NAG Library Routine Document

F08AVF (ZGELQF)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

F08AVF (ZGELQF) computes the

L Q

factorization of a complex

m

n

matrix.

2 Specification

SUBROUTINE F08AVF (

M, N, A, LDA, TAU, WORK, LWORK, INFO)

INTEGER	M, N, LDA, LWORK, INFO
COMPLEX (KIND=nag_wp)	A(LDA,), TAU(), WORK(max(1,LWORK))

The routine may be called by its LAPACK name zgelqf.

3 Description

F08AVF (ZGELQF) forms the

L Q

factorization of an arbitrary rectangular complex

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 (with real diagonal elements) and

Q

is an

n

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). Routines are provided to work with

Q

in this representation (see Section 8).

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 Parameters

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, $*$ ) – COMPLEX (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 unitary 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 unitary matrix $Q$ and the remaining elements are overwritten by the corresponding elements of the $m$ by $n$ lower trapezoidal matrix $L$ .
The diagonal elements of $L$ are real.
4: LDA – INTEGERInput: On entry: the first dimension of the array A as declared in the (sub)program from which F08AVF (ZGELQF) is called.
Constraint: $LDA \geq \max (1, M)$ .
5: TAU( $*$ ) – COMPLEX (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 unitary matrix $Q$ .
6: WORK( $\max (1, LWORK)$ ) – COMPLEX (KIND=nag_wp) arrayWorkspace: On exit: if $INFO = 0$ , the real part of $WORK (1)$ 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 F08AVF (ZGELQF) 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

Errors or warnings detected by the routine:

$INFO < 0$: 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 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

F08AVF (ZGELQF) may be followed by a call to F08AWF (ZUNGLQ):

CALL ZUNGLQ(N,N,MIN(M,N),A,LDA,TAU,WORK,LWORK,INFO)

but note that the first dimension of the array A, specified by the parameter LDA, must be at least N, which may be larger than was required by F08AVF (ZGELQF).

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 ZUNGLQ(M,N,M,A,LDA,TAU,WORK,LWORK,INFO)

To apply

Q

to an arbitrary complex rectangular matrix

C

, F08AVF (ZGELQF) may be followed by a call to F08AXF (ZUNMLQ). For example,

CALL ZUNMLQ('Left','Conjugate Transpose',M,P,MIN(M,N),A,LDA,TAU, &
             C,LDC,WORK,LWORK,INFO)

forms the matrix product

C = Q^{H} C

, where

C

m

p

The real analogue of this routine is F08AHF (DGELQF).

9 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} 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}) .

NAG Library Routine DocumentF08AVF (ZGELQF)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

F08AVF (ZGELQF)