Arguments

A

(input/output) REAL or COMPLEX array, shape $(:,:)$.
On entry, the matrix $A$.
On exit, the first $\min(size$(A,1), $size$(A,2)) rows of $A$ are overwritten with its right singular vectors, stored rowwise.

B

(input/output) REAL or COMPLEX array, shape $(:,:)$ with $size$(B,1) $= \max(size$(A,1), $size$(A,2)) or shape $(:)$ with $size$(B) $= \max(size$(A,1), $size$(A,2)).
On entry, the matrix $B$.
On exit, the solution matrix $X$.
If $size$(A,1) $\geq size$(A,2) and RANK $= size$(A,2), the residual sum-of-squares for the solution in a column of B is given by the sum of squares of elements in rows $size$(A,2)$+ 1:size$(A,1) of that column.

RANK

Optional (output) INTEGER.
The effective rank of $A$, i.e., the number of singular values of $A$ which are greater than the product ${\bf RCOND}\times\sigma_1$, where $\sigma_1$ is the greatest singular value.

S

Optional (output) REAL array, shape $(:)$ with $size$(S) $= \min(size$(A,1), $size$(A,2)).
The singular values of $A$ in decreasing order.
The condition number of $A$ in the 2-norm is $\kappa_2(A) = \sigma_1/\sigma_{\min(size({\bf A},1),size({\bf A},2))}$.

RCOND

Optional (input) REAL.
${\bf RCOND}$ is used to determine the effective rank of $A$.
Singular values $\sigma_i \leq {\bf RCOND}\times\sigma_1$ are treated as zero.
Default value: $10\times \max(size({\bf A},1), size({\bf A},2))\times {\bf EPSILON}(1.0\_{\it wp})$, where wp is the working precision.

INFO

Optional (output) INTEGER.


If ${\bf INFO}$ is not present and an error occurs, then the program is terminated with an error message.

References: [1] and [17,9,20].