NAG CL Interfacef08afc (dorgqr)

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1Purpose

f08afc generates all or part of the real orthogonal matrix $Q$ from a $QR$ factorization computed by f08aec or f08bfc.

2Specification

 #include
 void f08afc (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: f08afc, nag_lapackeig_dorgqr or nag_dorgqr.

3Description

f08afc is intended to be used after a call to f08aec or f08bfc. which perform a $QR$ factorization of a real matrix $A$. The orthogonal matrix $Q$ is represented as a product of elementary reflectors.
This function may be used to generate $Q$ explicitly as a square matrix, or to form only its leading columns.
Usually $Q$ is determined from the $QR$ factorization of an $m×p$ matrix $A$ with $m\ge p$. The whole of $Q$ may be computed by :
`nag_lapackeig_dorgqr(order,m,m,p,a,pda,tau,&fail)`
(note that the array a must have at least $m$ columns) or its leading $p$ columns by :
`nag_lapackeig_dorgqr(order,m,p,p,a,pda,tau,&fail)`
The columns of $Q$ returned by the last call form an orthonormal basis for the space spanned by the columns of $A$; thus f08aec followed by f08afc can be used to orthogonalize the columns of $A$.
The information returned by the $QR$ factorization functions also yields the $QR$ factorization of the leading $k$ columns of $A$, where $k. The orthogonal matrix arising from this factorization can be computed by :
`nag_lapackeig_dorgqr(order,m,m,k,a,pda,tau,&fail)`
or its leading $k$ columns by :
`nag_lapackeig_dorgqr(order,m,k,k,a,pda,tau,&fail)`

4References

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 order of the orthogonal matrix $Q$.
Constraint: ${\mathbf{m}}\ge 0$.
3: $\mathbf{n}$Integer Input
On entry: $n$, the number of columns of the matrix $Q$.
Constraint: ${\mathbf{m}}\ge {\mathbf{n}}\ge 0$.
4: $\mathbf{k}$Integer Input
On entry: $k$, the number of elementary reflectors whose product defines the matrix $Q$.
Constraint: ${\mathbf{n}}\ge {\mathbf{k}}\ge 0$.
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 f08aec or f08bfc.
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: further details of the elementary reflectors, as returned by f08aec or f08bfc.
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 ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{m}}\ge {\mathbf{n}}\ge 0$.
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{k}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge {\mathbf{k}}\ge 0$.
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
 $‖E‖2 = O(ε) ,$
where $\epsilon$ is the machine precision.

8Parallelism and Performance

f08afc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08afc 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 $n=k$, the number is approximately $\frac{2}{3}{n}^{2}\left(3m-n\right)$.
The complex analogue of this function is f08atc.

10Example

This example forms the leading $4$ columns of the orthogonal matrix $Q$ from the $QR$ factorization of the matrix $A$, where
 $A = ( -0.57 -1.28 -0.39 0.25 -1.93 1.08 -0.31 -2.14 2.30 0.24 0.40 -0.35 -1.93 0.64 -0.66 0.08 0.15 0.30 0.15 -2.13 -0.02 1.03 -1.43 0.50 ) .$
The columns of $Q$ form an orthonormal basis for the space spanned by the columns of $A$.

10.1Program Text

Program Text (f08afce.c)

10.2Program Data

Program Data (f08afce.d)

10.3Program Results

Program Results (f08afce.r)