | ) | [inline] |
Default Constructor.
The default constructor is useful in cases in which the user intends to perform decompositions via JacobiSVD::compute(const MatrixType&).
Default Constructor with memory preallocation.
Like the default constructor but with preallocation of the internal data according to the specified problem size.
JacobiSVD | ( | const MatrixType & | matrix, |
unsigned int | computationOptions = 0 |
||
) | [inline] |
Constructor performing the decomposition of given matrix.
matrix | the matrix to decompose |
computationOptions | optional parameter allowing to specify if you want full or thin U or V unitaries to be computed. By default, none is computed. This is a bit-field, the possible bits are ComputeFullU, ComputeThinU, ComputeFullV, ComputeThinV. |
Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not available with the (non-default) FullPivHouseholderQR preconditioner.
References JacobiSVD< _MatrixType, QRPreconditioner >::compute().
JacobiSVD< MatrixType, QRPreconditioner > & compute | ( | const MatrixType & | matrix, |
unsigned int | computationOptions | ||
) |
Method performing the decomposition of given matrix using custom options.
matrix | the matrix to decompose |
computationOptions | optional parameter allowing to specify if you want full or thin U or V unitaries to be computed. By default, none is computed. This is a bit-field, the possible bits are ComputeFullU, ComputeThinU, ComputeFullV, ComputeThinV. |
Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not available with the (non-default) FullPivHouseholderQR preconditioner.
References abs(), p, q, Eigen::internal::real_2x2_jacobi_svd(), and JacobiRotation< Scalar >::transpose().
Referenced by JacobiSVD< _MatrixType, QRPreconditioner >::compute(), and JacobiSVD< _MatrixType, QRPreconditioner >::JacobiSVD().
JacobiSVD& compute | ( | const MatrixType & | matrix | ) | [inline] |
Method performing the decomposition of given matrix using current options.
matrix | the matrix to decompose |
This method uses the current computationOptions, as already passed to the constructor or to compute(const MatrixType&, unsigned int).
References JacobiSVD< _MatrixType, QRPreconditioner >::compute(), and JacobiSVD< _MatrixType, QRPreconditioner >::m_computationOptions.
References JacobiSVD< _MatrixType, QRPreconditioner >::m_computeFullU, and JacobiSVD< _MatrixType, QRPreconditioner >::m_computeThinU.
Referenced by JacobiSVD< _MatrixType, QRPreconditioner >::matrixU(), and JacobiSVD< _MatrixType, QRPreconditioner >::solve().
References JacobiSVD< _MatrixType, QRPreconditioner >::m_computeFullV, and JacobiSVD< _MatrixType, QRPreconditioner >::m_computeThinV.
Referenced by JacobiSVD< _MatrixType, QRPreconditioner >::matrixV(), and JacobiSVD< _MatrixType, QRPreconditioner >::solve().
const MatrixUType& matrixU | ( | ) | const [inline] |
For the SVD decomposition of a n-by-p matrix, letting m be the minimum of n and p, the U matrix is n-by-n if you asked for ComputeFullU, and is n-by-m if you asked for ComputeThinU.
The m first columns of U are the left singular vectors of the matrix being decomposed.
This method asserts that you asked for U to be computed.
References JacobiSVD< _MatrixType, QRPreconditioner >::computeU(), JacobiSVD< _MatrixType, QRPreconditioner >::m_isInitialized, and JacobiSVD< _MatrixType, QRPreconditioner >::m_matrixU.
Referenced by Transform< _Scalar, _Dim, _Mode, _Options >::computeRotationScaling(), Transform< _Scalar, _Dim, _Mode, _Options >::computeScalingRotation(), and Eigen::umeyama().
const MatrixVType& matrixV | ( | ) | const [inline] |
For the SVD decomposition of a n-by-p matrix, letting m be the minimum of n and p, the V matrix is p-by-p if you asked for ComputeFullV, and is p-by-m if you asked for ComputeThinV.
