NAG Library Routine Document

f08bff (dgeqp3)


    1  Purpose
    7  Accuracy


f08bff (dgeqp3) computes the QR factorization, with column pivoting, of a real m by n matrix.


Fortran Interface
Subroutine f08bff ( m, n, a, lda, jpvt, tau, work, lwork, info)
Integer, Intent (In):: m, n, lda, lwork
Integer, Intent (Inout):: jpvt(*)
Integer, Intent (Out):: info
Real (Kind=nag_wp), Intent (Inout):: a(lda,*), tau(*)
Real (Kind=nag_wp), Intent (Out):: work(max(1,lwork))
C Header Interface
#include nagmk26.h
void  f08bff_ (const Integer *m, const Integer *n, double a[], const Integer *lda, Integer jpvt[], double tau[], double work[], const Integer *lwork, Integer *info)
The routine may be called by its LAPACK name dgeqp3.


f08bff (dgeqp3) forms the QR factorization, with column pivoting, of an arbitrary rectangular real m by n matrix.
If mn, the factorization is given by:
AP= Q R 0 ,  
where R is an n by n upper triangular matrix, Q is an m by m orthogonal matrix and P is an n by n permutation matrix. It is sometimes more convenient to write the factorization as
AP= Q1 Q2 R 0 ,  
which reduces to
AP= Q1 R ,  
where Q1 consists of the first n columns of Q, and Q2 the remaining m-n columns.
If m<n, R is trapezoidal, and the factorization can be written
AP= Q R1 R2 ,  
where R1 is upper triangular and R2 is rectangular.
The matrix Q is not formed explicitly but is represented as a product of minm,n elementary reflectors (see the F08 Chapter Introduction for details). Routines are provided to work with Q in this representation (see Section 9).
Note also that for any k<n, the information returned in the first k columns of the array a represents a QR factorization of the first k columns of the permuted matrix AP.
The routine allows specified columns of A to be moved to the leading columns of AP at the start of the factorization and fixed there. The remaining columns are free to be interchanged so that at the ith stage the pivot column is chosen to be the column which maximizes the 2-norm of elements i to m over columns i to n.


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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore


1:     m – IntegerInput
On entry: m, the number of rows of the matrix A.
Constraint: m0.
2:     n – IntegerInput
On entry: n, the number of columns of the matrix A.
Constraint: n0.
3:     alda* – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a must be at least max1,n.
On entry: the m by n matrix A.
On exit: if mn, the elements below the diagonal are overwritten by details of the orthogonal matrix Q and the upper triangle is overwritten by the corresponding elements of the n by n upper triangular matrix R.
If m<n, the strictly lower triangular part is overwritten by details of the orthogonal matrix Q and the remaining elements are overwritten by the corresponding elements of the m by n upper trapezoidal matrix R.
4:     lda – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08bff (dgeqp3) is called.
Constraint: ldamax1,m.
5:     jpvt* – Integer arrayInput/Output
Note: the dimension of the array jpvt must be at least max1,n.
On entry: if jpvtj0, the j th column of A is moved to the beginning of AP before the decomposition is computed and is fixed in place during the computation. Otherwise, the j th column of A is a free column (i.e., one which may be interchanged during the computation with any other free column).
On exit: details of the permutation matrix P. More precisely, if jpvtj=k, the kth column of A is moved to become the j th column of AP; in other words, the columns of AP are the columns of A in the order jpvt1,jpvt2,,jpvtn.
6:     tau* – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array tau must be at least max1,minm,n.
On exit: the scalar factors of the elementary reflectors.
7:     workmax1,lwork – Real (Kind=nag_wp) arrayWorkspace
On exit: if info=0, work1 contains the minimum value of lwork required for optimal performance.
8:     lwork – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08bff (dgeqp3) is called.
If lwork=-1, a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.
Suggested value: for optimal performance, lwork2×n+n+1×nb, where nb is the optimal block size.
Constraint: lwork3×n+1 or lwork=-1.
9:     info – IntegerOutput
On exit: info=0 unless the routine detects an error (see Section 6).

Error Indicators and Warnings

If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.


The computed factorization is the exact factorization of a nearby matrix A+E, where
E2 = Oε A2 ,  
and ε is the machine precision.

Parallelism and Performance

f08bff (dgeqp3) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08bff (dgeqp3) 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

Further Comments

The total number of floating-point operations is approximately 23 n2 3m-n  if mn or 23 m2 3n-m  if m<n.
To form the orthogonal matrix Q f08bff (dgeqp3) may be followed by a call to f08aff (dorgqr):
Call dorgqr(m,m,min(m,n),a,lda,tau,work,lwork,info)
but note that the second dimension of the array a must be at least m, which may be larger than was required by f08bff (dgeqp3).
When mn, it is often only the first n columns of Q that are required, and they may be formed by the call:
Call dorgqr(m,n,n,a,lda,tau,work,lwork,info)
To apply Q to an arbitrary real rectangular matrix C, f08bff (dgeqp3) may be followed by a call to f08agf (dormqr). For example,
Call dormqr('Left','Transpose',m,p,min(m,n),a,lda,tau,c,ldc,work, &
forms C=QTC, where C is m by p.
To compute a QR factorization without column pivoting, use f08aef (dgeqrf).
The complex analogue of this routine is f08btf (zgeqp3).


This example solves the linear least squares problems
minx bj - Axj 2 ,   j=1,2  
for the basic solutions x1 and x2, where
A = -0.09 0.14 -0.46 0.68 1.29 -1.56 0.20 0.29 1.09 0.51 -1.48 -0.43 0.89 -0.71 -0.96 -1.09 0.84 0.77 2.11 -1.27 0.08 0.55 -1.13 0.14 1.74 -1.59 -0.72 1.06 1.24 0.34   and  B= 7.4 2.7 4.2 -3.0 -8.3 -9.6 1.8 1.1 8.6 4.0 2.1 -5.7  
and bj is the jth column of the matrix B. The solution is obtained by first obtaining a QR factorization with column pivoting of the matrix A. A tolerance of 0.01 is used to estimate the rank of A from the upper triangular factor, R.
Note that the block size (NB) of 64 assumed in this example is not realistic for such a small problem, but should be suitable for large problems.

Program Text

Program Text (f08bffe.f90)

Program Data

Program Data (f08bffe.d)

Program Results

Program Results (f08bffe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017