This step required $O(n^3)$ work.What you say, that is, the singular values are the absolute values of the eigenvalues, makes only sense for normal matrices, e.g., Hermitian ones. Since $U_A B V_A^H = A$, $A = (U_A U_B) \Sigma (V_A V_B)^H$, so that the SVD can be explicitly formed after applying the Householder transforms which implicitly define $U_A$ and $V_A$ to $U_B$ and $V_B$, respectively. This step will require $O(n^2)$ work when MRRR is made stable for SVD, but currently requires as much as $O(n^3)$ work. The bidiagonal SVD yields $\$ such that $B = U_B \Sigma V_B^H$. LAPACK currently uses a divide and conquer approach for the bidiagonal SVD. Though the best-known algorithm for real symmetric tridiagonal EVP's (MRRR) is not yet, to my knowledge, stable for the bidiagonal SVD, there is an interesting discussion here. A variation on an algorithm for computing the eigenvalue decomposition of a real symmetric tridiagonal matrix is used to compute a bidiagonal SVD.On exit of this subroutine, the Householder vectors (normalized so that their first entry is implicitly one) for $U_A$ and $V_A$ are respectively stored in the portions of $B$ below and to the right of the main and super-diagonal. Implicitly computing unitary matrices $U_A$ and $V_A$, as products of Householder transforms, such that the general matrix $A$ is reduced to a real, upper bidiagonal matrix $B := U_A^H A V_A$.LAPACK's SVD routines, such as zgesvd, essentially proceed in three stages for square matrices: In general, there is no obvious relationship between the singular values of a given matrix and its eigenvalues, except in the case of normal matrices, where the singular values are the absolute value of the eigenvalues.ĭue to the wording of your question, I am assuming that your matrix is square. Cleve Moler's Numerical Computing with MATLAB gives a nice overview of the differences between singular values and eigenvalues. The SVD computes the singular values of a matrix. These two sets of quantities are not the same. To get background on this step, you'll need to consult Matrix Computations by Golub and Van Loan, or Applied Numerical Linear Algebra by Jim Demmel.įinally, you should not confuse singular values with eigenvalues. Then, xGESDD uses a divide-and-conquer algorithm to calculate the singular values and left and right singular vectors from the bidiagonal matrix. Lecture 31 in Numerical Linear Algebra by Trefethen and Bau gives an overview of this process. The algorithm used to reduce to bidiagonal form in LAPACK is probably the Lawson-Hanson-Chan algorithm, and it does use QR factorization at one point. First, the matrix to be decomposed is reduced to bidiagonal form. Singular value decompositions proceed in two stages. Janneb is correct that is a wrapper around xGESDD in LAPACK. SVD can also be used to solve linear systems, but it will be more expensive than QR factorization. For rectangular systems, QR factorization is a better choice because it will yield least-squares solutions to overdetermined linear systems. However, it requires more floating-point operations than LU factorization by a factor of 2 for square matrices (I believe JackPoulson may correct me on that). QR factorization is a more stable numerical algorithm for solving linear systems, which is probably why it gives you such precise results. There are some pathological cases for which LU factorization with partial pivoting is unstable (see Lecture 22 in Numerical Linear Algebra by Trefethen and Bau for details). However, for solving a linear system, LU factorization (with partial pivoting, which is the standard implementation in LAPACK) is extremely reliable in practice. The SVD is a more accurate, reliable algorithm for the determination of numerical rank, but it requires more floating-point operations. Instead, use a rank-revealing QR decomposition (such as xGEQPX or xGEPQY in LAPACK, where x is C, D, S, or Z, though those routines are difficult to track down see JedBrown's answer on a related question), or use an SVD (singular value decomposition, such as xGESDD or xGESVD, where x is again C, D, S, or Z). LU factorization is unreliable for this purpose in floating-point arithmetic. There are a number of issues in your question.ĭo not use Gaussian Elimination (LU factorization) to calculate the numerical rank of a matrix.
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