NAG CL Interface
f08cec (dgeqlf)

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1 Purpose

f08cec computes a QL factorization of a real m×n matrix A.

2 Specification

#include <nag.h>
void  f08cec (Nag_OrderType order, Integer m, Integer n, double a[], Integer pda, double tau[], NagError *fail)
The function may be called by the names: f08cec, nag_lapackeig_dgeqlf or nag_dgeqlf.

3 Description

f08cec 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 3.4.6 in the F08 Chapter Introduction for details). Functions 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 https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Arguments

1: 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 order=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: order=Nag_RowMajor or Nag_ColMajor.
2: m Integer Input
On entry: m, the number of rows of the matrix A.
Constraint: m0.
3: n Integer Input
On entry: n, the number of columns of the matrix A.
Constraint: n0.
4: a[dim] double Input/Output
Note: the dimension, dim, of the array a must be at least
  • max(1,pda×n) when order=Nag_ColMajor;
  • max(1,m×pda) when order=Nag_RowMajor.
where A(i,j) appears in this document, it refers to the array element
  • a[(j-1)×pda+i-1] when order=Nag_ColMajor;
  • a[(i-1)×pda+j-1] when order=Nag_RowMajor.
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 3.4.6 in the F08 Chapter Introduction).
5: pda Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraints:
  • if order=Nag_ColMajor, pdamax(1,m);
  • if order=Nag_RowMajor, pdamax(1,n).
6: tau[dim] double Output
Note: the dimension, dim, 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).
7: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error 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.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, m=value.
Constraint: m0.
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
NE_INT_2
On entry, pda=value and m=value.
Constraint: pdamax(1,m).
On entry, pda=value and n=value.
Constraint: pdamax(1,n).
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.

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.
f08cec 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.

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 f08cec may be followed by a call to f08cfc :
nag_lapackeig_dorgql(order,m,m,MIN(m,n),&a,pda,tau,&fail)
but note that the second dimension of the array a must be at least m, which may be larger than was required by f08cec.
When mn, it is often only the first n columns of Q that are required, and they may be formed by the call:
nag_lapackeig_dorgql(order,m,n,n,&a,pda,tau,&fail)
To apply Q to an arbitrary m×p real rectangular matrix C, f08cec may be followed by a call to f08cgc . For example,
nag_lapackeig_dormql(order,Nag_LeftSide,Nag_Trans,m,p,MIN(m,n),&a,pda,tau,
&c,pdc,&fail)
forms the matrix product C=QTC.
The complex analogue of this function is f08csc.

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.

10.1 Program Text

Program Text (f08cece.c)

10.2 Program Data

Program Data (f08cece.d)

10.3 Program Results

Program Results (f08cece.r)