# NAG CL Interfacef08cjc (dorgrq)

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## 1Purpose

f08cjc generates all or part of the real $n×n$ orthogonal matrix $Q$ from an $RQ$ factorization computed by f08chc.

## 2Specification

 #include
 void f08cjc (Nag_OrderType order, Integer m, Integer n, Integer k, double a[], Integer pda, const double tau[], NagError *fail)
The function may be called by the names: f08cjc, nag_lapackeig_dorgrq or nag_dorgrq.

## 3Description

f08cjc is intended to be used following a call to f08chc, which performs an $RQ$ factorization of a real matrix $A$ and represents the orthogonal matrix $Q$ as a product of $k$ elementary reflectors of order $n$.
This function may be used to generate $Q$ explicitly as a square matrix, or to form only its trailing rows.
Usually $Q$ is determined from the $RQ$ factorization of a $p×n$ matrix $A$ with $p\le n$. The whole of $Q$ may be computed by :
`nag_lapackeig_dorgrq(order,n,n,p,a,pda,tau,info)`
(note that the matrix $A$ must have at least $n$ rows) or its trailing $p$ rows by :
`nag_lapackeig_dorgrq(order,p,n,p,a,pda,tau,info)`
The rows of $Q$ returned by the last call form an orthonormal basis for the space spanned by the rows of $A$; thus f08chc followed by f08cjc can be used to orthogonalize the rows of $A$.
The information returned by f08chc also yields the $RQ$ factorization of the trailing $k$ rows of $A$, where $k. The orthogonal matrix arising from this factorization can be computed by :
`nag_lapackeig_dorgrq(order,n,n,k,a,pda,tau,info)`
or its leading $k$ columns by :
`nag_lapackeig_dorgrq(order,k,n,k,a,pda,tau,info)`

## 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 $Q$.
Constraint: ${\mathbf{m}}\ge 0$.
3: $\mathbf{n}$Integer Input
On entry: $n$, the number of columns of the matrix $Q$.
Constraint: ${\mathbf{n}}\ge {\mathbf{m}}$.
4: $\mathbf{k}$Integer Input
On entry: $k$, the number of elementary reflectors whose product defines the matrix $Q$.
Constraint: ${\mathbf{m}}\ge {\mathbf{k}}\ge 0$.
5: $\mathbf{a}\left[\mathit{dim}\right]$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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: details of the vectors which define the elementary reflectors, as returned by f08chc.
On exit: the $m×n$ matrix $Q$.
6: $\mathbf{pda}$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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
7: $\mathbf{tau}\left[\mathit{dim}\right]$const double Input
Note: the dimension, dim, of the array tau must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
On entry: ${\mathbf{tau}}\left[i-1\right]$ must contain the scalar factor of the elementary reflector ${H}_{i}$, as returned by f08chc.
8: $\mathbf{fail}$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, ${\mathbf{pda}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pda}}>0$.
NE_INT_2
On entry, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{k}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{m}}\ge {\mathbf{k}}\ge 0$.
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge {\mathbf{m}}$.
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 matrix $Q$ differs from an exactly orthogonal matrix by a matrix $E$ such that
 $‖E‖2 = O⁡ε$
and $\epsilon$ is the machine precision.

## 8Parallelism and Performance

f08cjc 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 $4mnk-2\left(m+n\right){k}^{2}+\frac{4}{3}{k}^{3}$; when $m=k$ this becomes $\frac{2}{3}{m}^{2}\left(3n-m\right)$.
The complex analogue of this function is f08cwc.

## 10Example

This example generates the first four rows of the matrix $Q$ of the $RQ$ factorization of $A$ as returned by f08chc, where
 $A = ( -0.57 -1.93 2.30 -1.93 0.15 -0.02 -1.28 1.08 0.24 0.64 0.30 1.03 -0.39 -0.31 0.40 -0.66 0.15 -1.43 0.25 -2.14 -0.35 0.08 -2.13 0.50 ) .$

### 10.1Program Text

Program Text (f08cjce.c)

### 10.2Program Data

Program Data (f08cjce.d)

### 10.3Program Results

Program Results (f08cjce.r)