Function givens$($A$,$i$,$k$).$ The function takes a matrix of size m $\backslash$times n$,$ m n and returns the Givens rotation Ri$,$k such that the multiplication RT A annihilates theelement in the i$-$th column and k$-$th row. Function householder$($x$),$ that takes a vector x from the space Rm and computes the Householder reflection P such that it annihilates all components of the vector except the first one. Function specialQR$($A$,$ extended$),$ that takes a matrix A Rm$\backslash$times n and a logical value extended. It calculates the QR decomposition of matrix A as follows:$($a$)$ Iterate through all columns of matrix A$.($b$)$ For each column, check how many non$-$zero elements are below the main diago$-$ nal. If there is only one non$-$zero element, eliminate it using a Givens rotation; otherwise, use Householder reflection.$($c$)$ Return the matrices Q and R$.$ If the value of the extended is true, return the extended QR decomposition; otherwise, return the regular QR decomposition.