NAG Library Routine Document

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 <nagmk26.h>

void	f08ahf_ (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 its LAPACK name dgelqf.

3

Description

f08ahf (dgelqf) forms the

L Q

factorization of an arbitrary rectangular real

m

n

matrix. No pivoting is performed.

m \leq n

, the factorization is given by:

A = (\begin{array}{r} L & 0 \end{array}) 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 = (\begin{array}{r} L & 0 \end{array}) (\begin{array}{r} Q_{1} \\ Q_{2} \end{array})

which reduces to

A = L Q_{1},

where

Q_{1}

consists of the first

m

rows of

Q

, and

Q_{2}

the remaining

n - m

rows.

m > n

L

is trapezoidal, and the factorization can be written

A = (\begin{array}{r} L_{1} \\ L_{2} \end{array}) Q

where

L_{1}

is lower triangular and

L_{2}

is rectangular.

The

L Q

factorization of

A

is essentially the same as the

Q R

factorization of

A^{T}

, since

A = (\begin{array}{r} L & 0 \end{array}) Q \Leftrightarrow A^{T} = Q^{T} (\begin{array}{r} L^{T} \\ 0 \end{array}) .

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). Routines 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

L Q

factorization of the first

k

rows of the original matrix

A

4

References

None.

5

Arguments

1: $m$ – IntegerInput: On entry: $m$ , the number of rows of the matrix $A$ .

Constraint: $m \geq 0$ .
2: $n$ – IntegerInput: On entry: $n$ , the number of columns of the matrix $A$ .

Constraint: $n \geq 0$ .
3: $a (lda *)$ – Real (Kind=nag_wp) arrayInput/Output: Note: the second dimension of the array a must be at least $\max (1, n)$ .

On entry: the $m$ by $n$ matrix $A$ .

On exit: if $m \leq n$ , 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 $m$ by $m$ 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 $m$ by $n$ lower trapezoidal matrix $L$ .
4: $lda$ – IntegerInput: On entry: the first dimension of the array a as declared in the (sub)program from which f08ahf (dgelqf) is called.

Constraint: $lda \geq \max (1, m)$ .
5: $tau (*)$ – Real (Kind=nag_wp) arrayOutput: Note: the dimension of the array tau must be at least $\max (1, \min (m, n))$ .

On exit: further details of the orthogonal matrix $Q$ .
6: $work (\max (1, lwork))$ – Real (Kind=nag_wp) arrayWorkspace: On exit: if $info = 0$ , $work (1)$ 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 f08ahf (dgelqf) 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, $lwork \geq m \times nb$ , where $nb$ is the optimal block size.

Constraint: $lwork \geq \max (1, m)$ or $lwork = - 1$ .
8: $info$ – IntegerOutput: On exit: $info = 0$ unless the routine detects an error (see Section 6).

6

Error Indicators and Warnings

$info < 0$: 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

{‖E‖}_{2} = O (ε) {‖A‖}_{2},

and

ε

is the machine precision.

8

Parallelism and Performance

f08ahf (dgelqf) 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

\frac{2}{3} m^{2} (3 n - m)

m \leq n

\frac{2}{3} n^{2} (3 m - n)

m > n

To form the orthogonal matrix

Q

f08ahf (dgelqf) may be followed by a call to f08ajf (dorglq):

Call dorglq(n,n,min(m,n),a,lda,tau,work,lwork,info)

but note that the first dimension of the array a, specified by the argument lda, must be at least n, which may be larger than was required by f08ahf (dgelqf).

When

m \leq n

, it is often only the first

m

rows of

Q

that are required, and they may be formed by the call:

Call dorglq(m,n,m,a,lda,tau,work,lwork,info)

To apply

Q

to an arbitrary real rectangular matrix

C

, f08ahf (dgelqf) may be followed by a call to f08akf (dormlq). For example,

Call dormlq('Left','Transpose',m,p,min(m,n),a,lda,tau,c,ldc, &
              work,lwork,info)

forms the matrix product

C = Q^{T} C

, where

C

m

p

The complex analogue of this routine is f08avf (zgelqf).

10

Example

This example finds the minimum norm solutions of the under-determined systems of linear equations

A x_{1} = b_{1} and A x_{2} = b_{2}

where

b_{1}

and

b_{2}

are the columns of the matrix

B

A = (\begin{array}{r} - 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 \end{array}) and B = (\begin{array}{r} - 2.87 & - 5.23 \\ 1.63 & 0.29 \\ - 3.52 & 4.76 \\ 0.45 & - 8.41 \end{array}) .

NAG Library Routine Document

f08ahf (dgelqf)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

10.2

Program Data

10.3

Program Results