Integer, Intent (In)	::	m, n, lda, lwork
Integer, Intent (Inout)	::	jpvt(*)
Integer, Intent (Out)	::	info
Real (Kind=nag_wp), Intent (Inout)	::	rwork(*)
Complex (Kind=nag_wp), Intent (Inout)	::	a(lda,), tau()
Complex (Kind=nag_wp), Intent (Out)	::	work(max(1,lwork))

C Header Interface

#include <nag.h>

void	f08btf_ (const Integer m, const Integer n, Complex a[], const Integer lda, Integer jpvt[], Complex tau[], Complex work[], const Integer lwork, double rwork[], Integer *info)

The routine may be called by the names f08btf, nagf_lapackeig_zgeqp3 or its LAPACK name zgeqp3.

3 Description

f08btf forms the

Q R

factorization, with column pivoting, of an arbitrary rectangular complex

m \times n

matrix.

m \geq n

, the factorization is given by:

A P = Q (\begin{array}{r} R \\ 0 \end{array}),

where

R

is an

n \times n

upper triangular matrix (with real diagonal elements),

Q

is an

m \times m

unitary matrix and

P

is an

n \times n

permutation matrix. It is sometimes more convenient to write the factorization as

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

which reduces to

A P = Q_{1} R,

where

Q_{1}

consists of the first

n

columns of

Q

, and

Q_{2}

the remaining

m - n

columns.

m < n

R

is trapezoidal, and the factorization can be written

A P = Q (\begin{array}{r} R_{1} & R_{2} \end{array}),

where

R_{1}

is upper triangular and

R_{2}

is rectangular.

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 < n

, the information returned in the first

k

columns of the array a represents a

Q R

factorization of the first

k

columns of the permuted matrix

A P

The routine allows specified columns of

A

to be moved to the leading columns of

A P

at the start of the factorization and fixed there. The remaining columns are free to be interchanged so that at the

i

th stage the pivot column is chosen to be the column which maximizes the

2

-norm of elements

i

m

over columns

i

n

4 References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia https://www.netlib.org/lapack/lug

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Arguments

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

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

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

On entry: the $m \times n$ matrix $A$ .

On exit: if $m \geq n$ , the elements below the diagonal are overwritten by details of the unitary matrix $Q$ and the upper triangle is overwritten by the corresponding elements of the $n \times n$ upper triangular matrix $R$ .
If $m < n$ , the strictly lower triangular part is overwritten by details of the unitary matrix $Q$ and the remaining elements are overwritten by the corresponding elements of the $m \times n$ upper trapezoidal matrix $R$ .

The diagonal elements of $R$ are real.
4: $lda$ – Integer Input: On entry: the first dimension of the array a as declared in the (sub)program from which f08btf is called.

Constraint: $lda \geq \max (1, m)$ .
5: $jpvt (*)$ – Integer array Input/Output: Note: the dimension of the array jpvt must be at least $\max (1, n)$ .

On entry: if $jpvt (j) \neq 0$ , the $j$ th column of $A$ is moved to the beginning of $A P$ before the decomposition is computed and is fixed in place during the computation. Otherwise, the $j$ th column of $A$ is a free column (i.e., one which may be interchanged during the computation with any other free column).

On exit: details of the permutation matrix $P$ . More precisely, if $jpvt (j) = k$ , the $k$ th column of $A$ is moved to become the $j$ th column of $A P$ ; in other words, the columns of $A P$ are the columns of $A$ in the order $jpvt (1), jpvt (2), \dots, jpvt (n)$ .
6: $tau (*)$ – Complex (Kind=nag_wp) array Output: Note: the dimension of the array tau must be at least $\max (1, \min (m, n))$ .

On exit: further details of the unitary matrix $Q$ .
7: $work (\max (1, lwork))$ – Complex (Kind=nag_wp) array Workspace: On exit: if $info = 0$ , the real part of $work (1)$ contains the minimum value of lwork required for optimal performance.
8: $lwork$ – Integer Input: On entry: the dimension of the array work as declared in the (sub)program from which f08btf 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, $lwork \geq (n + 1) \times nb$ , where $nb$ is the optimal block size.

Constraint: $lwork \geq n + 1$ or $lwork = −1$ .
9: $rwork (*)$ – Real (Kind=nag_wp) array Workspace: Note: the dimension of the array rwork must be at least $\max (1, 2 \times n)$ .
10: $info$ – Integer Output: 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

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

f08btf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08btf 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 real floating-point operations is approximately

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

m \geq n

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

m < n

To form the unitary matrix

Q

f08btf may be followed by a call to f08atf :

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

but note that the second dimension of the array a must be at least m, which may be larger than was required by f08btf.

When

m \geq n

, it is often only the first

n

columns of

Q

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

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

To apply

Q

to an arbitrary

m \times p

complex rectangular matrix

C

, f08btf may be followed by a call to f08auf . For example,

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

forms the matrix product

C = Q^{H} C

To compute a

Q R

factorization without column pivoting, use f08asf.

The real analogue of this routine is f08bff.

10 Example

This example solves the linear least squares problems

\min_{x} {‖ b_{j} - A x_{j} ‖}_{2}, j = 1, 2

for the basic solutions

x_{1}

and

x_{2}

, where

A = (\begin{array}{r} 0.47 - 0.34 i & - 0.40 + 0.54 i & 0.60 + 0.01 i & 0.80 - 1.02 i \\ - 0.32 - 0.23 i & - 0.05 + 0.20 i & - 0.26 - 0.44 i & - 0.43 + 0.17 i \\ 0.35 - 0.60 i & - 0.52 - 0.34 i & 0.87 - 0.11 i & - 0.34 - 0.09 i \\ 0.89 + 0.71 i & - 0.45 - 0.45 i & - 0.02 - 0.57 i & 1.14 - 0.78 i \\ - 0.19 + 0.06 i & 0.11 - 0.85 i & 1.44 + 0.80 i & 0.07 + 1.14 i \end{array})

and

B = (\begin{array}{r} - 1.08 - 2.59 i & 2.22 + 2.35 i \\ - 2.61 - 1.49 i & 1.62 - 1.48 i \\ 3.13 - 3.61 i & 1.65 + 3.43 i \\ 7.33 - 8.01 i & - 0.98 + 3.08 i \\ 9.12 + 7.63 i & - 2.84 + 2.78 i \end{array}) .

and

b_{j}

is the

j

th column of the matrix

B

. The solution is obtained by first obtaining a

Q R

factorization with column pivoting of the matrix

A

. A tolerance of

0.01

is used to estimate the rank of

A

from the upper triangular factor,

R

Note that the block size (NB) of

64

assumed in this example is not realistic for such a small problem, but should be suitable for large problems.

f08bt: FL CL CPP AD PY MB

NAG FL Interfacef08btf (zgeqp3)

▸▿ 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 FL Interface
f08btf (zgeqp3)