Purpose

LA_GESVD and LA_GESDD compute the singular values and, optionally, the left and/or right singular vectors from the singular value decomposition ($SVD$) of a real or complex $m \times n$ matrix $A$. The $SVD$ of $A$ is written

$$A = U \, \Sigma \, V^H$$

where $\Sigma$ is an $m \times n$ matrix which is zero except for its $\min(m,n)$ diagonal elements, $U$ is an $m \times m$ orthogonal (unitary) matrix, and $V$ is an $n\times n$ orthogonal (unitary) matrix. The diagonal elements of $\Sigma$, i.e., the values

$$\sigma_i \equiv \Sigma_{i,i}, \;\;\;\; i=1,2, \ldots, \min(m,n)$$

are the singular values of $A$; they are real and non-negative, and are returned in descending order. The first $\min(m,n)$ columns of $U$ and $V$ are the left and right singular vectors of $A$, respectively.
LA_GESDD solves the same problem as LA_GESVD but uses a divide and conquer method if singular vectors are desired. For large matrices it is usually much faster than LA_GESVD when singular vectors are desired, but uses more workspace.

Note: The routine returns $V^H$, not $V$.