template<typename MatrixType_, int Options_>
class Eigen::JacobiSVD< MatrixType_, Options_ >
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
Options
this optional parameter allows one to specify the type of QR decomposition that will be used internally for the R-SVD step for non-square matrices. Additionally, it allows one to specify whether to compute thin or full unitaries U and V. 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 singularvalues of A and the columns of U and V are known as the left and right singularvectors 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 thinU 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 thinU 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 is the matrix m:
-0.824 -0.199
0.924 -0.237
-0.0532 0.146
Its singular values are:
1.24
0.338
Its left singular vectors are the columns of the thin U matrix:
-0.657 -0.695
0.753 -0.579
-0.048 0.426
Its right singular vectors are the columns of the thin V matrix:
0.999 0.044
-0.044 0.999
Now consider this rhs vector:
1
0
0
A least-squares solution of m*x = rhs is:
-0.62
-2.03
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 QR preconditioners that can be set with Options template parameter 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.
One may also use the Options template parameter to specify how the unitaries should be computed. The options are ComputeThinU, ComputeThinV, ComputeFullU, ComputeFullV. It is not possible to request both the thin and full versions of a unitary. By default, unitaries will not be computed.
You can set the QRPreconditioner and unitary options together: JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner | ComputeThinU | ComputeFullV>
See also
MatrixBase::jacobiSvd()
Inheritance diagram for Eigen::JacobiSVD< MatrixType_, Options_ >:
Method performing the decomposition of given matrix. Computes Thin/Full unitaries U/V if specified using the Options template parameter or the class constructor.
Method performing the decomposition of given matrix. Computes Thin/Full unitaries U/V if specified using the Options template parameter or the class constructor.
This method has to determine which singular values should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).
The documentation for this class was generated from the following files:
![\[ A = U S V^* \]](form_199_dark.png)
Inheritance diagram for Eigen::JacobiSVD< MatrixType_, Options_ >:
1.14.0