 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ dgelq2()

 subroutine dgelq2 ( integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU, double precision, dimension( * ) WORK, integer INFO )

DGELQ2 computes the LQ factorization of a general rectangular matrix using an unblocked algorithm.

Purpose:
``` DGELQ2 computes an LQ factorization of a real m-by-n matrix A:

A = ( L 0 ) *  Q

where:

Q is a n-by-n orthogonal matrix;
L is a lower-triangular m-by-m matrix;
0 is a m-by-(n-m) zero matrix, if m < n.```
Parameters
 [in] M ``` M is INTEGER The number of rows of the matrix A. M >= 0.``` [in] N ``` N is INTEGER The number of columns of the matrix A. N >= 0.``` [in,out] A ``` A is DOUBLE PRECISION array, dimension (LDA,N) On entry, the m by n matrix A. On exit, the elements on and below the diagonal of the array contain the m by min(m,n) lower trapezoidal matrix L (L is lower triangular if m <= n); the elements above the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors (see Further Details).``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).``` [out] TAU ``` TAU is DOUBLE PRECISION array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details).``` [out] WORK ` WORK is DOUBLE PRECISION array, dimension (M)` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value```
Further Details:
```  The matrix Q is represented as a product of elementary reflectors

Q = H(k) . . . H(2) H(1), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i,i+1:n),
and tau in TAU(i).```

Definition at line 128 of file dgelq2.f.

129 *
130 * -- LAPACK computational routine --
131 * -- LAPACK is a software package provided by Univ. of Tennessee, --
132 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
133 *
134 * .. Scalar Arguments ..
135  INTEGER INFO, LDA, M, N
136 * ..
137 * .. Array Arguments ..
138  DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
139 * ..
140 *
141 * =====================================================================
142 *
143 * .. Parameters ..
144  DOUBLE PRECISION ONE
145  parameter( one = 1.0d+0 )
146 * ..
147 * .. Local Scalars ..
148  INTEGER I, K
149  DOUBLE PRECISION AII
150 * ..
151 * .. External Subroutines ..
152  EXTERNAL dlarf, dlarfg, xerbla
153 * ..
154 * .. Intrinsic Functions ..
155  INTRINSIC max, min
156 * ..
157 * .. Executable Statements ..
158 *
159 * Test the input arguments
160 *
161  info = 0
162  IF( m.LT.0 ) THEN
163  info = -1
164  ELSE IF( n.LT.0 ) THEN
165  info = -2
166  ELSE IF( lda.LT.max( 1, m ) ) THEN
167  info = -4
168  END IF
169  IF( info.NE.0 ) THEN
170  CALL xerbla( 'DGELQ2', -info )
171  RETURN
172  END IF
173 *
174  k = min( m, n )
175 *
176  DO 10 i = 1, k
177 *
178 * Generate elementary reflector H(i) to annihilate A(i,i+1:n)
179 *
180  CALL dlarfg( n-i+1, a( i, i ), a( i, min( i+1, n ) ), lda,
181  \$ tau( i ) )
182  IF( i.LT.m ) THEN
183 *
184 * Apply H(i) to A(i+1:m,i:n) from the right
185 *
186  aii = a( i, i )
187  a( i, i ) = one
188  CALL dlarf( 'Right', m-i, n-i+1, a( i, i ), lda, tau( i ),
189  \$ a( i+1, i ), lda, work )
190  a( i, i ) = aii
191  END IF
192  10 CONTINUE
193  RETURN
194 *
195 * End of DGELQ2
196 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine dlarf(SIDE, M, N, V, INCV, TAU, C, LDC, WORK)
DLARF applies an elementary reflector to a general rectangular matrix.
Definition: dlarf.f:124
subroutine dlarfg(N, ALPHA, X, INCX, TAU)
DLARFG generates an elementary reflector (Householder matrix).
Definition: dlarfg.f:106
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