NAG C Library Function Document

1Purpose

nag_dggqrf (f08zec) computes a generalized $QR$ factorization of a real matrix pair $\left(A,B\right)$, where $A$ is an $n$ by $m$ matrix and $B$ is an $n$ by $p$ matrix.

2Specification

 #include #include
 void nag_dggqrf (Nag_OrderType order, Integer n, Integer m, Integer p, double a[], Integer pda, double taua[], double b[], Integer pdb, double taub[], NagError *fail)

3Description

nag_dggqrf (f08zec) forms the generalized $QR$ factorization of an $n$ by $m$ matrix $A$ and an $n$ by $p$ matrix $B$
 $A =QR , B=QTZ ,$
where $Q$ is an $n$ by $n$ orthogonal matrix, $Z$ is a $p$ by $p$ orthogonal matrix and $R$ and $T$ are of the form
 $R = mmR11n-m0() , if ​n≥m; nm-nnR11R12() , if ​n
with ${R}_{11}$ upper triangular,
 $T = p-nnn0T12() , if ​n≤p, pn-pT11pT21() , if ​n>p,$
with ${T}_{12}$ or ${T}_{21}$ upper triangular.
In particular, if $B$ is square and nonsingular, the generalized $QR$ factorization of $A$ and $B$ implicitly gives the $QR$ factorization of ${B}^{-1}A$ as
 $B-1A= ZT T-1 R .$

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 http://www.netlib.org/lapack/lug
Anderson E, Bai Z and Dongarra J (1992) Generalized QR factorization and its applications Linear Algebra Appl. (Volume 162–164) 243–271
Hammarling S (1987) The numerical solution of the general Gauss-Markov linear model Mathematics in Signal Processing (eds T S Durrani, J B Abbiss, J E Hudson, R N Madan, J G McWhirter and T A Moore) 441–456 Oxford University Press
Paige C C (1990) Some aspects of generalized $QR$ factorizations . In Reliable Numerical Computation (eds M G Cox and S Hammarling) 73–91 Oxford University Press

5Arguments

1:    $\mathbf{order}$Nag_OrderTypeInput
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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of rows of the matrices $A$ and $B$.
Constraint: ${\mathbf{n}}\ge 0$.
3:    $\mathbf{m}$IntegerInput
On entry: $m$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
4:    $\mathbf{p}$IntegerInput
On entry: $p$, the number of columns of the matrix $B$.
Constraint: ${\mathbf{p}}\ge 0$.
5:    $\mathbf{a}\left[\mathit{dim}\right]$doubleInput/Output
Note: the dimension, dim, of the array a must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×{\mathbf{m}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×{\mathbf{pda}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
The $\left(i,j\right)$th element of the matrix $A$ is stored in
• ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$ 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 $n$ by $m$ matrix $A$.
On exit: the elements on and above the diagonal of the array contain the $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,m\right)$ by $m$ upper trapezoidal matrix $R$ ($R$ is upper triangular if $n\ge m$); the elements below the diagonal, with the array taua, represent the orthogonal matrix $Q$ as a product of $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,m\right)$ elementary reflectors (see Section 3.3.6 in the f08 Chapter Introduction).
6:    $\mathbf{pda}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
7:    $\mathbf{taua}\left[\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{n}},{\mathbf{m}}\right)\right]$doubleOutput
On exit: the scalar factors of the elementary reflectors which represent the orthogonal matrix $Q$.
8:    $\mathbf{b}\left[\mathit{dim}\right]$doubleInput/Output
Note: the dimension, dim, of the array b must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdb}}×{\mathbf{p}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×{\mathbf{pdb}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
Where ${\mathbf{B}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{b}}\left[\left(j-1\right)×{\mathbf{pdb}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{b}}\left[\left(i-1\right)×{\mathbf{pdb}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the $n$ by $p$ matrix $B$.
On exit: if $n\le p$, the upper triangle of the subarray ${\mathbf{B}}\left(1:n,p-n+1:p\right)$ contains the $n$ by $n$ upper triangular matrix ${T}_{12}$.
If $n>p$, the elements on and above the $\left(n-p\right)$th subdiagonal contain the $n$ by $p$ upper trapezoidal matrix $T$; the remaining elements, with the array taub, represent the orthogonal matrix $Z$ as a product of elementary reflectors (see Section 3.3.6 in the f08 Chapter Introduction).
9:    $\mathbf{pdb}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$.
10:  $\mathbf{taub}\left[\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{n}},{\mathbf{p}}\right)\right]$doubleOutput
On exit: the scalar factors of the elementary reflectors which represent the orthogonal matrix $Z$.
11:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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, ${\mathbf{p}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{p}}\ge 0$.
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}>0$.
On entry, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdb}}>0$.
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: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$ and ${\mathbf{p}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$.
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

7Accuracy

The computed generalized $QR$ factorization is the exact factorization for nearby matrices $\left(A+E\right)$ and $\left(B+F\right)$, where
 $E2 = O⁡ε A2 and F2= O⁡ε B2 ,$
and $\epsilon$ is the machine precision.

8Parallelism and Performance

nag_dggqrf (f08zec) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dggqrf (f08zec) 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 orthogonal matrices $Q$ and $Z$ may be formed explicitly by calls to nag_dorgqr (f08afc) and nag_dorgrq (f08cjc) respectively. nag_dormqr (f08agc) may be used to multiply $Q$ by another matrix and nag_dormrq (f08ckc) may be used to multiply $Z$ by another matrix.
The complex analogue of this function is nag_zggqrf (f08zsc).

10Example

This example solves the general Gauss–Markov linear model problem
 $minx y2 subject to d=Ax+By$
where
 $A = -0.57 -1.28 -0.39 -1.93 1.08 -0.31 2.30 0.24 -0.40 -0.02 1.03 -1.43 , B= 0.5 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 2.0 0.0 0.0 0.0 0.0 5.0 and d= 1.32 -4.00 5.52 3.24 .$
The solution is obtained by first computing a generalized $QR$ factorization of the matrix pair $\left(A,B\right)$. The example illustrates the general solution process, although the above data corresponds to a simple weighted least squares problem.

10.1Program Text

Program Text (f08zece.c)

10.2Program Data

Program Data (f08zece.d)

10.3Program Results

Program Results (f08zece.r)