template<typename _MatrixType, int QRPreconditioner>
JacobiSVD class
Two-sided Jacobi SVD decomposition of a rectangular matrix.
| Template parameters | |
|---|---|
| _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. |
Contents
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:
- ColPivHouseholderQRPreconditioner is the default. In practice it's very safe. It uses column-pivoting QR.
- FullPivHouseholderQRPreconditioner, is the safest and slowest. It uses full-pivoting QR. Contrary to other QRs, it doesn't allow computing thin unitaries.
- HouseholderQRPreconditioner is the fastest, and less safe and accurate than the pivoting variants. It uses non-pivoting QR. This is very similar in safety and accuracy to the bidiagonalization process used by bidiagonalizing SVD algorithms (since bidiagonalization is inherently non-pivoting). However the resulting SVD is still more reliable than bidiagonalizing SVDs because the Jacobi-based iterarive process is more reliable than the optimized bidiagonal SVD iterations.
- NoQRPreconditioner allows not to use a QR preconditioner at all. This is useful if you know that you will only be computing JacobiSVD decompositions of square matrices. Non-square matrices require a QR preconditioner. Using this option will result in faster compilation and smaller executable code. It won't significantly speed up computation, since JacobiSVD is always checking if QR preconditioning is needed before applying it anyway.
Base classes
-
template<typename Derived>class SVDBase
- Base class of SVD algorithms.
Public types
- enum (anonymous) { RowsAtCompileTime = MatrixType::RowsAtCompileTime, ColsAtCompileTime = MatrixType::ColsAtCompileTime, DiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_DYNAMIC(RowsAtCompileTime,ColsAtCompileTime), MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime, MaxDiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(MaxRowsAtCompileTime,MaxColsAtCompileTime), MatrixOptions = MatrixType::Options }
- using ColType = internal::plain_col_type<MatrixType>::type
- using MatrixType = _MatrixType
-
using MatrixUType = Base::
MatrixUType -
using MatrixVType = Base::
MatrixVType - using RealScalar = NumTraits<typename MatrixType::Scalar>::Real
- using RowType = internal::plain_row_type<MatrixType>::type
- using Scalar = MatrixType::Scalar
- using SingularValuesType = Base::SingularValuesType
- using WorkMatrixType = Matrix<Scalar, DiagSizeAtCompileTime, DiagSizeAtCompileTime, MatrixOptions, MaxDiagSizeAtCompileTime, MaxDiagSizeAtCompileTime>
Constructors, destructors, conversion operators
Public functions
- auto cols() const -> Index
- auto compute(const MatrixType& matrix, unsigned int computationOptions) -> JacobiSVD&
- Method performing the decomposition of given matrix using custom options.
- auto compute(const MatrixType& matrix) -> JacobiSVD&
- Method performing the decomposition of given matrix using current options.
- auto computeU() const -> bool
- auto computeV() const -> bool
- auto rank() const -> Index
- auto rows() const -> Index
Protected variables
- Index m_cols
- unsigned int m_computationOptions
- bool m_computeFullU
- bool m_computeFullV
- bool m_computeThinU
- bool m_computeThinV
- Index m_diagSize
- bool m_isAllocated
- bool m_isInitialized
- MatrixUType m_matrixU
- MatrixVType m_matrixV
- Index m_nonzeroSingularValues
- RealScalar m_prescribedThreshold
- internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreColsThanRows> m_qr_precond_morecols
- internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreRowsThanCols> m_qr_precond_morerows
- Index m_rows
- MatrixType m_scaledMatrix
- SingularValuesType m_singularValues
- bool m_usePrescribedThreshold
- WorkMatrixType m_workMatrix
Enum documentation
template<typename _MatrixType, int QRPreconditioner>
enum Eigen::
| Enumerators | |
|---|---|
| RowsAtCompileTime | |
| ColsAtCompileTime | |
| DiagSizeAtCompileTime | |
| MaxRowsAtCompileTime | |
| MaxColsAtCompileTime | |
| MaxDiagSizeAtCompileTime | |
| MatrixOptions |
Typedef documentation
template<typename _MatrixType, int QRPreconditioner>
typedef internal::plain_col_type<MatrixType>::type Eigen::
template<typename _MatrixType, int QRPreconditioner>
typedef _MatrixType Eigen::
template<typename _MatrixType, int QRPreconditioner>
typedef Base::
template<typename _MatrixType, int QRPreconditioner>
typedef Base::
template<typename _MatrixType, int QRPreconditioner>
typedef NumTraits<typename MatrixType::Scalar>::Real Eigen::
template<typename _MatrixType, int QRPreconditioner>
typedef internal::plain_row_type<MatrixType>::type Eigen::
template<typename _MatrixType, int QRPreconditioner>
typedef MatrixType::Scalar Eigen::
template<typename _MatrixType, int QRPreconditioner>
typedef Base::SingularValuesType Eigen::
template<typename _MatrixType, int QRPreconditioner>
typedef Matrix<Scalar, DiagSizeAtCompileTime, DiagSizeAtCompileTime, MatrixOptions, MaxDiagSizeAtCompileTime, MaxDiagSizeAtCompileTime> Eigen::
Function documentation
template<typename _MatrixType, int QRPreconditioner>
Eigen::
Default Constructor.
The default constructor is useful in cases in which the user intends to perform decompositions via JacobiSVD::
template<typename _MatrixType, int QRPreconditioner>
Eigen::
Default Constructor with memory preallocation.
Like the default constructor but with preallocation of the internal data according to the specified problem size.
template<typename _MatrixType, int QRPreconditioner>
Eigen::
Constructor performing the decomposition of given matrix.
| Parameters | |
|---|---|
| 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.
template<typename _MatrixType, int QRPreconditioner>
JacobiSVD& Eigen::
Method performing the decomposition of given matrix using custom options.
| Parameters | |
|---|---|
| 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.
template<typename _MatrixType, int QRPreconditioner>
JacobiSVD& Eigen::
Method performing the decomposition of given matrix using current options.
| Parameters | |
|---|---|
| matrix | the matrix to decompose |
This method uses the current computationOptions, as already passed to the constructor or to compute(const MatrixType&, unsigned int).
template<typename _MatrixType, int QRPreconditioner>
bool Eigen::
| Returns | true if U (full or thin) is asked for in this SVD decomposition |
|---|
template<typename _MatrixType, int QRPreconditioner>
bool Eigen::
| Returns | true if V (full or thin) is asked for in this SVD decomposition |
|---|