NAG CL Interface
f08ahc (dgelqf)

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1 Purpose

f08ahc computes the LQ factorization of a real m×n matrix.

2 Specification

#include <nag.h>
void  f08ahc (Nag_OrderType order, Integer m, Integer n, double a[], Integer pda, double tau[], NagError *fail)
The function may be called by the names: f08ahc, nag_lapackeig_dgelqf or nag_dgelqf.

3 Description

f08ahc forms the LQ factorization of an arbitrary rectangular real m×n matrix. No pivoting is performed.
If mn, the factorization is given by:
A = ( L 0 ) Q  
where L is an m×m lower triangular matrix and Q is an n×n orthogonal matrix. It is sometimes more convenient to write the factorization as
A = ( L 0 ) ( Q1 Q2 )  
which reduces to
A = LQ1 ,  
where Q1 consists of the first m rows of Q, and Q2 the remaining n-m rows.
If m>n, L is trapezoidal, and the factorization can be written
A = ( L1 L2 ) Q  
where L1 is lower triangular and L2 is rectangular.
The LQ factorization of A is essentially the same as the QR factorization of AT, since
A = ( L 0 ) QAT= QT ( LT 0 ) .  
The matrix Q is not formed explicitly but is represented as a product of min(m,n) elementary reflectors (see 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<m, the information returned in the first k rows of the array a represents an LQ factorization of the first k rows of the original matrix A.

4 References

None.

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.
The (i,j)th element of the matrix A is stored in
  • 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 elements above the diagonal are overwritten by details of the orthogonal matrix Q and the lower triangle is overwritten by the corresponding elements of the m×m lower triangular matrix L.
If m>n, the strictly upper triangular part is overwritten by details of the orthogonal matrix Q and the remaining elements are overwritten by the corresponding elements of the m×n lower trapezoidal matrix L.
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: further details of the orthogonal matrix Q.
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

f08ahc 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 m2 (3n-m) if mn or 23 n2 (3m-n) if m>n.
To form the orthogonal matrix Q f08ahc may be followed by a call to f08ajc :
nag_lapackeig_dorglq(order,n,n,MIN(m,n),&a,pda,tau,&fail)
but note that the first dimension of the array a must be at least n, which may be larger than was required by f08ahc.
When mn, it is often only the first m rows of Q that are required, and they may be formed by the call:
nag_lapackeig_dorglq(order,m,n,m,&a,pda,tau,&fail)
To apply Q to an arbitrary m×p real rectangular matrix C, f08ahc may be followed by a call to f08akc . For example,
nag_lapackeig_dormlq(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 f08avc.

10 Example

This example finds the minimum norm solutions of the underdetermined systems of linear equations
Ax1= b1   and   Ax2= b2  
where b1 and b2 are the columns of the matrix B,
A = ( -5.42 3.28 -3.68 0.27 2.06 0.46 -1.65 -3.40 -3.20 -1.03 -4.06 -0.01 -0.37 2.35 1.90 4.31 -1.76 1.13 -3.15 -0.11 1.99 -2.70 0.26 4.50 )   and   B= ( -2.87 -5.23 1.63 0.29 -3.52 4.76 0.45 -8.41 ) .  

10.1 Program Text

Program Text (f08ahce.c)

10.2 Program Data

Program Data (f08ahce.d)

10.3 Program Results

Program Results (f08ahce.r)