void	f08zec (Nag_OrderType order, Integer n, Integer m, Integer p, double a[], Integer pda, double taua[], double b[], Integer pdb, double taub[], NagError *fail)

The function may be called by the names: f08zec, nag_lapackeig_dggqrf or nag_dggqrf.

3 Description

f08zec forms the generalized

Q R

factorization of an

n \times m

matrix

A

and an

n \times p

matrix

B

A = Q R, B = Q T Z,

where

Q

is an

n \times n

orthogonal matrix,

Z

is a

p \times p

orthogonal matrix and

R

and

T

are of the form

R = {\begin{matrix} \begin{array}{rc} m \\ m & R_{11} \\ n - m & 0 \\ ( & ) \end{array},   if ​ n \geq m; \\ \begin{array}{rc} n & m - n \\ n & R_{11} & R_{12} \\ ( & ) \end{array},   if ​ n < m, \end{matrix}

with

R_{11}

upper triangular,

T = {\begin{matrix} \begin{array}{rc} p - n & n \\ n & 0 & T_{12} \\ ( & ) \end{array},   if ​ n \leq p, \\ \begin{array}{rc} p \\ n - p & T_{11} \\ p & T_{21} \\ ( & ) \end{array},   if ​ n > p, \end{matrix}

with

T_{12}

T_{21}

upper triangular.

In particular, if

B

is square and nonsingular, the generalized

Q R

factorization of

A

and

B

implicitly gives the

Q R

factorization of

B^{- 1} A

B^{- 1} A = Z^{T} (T^{- 1} R) .

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

Anderson E, Bai Z and Dongarra J (1992) Generalized

Q R

factorization and its applications Linear Algebra Appl. (Volume 162–164) 243–271

Hammarling S (1987) The numerical solution of the general Gauss-Markov linear model Mathematics in Signal Processing (eds T S Durrani, J B Abbiss, J E Hudson, R N Madan, J G McWhirter and T A Moore) 441–456 Oxford University Press

Paige C C (1990) Some aspects of generalized

Q R

factorizations . In Reliable Numerical Computation (eds M G Cox and S Hammarling) 73–91 Oxford University Press

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: $n$ – Integer Input

On entry:

n

, the number of rows of the matrices

A

and

B

Constraint:

n \geq 0

3: $m$ – Integer Input

On entry:

m

, the number of columns of the matrix

A

Constraint:

m \geq 0

4: $p$ – Integer Input

On entry:

p

, the number of columns of the matrix

B

Constraint:

p \geq 0

5: $a [\dim]$ – double Input/Output

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

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

n \times m

matrix

A

On exit: the elements on and above the diagonal of the array contain the

\min (n, m) \times m

upper trapezoidal matrix

R

(

R

is upper triangular if

n \geq m

); the elements below the diagonal, with the array taua, represent the orthogonal matrix

Q

as a product of

\min (n, m)

elementary reflectors (see Section 3.4.6 in the F08 Chapter Introduction).

6: $pda$ – Integer Input

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

Constraints:

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

7: $taua [\min (n, m)]$ – double Output

On exit: the scalar factors of the elementary reflectors which represent the orthogonal matrix

Q

8: $b [\dim]$ – double Input/Output

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

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

where

B (i, j)

appears in this document, it refers to the array element

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

On entry: the

n \times p

matrix

B

On exit: if

n \leq p

, the upper triangle of the subarray

B (1 : n, p - n + 1 : p)

contains the

n \times n

upper triangular matrix

T_{12}

n > p

, the elements on and above the

(n - p)

th subdiagonal contain the

n \times p

upper trapezoidal matrix

T

; the remaining elements, with the array taub, represent the orthogonal matrix

Z

as a product of elementary reflectors (see Section 3.4.6 in the F08 Chapter Introduction).

9: $pdb$ – Integer Input

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

Constraints:

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

10: $taub [\min (n, p)]$ – double Output

On exit: the scalar factors of the elementary reflectors which represent the orthogonal matrix

Z

11: $fail$ – 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, $p = ⟨ value ⟩$ .
Constraint: $p \geq 0$ .

On entry, $pda = ⟨ value ⟩$ .
Constraint: $pda > 0$ .

On entry, $pdb = ⟨ value ⟩$ .
Constraint: $pdb > 0$ .
NE_INT_2: On entry, $pda = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Constraint: $pda \geq \max (1, m)$ .

On entry, $pda = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pda \geq \max (1, n)$ .

On entry, $pdb = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pdb \geq \max (1, n)$ .

On entry, $pdb = ⟨ value ⟩$ and $p = ⟨ value ⟩$ .
Constraint: $pdb \geq \max (1, p)$ .
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 generalized

Q R

factorization is the exact factorization for nearby matrices

(A + E)

and

(B + F)

, where

{‖ E ‖}_{2} = O ε {‖ A ‖}_{2} and {‖ F ‖}_{2} = O ε {‖ B ‖}_{2},

and

ε

is the machine precision.

8 Parallelism and Performance

f08zec is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08zec 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 orthogonal matrices

Q

and

Z

may be formed explicitly by calls to f08afc and f08cjc respectively. f08agc may be used to multiply

Q

by another matrix and f08ckc may be used to multiply

Z

by another matrix.

The complex analogue of this function is f08zsc.

10 Example

This example solves the general Gauss–Markov linear model problem

\min_{x} {‖ y ‖}_{2} subject to d = A x + B y

where

A = (\begin{array}{r} - 0.57 & - 1.28 & - 0.39 \\ - 1.93 & 1.08 & - 0.31 \\ 2.30 & 0.24 & - 0.40 \\ - 0.02 & 1.03 & - 1.43 \end{array}), B = (\begin{array}{r} 0.5 & 0 & 0 & 0 \\ 0 & 1.0 & 0 & 0 \\ 0 & 0 & 2.0 & 0 \\ 0 & 0 & 0 & 5.0 \end{array}) and d = (\begin{array}{r} 1.32 \\ - 4.00 \\ 5.52 \\ 3.24 \end{array}) .

The solution is obtained by first computing a generalized

Q R

factorization of the matrix pair

(A, B)

. The example illustrates the general solution process, although the above data corresponds to a simple weighted least squares problem.

f08ze: FL CL CPP AD

NAG CL Interfacef08zec (dggqrf)

▸▿ 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
f08zec (dggqrf)