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NAG Toolbox: nag_lapack_dgelqf (f08ah)

Purpose

nag_lapack_dgelqf (f08ah) computes the LQLQ factorization of a real mm by nn matrix.

Syntax

[a, tau, info] = f08ah(a, 'm', m, 'n', n)
[a, tau, info] = nag_lapack_dgelqf(a, 'm', m, 'n', n)

Description

nag_lapack_dgelqf (f08ah) forms the LQLQ factorization of an arbitrary rectangular real mm by nn matrix. No pivoting is performed.
If mnmn, the factorization is given by:
A =
(L0)
Q
A = L 0 Q
where LL is an mm by mm lower triangular matrix and QQ is an nn by nn orthogonal matrix. It is sometimes more convenient to write the factorization as
A =
(L0)
(Q1)
Q2
A = L 0 Q1 Q2
which reduces to
A = LQ1 ,
A = LQ1 ,
where Q1Q1 consists of the first mm rows of QQ, and Q2Q2 the remaining nmn-m rows.
If m > nm>n, LL is trapezoidal, and the factorization can be written
A =
(L1)
L2
Q
A = L1 L2 Q
where L1L1 is lower triangular and L2L2 is rectangular.
The LQLQ factorization of AA is essentially the same as the QRQR factorization of ATAT, since
A =
(L0)
QAT = QT
(LT)
0
.
A = L 0 QAT= QT LT 0 .
The matrix QQ is not formed explicitly but is represented as a product of min (m,n)min(m,n) elementary reflectors (see the F08 Chapter Introduction for details). Functions are provided to work with QQ in this representation (see Section [Further Comments]).
Note also that for any k < mk<m, the information returned in the first kk rows of the array a represents an LQLQ factorization of the first kk rows of the original matrix AA.

References

None.

Parameters

Compulsory Input Parameters

1:     a(lda, : :) – double array
The first dimension of the array a must be at least max (1,m)max(1,m)
The second dimension of the array must be at least max (1,n)max(1,n)
The mm by nn matrix AA.

Optional Input Parameters

1:     m – int64int32nag_int scalar
Default: The first dimension of the array a.
mm, the number of rows of the matrix AA.
Constraint: m0m0.
2:     n – int64int32nag_int scalar
Default: The second dimension of the array a.
nn, the number of columns of the matrix AA.
Constraint: n0n0.

Input Parameters Omitted from the MATLAB Interface

lda work lwork

Output Parameters

1:     a(lda, : :) – double array
The first dimension of the array a will be max (1,m)max(1,m)
The second dimension of the array will be max (1,n)max(1,n)
ldamax (1,m)ldamax(1,m).
If mnmn, the elements above the diagonal store details of the orthogonal matrix QQ and the lower triangle stores the corresponding elements of the mm by mm lower triangular matrix LL.
If m > nm>n, the strictly upper triangular part stores details of the orthogonal matrix QQ and the remaining elements store the corresponding elements of the mm by nn lower trapezoidal matrix LL.
2:     tau( : :) – double array
Note: the dimension of the array tau must be at least max (1,min (m,n))max(1,min(m,n)).
Further details of the orthogonal matrix QQ.
3:     info – int64int32nag_int scalar
info = 0info=0 unless the function detects an error (see Section [Error Indicators and Warnings]).

Error Indicators and Warnings

  info = iinfo=-i
If info = iinfo=-i, parameter ii 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)(A+E), where
E2 = O(ε) A2 ,
E2 = O(ε) A2 ,
and εε is the machine precision.

Further Comments

The total number of floating point operations is approximately (2/3) m2 (3nm) 23 m2 (3n-m)  if mnmn or (2/3) n2 (3mn) 23 n2 (3m-n)  if m > nm>n.
To form the orthogonal matrix QQ nag_lapack_dgelqf (f08ah) may be followed by a call to nag_lapack_dorglq (f08aj):
[a, info] = f08aj(a, tau, 'k', min(m,n));
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 nag_lapack_dgelqf (f08ah).
When mnmn, it is often only the first mm rows of QQ that are required, and they may be formed by the call:
[a, info] = f08aj(a, tau, 'k', m);
To apply QQ to an arbitrary real rectangular matrix CC, nag_lapack_dgelqf (f08ah) may be followed by a call to nag_lapack_dormlq (f08ak). For example,
[c, info] = f08ak('Left', 'Transpose', a, tau, c, 'k', min(m, n));
forms the matrix product C = QTCC=QTC, where CC is mm by pp.
The complex analogue of this function is nag_lapack_zgelqf (f08av).

Example

function nag_lapack_dgelqf_example
a = [-5.42, 3.28, -3.68, 0.27, 2.06, 0.46;
     -1.65, -3.4, -3.2, -1.03, -4.06, -0.01;
     -0.37, 2.35, 1.9, 4.31, -1.76, 1.13;
     -3.15, -0.11, 1.99, -2.7, 0.26, 4.5];
b = [-2.87, -5.23;
      1.63,  0.29;
     -3.52,  4.76;
      0.45, -8.41;
      0,     0;
      0,     0];
% Compute the LQ factorization of a
[a, tau, info] = nag_lapack_dgelqf(a);

% solve l*y=b
l = tril(a(:, 1:4));
b(1:4,:) = inv(l)*b(1:4,:);

% Compute minimum-norm solution x = (q^t)*b in b
[b, info] = nag_lapack_dormlq('Left', 'Transpose', a, tau, b)
 

b =

    0.2371    0.7383
   -0.4575    0.0158
   -0.0085   -0.0161
   -0.5192    1.0768
    0.0239   -0.6436
   -0.0543   -0.6613


info =

                    0


function f08ah_example
a = [-5.42, 3.28, -3.68, 0.27, 2.06, 0.46;
     -1.65, -3.4, -3.2, -1.03, -4.06, -0.01;
     -0.37, 2.35, 1.9, 4.31, -1.76, 1.13;
     -3.15, -0.11, 1.99, -2.7, 0.26, 4.5];
b = [-2.87, -5.23;
      1.63,  0.29;
     -3.52,  4.76;
      0.45, -8.41;
      0,     0;
      0,     0];
% Compute the LQ factorization of a
[a, tau, info] = f08ah(a);

% solve l*y=b
l = tril(a(:, 1:4));
b(1:4,:) = inv(l)*b(1:4,:);

% Compute minimum-norm solution x = (q^t)*b in b
[b, info] = f08ak('Left', 'Transpose', a, tau, b)
 

b =

    0.2371    0.7383
   -0.4575    0.0158
   -0.0085   -0.0161
   -0.5192    1.0768
    0.0239   -0.6436
   -0.0543   -0.6613


info =

                    0



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