NAG Library Function Document

nag_dgelqf (f08ahc)

+− 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

nag_dgelqf (f08ahc) computes the

L Q

factorization of a real

m

n

matrix.

2 Specification

#include <nag.h>

#include <nagf08.h>

void	nag_dgelqf (Nag_OrderType order, Integer m, Integer n, double a[], Integer pda, double tau[], NagError *fail)

3 Description

nag_dgelqf (f08ahc) 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). 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

L Q

factorization of the first

k

rows of the original matrix

A

4 References

None.

5 Arguments

1: order – Nag_OrderTypeInput

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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: m – IntegerInput

On entry:

m

, the number of rows of the matrix

A

Constraint:

m \geq 0

3: n – IntegerInput

On entry:

n

, the number of columns of the matrix

A

Constraint:

n \geq 0

4: a[ $\dim$ ] – doubleInput/Output

Note: the dimension, dim, of the array a must be at least

$\max (1, pda \times n)$ when $order = Nag_ColMajor$ ;
$\max (1, m \times pda)$ when $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

A

is stored in

$a [(j - 1) \times pda + i - 1]$ when $order = Nag_ColMajor$ ;
$a [(i - 1) \times pda + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

m

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

m

lower triangular matrix

L

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 – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array a.

Constraints:

if $order = Nag_ColMajor$ , $pda \geq \max (1, m)$ ;
if $order = Nag_RowMajor$ , $pda \geq \max (1, n)$ .

6: tau[ $\dim$ ] – doubleOutput

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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT: On entry, $m = ⟨value⟩$ .
Constraint: $m \geq 0$ .
On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 0$ .
On entry, $pda = ⟨value⟩$ .
Constraint: $pda > 0$ .
NE_INT_2: On entry, $pda = ⟨value⟩$ and $m = ⟨value⟩$ .
Constraint: $pda \geq \max (1, m)$ .
On entry, $pda = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $pda \geq \max (1, n)$ .
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.

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

nag_dgelqf (f08ahc) is not threaded by NAG in any implementation.

nag_dgelqf (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 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

nag_dgelqf (f08ahc) may be followed by a call to nag_dorglq (f08ajc):

nag_dorglq(order,n,n,MIN(m,n),&a,pda,tau,&fail)

but note that the first dimension of the array a, specified by the argument pda, must be at least n, which may be larger than was required by nag_dgelqf (f08ahc).

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:

nag_dorglq(order,m,n,m,&a,pda,tau,&fail)

To apply

Q

to an arbitrary real rectangular matrix

C

, nag_dgelqf (f08ahc) may be followed by a call to nag_dormlq (f08akc). For example,

nag_dormlq(order,Nag_LeftSide,Nag_Trans,m,p,MIN(m,n),&a,pda,
  tau,&c,pdc,&fail)

forms the matrix product

C = Q^{T} C

, where

C

m

p

The complex analogue of this function is nag_zgelqf (f08avc).

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 Function Documentnag_dgelqf (f08ahc)

+− 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

NAG Library Function Document

nag_dgelqf (f08ahc)