NAG Library Function Document

nag_dggrqf (f08zfc)

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

3 Description

nag_dggrqf (f08zfc) forms the generalized

R Q

factorization of an

m

n

matrix

A

and a

p

n

matrix

B

A = R Q, B = Z T Q,

where

Q

is an

n

n

orthogonal matrix,

Z

is a

p

p

orthogonal matrix and

R

and

T

are of the form

R = \{\begin{matrix} \begin{array}{rc} n - m & m \\ m & ( & 0 & R_{12} & ) \end{array};   if ​ m \leq n, \\ \begin{array}{rc} n \\ m - n & ( & R_{11} & ) \\ n & R_{21} \end{array};   if ​ m > n, \end{matrix}

with

R_{12}

R_{21}

upper triangular,

T = \{\begin{matrix} \begin{array}{rc} n \\ n & ( & T_{11} & ) \\ p - n & 0 \end{array};   if ​ p \geq n, \\ \begin{array}{rc} p & n - p \\ p & ( & T_{11} & T_{12} & ) \end{array};   if ​ p < n, \end{matrix}

with

T_{11}

upper triangular.

In particular, if

B

is square and nonsingular, the generalized

R Q

factorization of

A

and

B

implicitly gives the

R Q

factorization of

A B^{- 1}

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

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 http://www.netlib.org/lapack/lug

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

or Nag_ColMajor.

2: m – IntegerInput

On entry:

m

, the number of rows of the matrix

A

Constraint:

m \geq 0

3: p – IntegerInput

On entry:

p

, the number of rows of the matrix

B

Constraint:

p \geq 0

4: n – IntegerInput

On entry:

n

, the number of columns of the matrices

A

and

B

Constraint:

n \geq 0

5: 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 $order = Nag_RowMajor$ .

Where

A (i, j)

appears in this document, it refers to the array element

$a [(j - 1) \times pda + i - 1]$ 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 upper triangle of the subarray

A (1 : m, n - m + 1 : n)

contains the

m

m

upper triangular matrix

R_{12}

m \geq n

, the elements on and above the

(m - n)

th subdiagonal contain the

m

n

upper trapezoidal matrix

R

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

Q

as a product of

\min (m, n)

elementary reflectors (see Section 3.3.6 in the f08 Chapter Introduction).

6: 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 $order = Nag_RowMajor$ , $pda \geq \max (1, n)$ .

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

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

Q

8: b[ $\dim$ ] – doubleInput/Output

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

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

The

(i, j)

th element of the matrix

B

is stored in

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

On entry: the

p

n

matrix

B

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

\min (p, n)

n

upper trapezoidal matrix

T

(

T

is upper triangular if

p \geq n

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

Z

as a product of elementary reflectors (see Section 3.3.6 in the f08 Chapter Introduction).

9: pdb – IntegerInput

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

Constraints:

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

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

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 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, $p = 〈value〉$ .
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.

7 Accuracy

The computed generalized

R Q

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 Further Comments

The orthogonal matrices

Q

and

Z

may be formed explicitly by calls to nag_dorgrq (f08cjc) and nag_dorgqr (f08afc) respectively. nag_dormrq (f08ckc) may be used to multiply

Q

by another matrix and nag_dormqr (f08agc) may be used to multiply

Z

by another matrix.

The complex analogue of this function is nag_zggrqf (f08ztc).

9 Example

This example solves the linear equality constrained least squares problem

\min_{x} {‖c - A x‖}_{2} subject to B x = d

where

A = (\begin{array}{r} - 0.57 & - 1.28 & - 0.39 & 0.25 \\ - 1.93 & 1.08 & - 0.31 & - 2.14 \\ 2.30 & 0.24 & 0.40 & - 0.35 \\ - 1.93 & 0.64 & - 0.66 & 0.08 \\ 0.15 & 0.30 & 0.15 & - 2.13 \\ - 0.02 & 1.03 & - 1.43 & 0.50 \end{array}), B = (\begin{array}{r} 1 & 0 & - 1 & 0 \\ 0 & 1 & 0 & - 1 \end{array}),

c = (\begin{array}{r} - 1.50 \\ - 2.14 \\ 1.23 \\ - 0.54 \\ - 1.68 \\ 0.82 \end{array}) and d = (\begin{matrix} 0 \\ 0 \end{matrix}) .

The constraints

B x = d

correspond to

x_{1} = x_{3}

and

x_{2} = x_{4}

The solution is obtained by first computing a generalized

R Q

factorization of the matrix pair

(B, A)

. The example illustrates the general solution process.

NAG Library Function Documentnag_dggrqf (f08zfc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

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 Function Document

nag_dggrqf (f08zfc)