NAG FL Interface
f08cef (dgeqlf)

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1 Purpose

f08cef computes a QL factorization of a real m×n matrix A.

2 Specification

Fortran Interface
Subroutine f08cef ( m, n, a, lda, tau, work, lwork, info)
Integer, Intent (In) :: m, n, lda, lwork
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 <nag.h>
void  f08cef_ (const Integer *m, const Integer *n, double a[], const Integer *lda, double tau[], double work[], const Integer *lwork, Integer *info)
The routine may be called by the names f08cef, nagf_lapackeig_dgeqlf or its LAPACK name dgeqlf.

3 Description

f08cef forms the QL factorization of an arbitrary rectangular real m×n matrix.
If mn, the factorization is given by:
A = Q ( 0 L ) ,  
where L is an n×n lower triangular matrix and Q is an m×m orthogonal matrix. If m<n the factorization is given by
A = QL ,  
where L is an m×n lower trapezoidal matrix and Q is again an m×m orthogonal matrix. In the case where m>n the factorization can be expressed as
A = ( Q1 Q2 ) ( 0 L ) = Q2 L ,  
where Q1 consists of the first m-n columns of Q, and Q2 the remaining n columns.
The matrix Q is not formed explicitly but is represented as a product of min(m,n) elementary reflectors (see Section 3.3.6 in 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 last k columns of the array a represents a QL factorization of the last k columns of the original matrix A.

4 References

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

5 Arguments

1: m Integer Input
On entry: m, the number of rows of the matrix A.
Constraint: m0.
2: n Integer Input
On entry: n, the number of columns of the matrix A.
Constraint: n0.
3: a(lda,*) Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least max(1,n).
On entry: the m×n matrix A.
On exit: if mn, the lower triangle of the subarray a(m-n+1:m,1:n) contains the n×n lower triangular matrix L.
If mn, the elements on and below the (n-m)th superdiagonal contain the m×n lower trapezoidal matrix L. The remaining elements, with the array tau, represent the orthogonal matrix Q as a product of elementary reflectors (see Section 3.3.6 in the F08 Chapter Introduction).
4: lda Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f08cef is called.
Constraint: ldamax(1,m).
5: tau(*) Real (Kind=nag_wp) array Output
Note: the dimension of the array tau must be at least max(1,min(m,n)).
On exit: the scalar factors of the elementary reflectors (see Section 9).
6: work(max(1,lwork)) Real (Kind=nag_wp) array Workspace
On exit: if info=0, work(1) contains the minimum value of lwork required for optimal performance.
7: lwork Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f08cef 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: lworkmax(1,n) or lwork=−1.
8: info Integer Output
On exit: info=0 unless the routine detects an error (see Section 6).

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.

7 Accuracy

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

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08cef 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.

9 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 f08cef may be followed by a call to f08cff :
Call dorgql(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 f08cef.
When mn, it is often only the first n columns of Q that are required, and they may be formed by the call:
Call dorgql(m,n,n,a,lda,tau,work,lwork,info)
To apply Q to an arbitrary m×p real rectangular matrix C, f08cef may be followed by a call to f08cgf . For example,
Call dormql('Left','Transpose',m,p,min(m,n),a,lda,tau,c,ldc,work, &
forms the matrix product C=QTC.
The complex analogue of this routine is f08csf.

10 Example

This example solves the linear least squares problems
minx bj-Axj2 , ​ j=1,2  
for x1 and x2, where bj is the jth column 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= ( -2.67 0.41 -0.55 -3.10 3.34 -4.01 -0.77 2.76 0.48 -6.17 4.10 0.21 ) .  
The solution is obtained by first obtaining a QL factorization of the matrix A.
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.

10.1 Program Text

Program Text (f08cefe.f90)

10.2 Program Data

Program Data (f08cefe.d)

10.3 Program Results

Program Results (f08cefe.r)