f08av: FL CL CPP AD PY MB

NAG CL Interface
f08avc (zgelqf)

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NAG Library Manual, Mark 30.1

Interfaces: FL CL CPP AD PY MB

NAG CL Interface Introduction

F08 (Lapackeig) Chapter Contents

F08 (Lapackeig) Chapter Introduction

f08av: FL CL CPP AD PY MB

▸▿ 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

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1 Purpose

f08avc computes the

L Q

factorization of a complex

m \times n

matrix.

2 Specification

#include <nag.h>

void	f08avc (Nag_OrderType order, Integer m, Integer n, Complex a[], Integer pda, Complex tau[], NagError *fail)

The function may be called by the names: f08avc, nag_lapackeig_zgelqf or nag_zgelqf.

3 Description

f08avc forms the

L Q

factorization of an arbitrary rectangular complex

m \times 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 \times m

lower triangular matrix (with real diagonal elements) and

Q

is an

n \times n

unitary 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^{H}

, since

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

Nag_ColMajor

2: $m$ – Integer Input

On entry:

m

, the number of rows of the matrix

A

Constraint:

m \geq 0

3: $n$ – Integer Input

On entry:

n

, the number of columns of the matrix

A

Constraint:

n \geq 0

4: $a [\dim]$ – Complex Input/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 \times n

matrix

A

On exit: if

m \leq n

, the elements above the diagonal are overwritten by details of the unitary matrix

Q

and the lower triangle is overwritten by the corresponding elements of the

m \times m

lower triangular matrix

L

m > n

, the strictly upper triangular part is overwritten by details of the unitary matrix

Q

and the remaining elements are overwritten by the corresponding elements of the

m \times n

lower trapezoidal matrix

L

The diagonal elements of

L

are real.

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

6: $tau [\dim]$ – Complex Output

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

\max (1, \min (m, n))

On exit: further details of the unitary 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: $m \geq 0$ .

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.
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

{‖ E ‖}_{2} = O (ε) {‖ A ‖}_{2},

and

ε

is the machine precision.

8 Parallelism and Performance

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

f08avc 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 real floating-point operations is approximately

\frac{8}{3} m^{2} (3 n - m)

m \leq n

\frac{8}{3} n^{2} (3 m - n)

m > n

To form the unitary matrix

Q

f08avc may be followed by a call to f08awc :

nag_lapackeig_zunglq(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 f08avc.

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

To apply

Q

to an arbitrary

m \times p

complex rectangular matrix

C

, f08avc may be followed by a call to f08axc . For example,

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

forms the matrix product

C = Q^{H} C

The real analogue of this function is f08ahc.

10 Example

This example finds the minimum norm solutions of the underdetermined 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} 0.28 - 0.36 i & 0.50 - 0.86 i & - 0.77 - 0.48 i & 1.58 + 0.66 i \\ - 0.50 - 1.10 i & - 1.21 + 0.76 i & - 0.32 - 0.24 i & - 0.27 - 1.15 i \\ 0.36 - 0.51 i & - 0.07 + 1.33 i & - 0.75 + 0.47 i & - 0.08 + 1.01 i \end{array})

and

B = (\begin{array}{r} - 1.35 + 0.19 i & 4.83 - 2.67 i \\ 9.41 - 3.56 i & - 7.28 + 3.34 i \\ - 7.57 + 6.93 i & 0.62 + 4.53 i \end{array}) .

f08av: FL CL CPP AD PY MB

NAG CL Interfacef08avc (zgelqf)

▸▿ 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 CL Interface
f08avc (zgelqf)