# NAG CL Interfacef08cec (dgeqlf)

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## 1Purpose

f08cec computes a $QL$ factorization of a real $m×n$ matrix $A$.

## 2Specification

 #include
 void f08cec (Nag_OrderType order, Integer m, Integer n, double a[], Integer pda, double tau[], NagError *fail)
The function may be called by the names: f08cec, nag_lapackeig_dgeqlf or nag_dgeqlf.

## 3Description

f08cec forms the $QL$ factorization of an arbitrary rectangular real $m×n$ matrix.
If $m\ge n$, the factorization is given by:
 $A = Q ( 0 L ) ,$
where $L$ is an $n×n$ lower triangular matrix and $Q$ is an $m×m$ orthogonal matrix. If $m the factorization is given by
 $A = QL ,$
where $L$ is an $m×n$ lower trapezoidal matrix and $Q$ is again an $m×m$ orthogonal matrix. In the case where $m>n$ the factorization can be expressed as
 $A = ( Q1 Q2 ) ( 0 L ) = Q2 L ,$
where ${Q}_{1}$ consists of the first $m-n$ columns of $Q$, and ${Q}_{2}$ the remaining $n$ columns.
The matrix $Q$ is not formed explicitly but is represented as a product of $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ elementary reflectors (see Section 3.4.6 in the F08 Chapter Introduction for details). Functions are provided to work with $Q$ in this representation (see Section 9).
Note also that for any $k, the information returned in the last $k$ columns of the array a represents a $QL$ factorization of the last $\mathrm{k}$ columns of the original matrix $A$.

## 4References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5Arguments

1: $\mathbf{order}$Nag_OrderType Input
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2: $\mathbf{m}$Integer Input
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
3: $\mathbf{n}$Integer Input
On entry: $n$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4: $\mathbf{a}\left[\mathit{dim}\right]$double Input/Output
Note: the dimension, dim, of the array a must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×{\mathbf{n}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}×{\mathbf{pda}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
where ${\mathbf{A}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the $m×n$ matrix $A$.
On exit: if $m\ge n$, the lower triangle of the subarray ${\mathbf{A}}\left(m-n+1:m,1:n\right)$ contains the $n×n$ lower triangular matrix $L$.
If $m\le n$, the elements on and below the $\left(n-m\right)$th superdiagonal contain the $m×n$ lower trapezoidal matrix $L$. The remaining elements, with the array tau, represent the orthogonal matrix $Q$ as a product of elementary reflectors (see Section 3.4.6 in the F08 Chapter Introduction).
5: $\mathbf{pda}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
6: $\mathbf{tau}\left[\mathit{dim}\right]$double Output
Note: the dimension, dim, of the array tau must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$.
On exit: the scalar factors of the elementary reflectors (see Section 9).
7: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{m}}\ge 0$.
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{pda}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pda}}>0$.
NE_INT_2
On entry, ${\mathbf{pda}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
On entry, ${\mathbf{pda}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

## 7Accuracy

The computed factorization is the exact factorization of a nearby matrix $\left(A+E\right)$, where
 $‖E‖2 = O(ε) ‖A‖2 ,$
and $\epsilon$ is the machine precision.

## 8Parallelism and Performance

f08cec makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The total number of floating-point operations is approximately $\frac{2}{3}{n}^{2}\left(3m-n\right)$ if $m\ge n$ or $\frac{2}{3}{m}^{2}\left(3n-m\right)$ if $m.
To form the orthogonal matrix $Q$ f08cec may be followed by a call to f08cfc :
`nag_lapackeig_dorgql(order,m,m,MIN(m,n),&a,pda,tau,&fail)`
but note that the second dimension of the array a must be at least m, which may be larger than was required by f08cec.
When $m\ge n$, it is often only the first $n$ columns of $Q$ that are required, and they may be formed by the call:
`nag_lapackeig_dorgql(order,m,n,n,&a,pda,tau,&fail)`
To apply $Q$ to an arbitrary $m×p$ real rectangular matrix $C$, f08cec may be followed by a call to f08cgc . For example,
```nag_lapackeig_dormql(order,Nag_LeftSide,Nag_Trans,m,p,MIN(m,n),&a,pda,tau,
&c,pdc,&fail)```
forms the matrix product $C={Q}^{\mathrm{T}}C$.
The complex analogue of this function is f08csc.

## 10Example

This example solves the linear least squares problems
 $minx ‖bj-Axj‖2 , ​ j=1,2$
for ${x}_{1}$ and ${x}_{2}$, where ${b}_{j}$ is the $j$th column of the matrix $B$,
 $A = ( -0.57 -1.28 -0.39 0.25 -1.93 1.08 -0.31 -2.14 2.30 0.24 0.40 -0.35 -1.93 0.64 -0.66 0.08 0.15 0.30 0.15 -2.13 -0.02 1.03 -1.43 0.50 ) and B= ( -2.67 0.41 -0.55 -3.10 3.34 -4.01 -0.77 2.76 0.48 -6.17 4.10 0.21 ) .$
The solution is obtained by first obtaining a $QL$ factorization of the matrix $A$.

### 10.1Program Text

Program Text (f08cece.c)

### 10.2Program Data

Program Data (f08cece.d)

### 10.3Program Results

Program Results (f08cece.r)