Integer type: int32 int64 nag_int show int32 show int32 show int64 show int64 show nag_int show nag_int

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_lapack_dgerqf (f08ch)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_lapack_dgerqf (f08ch) computes an RQ factorization of a real

m

n

matrix

A

Syntax

[a, tau, info] = f08ch(a, 'm', m, 'n', n)

[a, tau, info] = nag_lapack_dgerqf(a, 'm', m, 'n', n)

Description

nag_lapack_dgerqf (f08ch) forms the

R Q

factorization of an arbitrary rectangular real

m

n

matrix. If

m \leq n

, the factorization is given by

A = (\begin{matrix} 0 & R \end{matrix}) Q,

where

R

is an

m

m

lower triangular matrix and

Q

is an

n

n

orthogonal matrix. If

m > n

the factorization is given by

A = R Q,

where

R

is an

m

n

upper trapezoidal matrix and

Q

is again an

n

n

orthogonal matrix. In the case where

m < n

the factorization can be expressed as

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

where

Q_{1}

consists of the first

(n - m)

rows of

Q

and

Q_{2}

the remaining

m

rows.

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 Further Comments).

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

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

Parameters

Compulsory Input Parameters

1: $a (lda :)$ – double array: The first dimension of the array a must be at least $\max (1, m)$ .
The second dimension of the array a must be at least $\max (1, n)$ .

The $m$ by $n$ matrix $A$ .

Optional Input Parameters

1: $m$ – int64int32nag_int scalar: Default: the first dimension of the array a.
$m$ , the number of rows of the matrix $A$ .

Constraint: $m \geq 0$ .
2: $n$ – int64int32nag_int scalar: Default: the second dimension of the array a.
$n$ , the number of columns of the matrix $A$ .

Constraint: $n \geq 0$ .

Output Parameters

1: $a (lda :)$ – double array: The first dimension of the array a will be $\max (1, m)$ .
The second dimension of the array a will be $\max (1, n)$ .

If $m \leq n$ , the upper triangle of the subarray $a (1 : m, n - m + 1 : n)$ contains the $m$ by $m$ upper triangular matrix $R$ .
If $m \geq n$ , the elements on and above the $(m - n)$ th subdiagonal contain the $m$ by $n$ upper trapezoidal matrix $R$ ; the remaining elements, with the array tau, represent the orthogonal matrix $Q$ as a product of $\min (m, n)$ elementary reflectors (see Representation of orthogonal or unitary matrices in the F08 Chapter Introduction).
2: $tau (:)$ – double array: The dimension of the array tau will be $\max (1, \min (m, n))$

The scalar factors of the elementary reflectors.
3: $info$ – int64int32nag_int scalar: $info = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

$info = - i$: If $info = - i$ , parameter $i$ had an illegal value on entry. The parameters are numbered as follows:
1: m, 2: n, 3: a, 4: lda, 5: tau, 6: work, 7: lwork, 8: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

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.

Further Comments

The total number of floating-point operations is approximately

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

m \leq n

, or

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

m > n

To form the orthogonal matrix

Q

nag_lapack_dgerqf (f08ch) may be followed by a call to nag_lapack_dorgrq (f08cj):

[a, info] = f08cj(a, tau);

but note that the first dimension of the array a must be at least n, which may be larger than was required by nag_lapack_dgerqf (f08ch). 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:

[a, info] = f08cj(a(1:m,1:n), tau);

To apply

Q

to an arbitrary real rectangular matrix

C

, nag_lapack_dgerqf (f08ch) may be followed by a call to nag_lapack_dormrq (f08ck). For example:

[a, c, info] = f08ck('Left','Transpose', a, tau, c);

forms

C = Q^{T} C

, where

C

n

p

The complex analogue of this function is nag_lapack_zgerqf (f08cv).

Example

This example finds the minimum norm solution to the underdetermined equations

A x = b

where

A = (\begin{array}{r} - 5.42 & 3.28 & - 3.68 & 0.27 & 2.06 & 0.46 \\ - 1.65 & - 3.40 & - 3.20 & - 1.03 & - 4.06 & - 0.01 \\ - 0.37 & 2.35 & 1.90 & 4.31 & - 1.76 & 1.13 \\ - 3.15 & - 0.11 & 1.99 & - 2.70 & 0.26 & 4.50 \end{array}) and b = (\begin{array}{r} - 2.87 \\ 1.63 \\ - 3.52 \\ 0.45 \end{array}) .

The solution is obtained by first obtaining an

R Q

factorization of the matrix

A

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.

Open in the MATLAB editor: f08ch_example

function f08ch_example


fprintf('f08ch example results\n\n');

a = [-5.42,  3.28, -3.68,  0.27,  2.06,  0.46;
     -1.65, -3.40, -3.20, -1.03, -4.06, -0.01;
     -0.37,  2.35,  1.90,  4.31, -1.76,  1.13;
     -3.15, -0.11,  1.99, -2.70,  0.26,  4.50];
b = [-2.87;  1.63; -3.52;  0.45];
% Compute the RQ factorization of a
[a, tau, info] = f08ch(a);

% Solve R*y2 = b
c = zeros(6,1);
[c(3:6), info] = f07te( ...
			'Upper', 'No transpose', 'Non-Unit', a(:, 3:6), b);

if (info >0)
  fprintf('The upper triangular factor, R, of A is singular,\n');
  fprintf('the least squares solution could not be computed.\n');
else
  % Compute the minimum-norm solution x = (Q^T)*y
  [a, c, info] = f08ck( ...
			'Left', 'Transpose', a, tau, c);

  fprintf('Minimum-norm solution\n');
  disp(transpose(c));
end

f08ch example results

Minimum-norm solution
    0.2371   -0.4575   -0.0085   -0.5192    0.0239   -0.0543

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox