Arguments

A

(input/output) REAL or COMPLEX array, shape $(:,:)$.
On entry, the matrix $A$.
On exit, $A$ has been overwritten by details of its complete orthogonal factorization.

B

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

RANK

Optional (output) INTEGER.
The effective rank of $A$, i.e., the order of the submatrix $R_{11}$. This is the same as the order of the submatrix $T_{11}$ in the complete orthogonal factorization of $A$.

JPVT

Optional (input/output) INTEGER array, shape $(:)$ with $size$(JPVT) $=$ $size$(A,2).
On entry, if ${\bf JPVT}_i \neq 0$, the $i^{th}$ column of $A$ is an initial column, otherwise it is a free column. Before the $QR$ factorization of $A$, all initial columns are permuted to the leading positions; only the remaining free columns are moved as a result of column pivoting during the factorization.
On exit, if ${\bf JPVT}_i = k$, then the $i^{th}$ column of the matrix product $A\,P$ was the $k^{th}$ column of $A$.

RCOND

Optional (input) REAL.
${\bf RCOND}$ is used to determine the effective rank of $A$. This is defined as the order of the largest leading triangular submatrix $R_{11}$ in the $QR$ factorization of $A$, with pivoting, whose estimated condition number $< 1/{\bf RCOND}$.
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 [17,9,20,35].