F08AEF (pdf version)
Chapter Contents
Chapter Introduction
NAG Library Manual

NAG Fortran Library Routine Document

F08AEF (DGEQRF)

Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

     1 Purpose
     7 Accuracy

1  Purpose

F08AEF (DGEQRF) computes the QR factorization of a real m by n matrix.

2  Specification

SUBROUTINE F08AEF (M, N, A, LDA, TAU, WORK, LWORK, INFO)
INTEGERM, N, LDA, LWORK, INFO
double precisionA(LDA,*), TAU(*), WORK(*)
The routine may be called by its LAPACK name dgeqrf.

3  Description

F08AEF (DGEQRF) forms the QR factorization of an arbitrary rectangular real m by n matrix. No pivoting is performed.
If mn, the factorization is given by:
A=Q R 0 ,
where R is an n by n upper triangular matrix and Q is an m by m orthogonal matrix. It is sometimes more convenient to write the factorization as
A= Q1 Q2 R 0 ,
which reduces to
A=Q1R,
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
A=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 8).
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 original matrix A.

4  References

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

5  Parameters

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:     A(LDA,*) – double precision 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 F08AEF (DGEQRF) is called.
Constraint: LDAmax1,M.
5:     TAU(*) – double precision arrayOutput
Note: the dimension of the array TAU must be at least max1,minM,N.
On exit: further details of the orthogonal matrix Q.
6:     WORK(*) – double precision arrayWorkspace
Note: the dimension of the array WORK must be at least max1,LWORK.
On exit: if INFO=0, WORK1 contains the minimum value of LWORK required for optimal performance.
7:     LWORK – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)program from which F08AEF (DGEQRF) 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, LWORKN×nb, where nb is the optimal block size.
Constraint: LWORKmax1,N or LWORK=-1.
8:     INFO – INTEGEROutput

6  Error Indicators and Warnings

Errors or warnings detected by the routine:
INFO < 0
If INFO=-i, the ith parameter had an illegal value. An explanatory message is output, and execution of the program is terminated.

7  Accuracy

The computed factorization is the exact factorization of a nearby matrix A+E, where
E2=OεA2,
and ε is the machine precision.

8  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 F08AEF (DGEQRF) 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 F08AEF (DGEQRF).
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, F08AEF (DGEQRF) 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,
+             LWORK,INFO)
forms C=QTC, where C is m by p.
To compute a QR factorization with column pivoting, use F08BEF (DGEQPF).
The complex analogue of this routine is F08ASF (ZGEQRF).

9  Example

This example solves the linear least-squares problems
minimize Axi - bi 2 ,   i=1,2
where b1 and b2 are the columns 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= -3.15 2.19 -0.11 -3.64 1.99 0.57 -2.70 8.23 0.26 -6.35 4.50 -1.48 .

9.1  Program Text

Program Text (f08aefe.f)

9.2  Program Data

Program Data (f08aefe.d)

9.3  Program Results

Program Results (f08aefe.r)


F08AEF (pdf version)
Chapter Contents
Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2006