The m first columns of V are the right singular vectors of the matrix being decomposed.
This method asserts that you asked for V to be computed.
References JacobiSVD< _MatrixType, QRPreconditioner >::computeV(), JacobiSVD< _MatrixType, QRPreconditioner >::m_isInitialized, and JacobiSVD< _MatrixType, QRPreconditioner >::m_matrixV.
Referenced by Transform< _Scalar, _Dim, _Mode, _Options >::computeRotationScaling(), Transform< _Scalar, _Dim, _Mode, _Options >::computeScalingRotation(), QuaternionBase< Derived >::setFromTwoVectors(), and Eigen::umeyama().
Index nonzeroSingularValues | ( | ) | const [inline] |
References JacobiSVD< _MatrixType, QRPreconditioner >::m_isInitialized, and JacobiSVD< _MatrixType, QRPreconditioner >::m_nonzeroSingularValues.
const SingularValuesType& singularValues | ( | ) | const [inline] |
For the SVD decomposition of a n-by-p matrix, letting m be the minimum of n and p, the returned vector has size m. Singular values are always sorted in decreasing order.
References JacobiSVD< _MatrixType, QRPreconditioner >::m_isInitialized, and JacobiSVD< _MatrixType, QRPreconditioner >::m_singularValues.
Referenced by Transform< _Scalar, _Dim, _Mode, _Options >::computeRotationScaling(), Transform< _Scalar, _Dim, _Mode, _Options >::computeScalingRotation(), and Eigen::umeyama().
const internal::solve_retval<JacobiSVD, Rhs> solve | ( | const MatrixBase< Rhs > & | b | ) | const [inline] |
b | the right-hand-side of the equation to solve. |
References JacobiSVD< _MatrixType, QRPreconditioner >::computeU(), JacobiSVD< _MatrixType, QRPreconditioner >::computeV(), and JacobiSVD< _MatrixType, QRPreconditioner >::m_isInitialized.
Referenced by JacobiSVD< _MatrixType, QRPreconditioner >::cols().
unsigned int m_computationOptions [protected] |
Referenced by JacobiSVD< _MatrixType, QRPreconditioner >::compute().
bool m_computeFullU [protected] |
Referenced by JacobiSVD< _MatrixType, QRPreconditioner >::computeU().
bool m_computeFullV [protected] |
Referenced by JacobiSVD< _MatrixType, QRPreconditioner >::computeV().
bool m_computeThinU [protected] |
Referenced by JacobiSVD< _MatrixType, QRPreconditioner >::computeU().
bool m_computeThinV [protected] |
Referenced by JacobiSVD< _MatrixType, QRPreconditioner >::computeV().
Index m_diagSize [protected] |
bool m_isAllocated [protected] |
bool m_isInitialized [protected] |
Referenced by JacobiSVD< _MatrixType, QRPreconditioner >::matrixU(), JacobiSVD< _MatrixType, QRPreconditioner >::matrixV(), JacobiSVD< _MatrixType, QRPreconditioner >::nonzeroSingularValues(), JacobiSVD< _MatrixType, QRPreconditioner >::singularValues(), and JacobiSVD< _MatrixType, QRPreconditioner >::solve().
MatrixUType m_matrixU [protected] |
Referenced by JacobiSVD< _MatrixType, QRPreconditioner >::matrixU().
MatrixVType m_matrixV [protected] |
Referenced by JacobiSVD< _MatrixType, QRPreconditioner >::matrixV().
Index m_nonzeroSingularValues [protected] |
internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreColsThanRows> m_qr_precond_morecols [protected] |
internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreRowsThanCols> m_qr_precond_morerows [protected] |
Referenced by JacobiSVD< _MatrixType, QRPreconditioner >::rows().
SingularValuesType m_singularValues [protected] |
Referenced by JacobiSVD< _MatrixType, QRPreconditioner >::singularValues().
WorkMatrixType m_workMatrix [protected] |