1 SUBROUTINE DGELQF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
2 *
3 * -- LAPACK routine (version 3.3.1) --
4 * -- LAPACK is a software package provided by Univ. of Tennessee, --
5 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
6 * -- April 2011 --
7 *
8 * .. Scalar Arguments ..
9 INTEGER INFO, LDA, LWORK, M, N
10 * ..
11 * .. Array Arguments ..
12 DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
13 * ..
14 *
15 * Purpose
16 * =======
17 *
18 * DGELQF computes an LQ factorization of a real M-by-N matrix A:
19 * A = L * Q.
20 *
21 * Arguments
22 * =========
23 *
24 * M (input) INTEGER
25 * The number of rows of the matrix A. M >= 0.
26 *
27 * N (input) INTEGER
28 * The number of columns of the matrix A. N >= 0.
29 *
30 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
31 * On entry, the M-by-N matrix A.
32 * On exit, the elements on and below the diagonal of the array
33 * contain the m-by-min(m,n) lower trapezoidal matrix L (L is
34 * lower triangular if m <= n); the elements above the diagonal,
35 * with the array TAU, represent the orthogonal matrix Q as a
36 * product of elementary reflectors (see Further Details).
37 *
38 * LDA (input) INTEGER
39 * The leading dimension of the array A. LDA >= max(1,M).
40 *
41 * TAU (output) DOUBLE PRECISION array, dimension (min(M,N))
42 * The scalar factors of the elementary reflectors (see Further
43 * Details).
44 *
45 * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
46 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
47 *
48 * LWORK (input) INTEGER
49 * The dimension of the array WORK. LWORK >= max(1,M).
50 * For optimum performance LWORK >= M*NB, where NB is the
51 * optimal blocksize.
52 *
53 * If LWORK = -1, then a workspace query is assumed; the routine
54 * only calculates the optimal size of the WORK array, returns
55 * this value as the first entry of the WORK array, and no error
56 * message related to LWORK is issued by XERBLA.
57 *
58 * INFO (output) INTEGER
59 * = 0: successful exit
60 * < 0: if INFO = -i, the i-th argument had an illegal value
61 *
62 * Further Details
63 * ===============
64 *
65 * The matrix Q is represented as a product of elementary reflectors
66 *
67 * Q = H(k) . . . H(2) H(1), where k = min(m,n).
68 *
69 * Each H(i) has the form
70 *
71 * H(i) = I - tau * v * v**T
72 *
73 * where tau is a real scalar, and v is a real vector with
74 * v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i,i+1:n),
75 * and tau in TAU(i).
76 *
77 * =====================================================================
78 *
79 * .. Local Scalars ..
80 LOGICAL LQUERY
81 INTEGER I, IB, IINFO, IWS, K, LDWORK, LWKOPT, NB,
82 $ NBMIN, NX
83 * ..
84 * .. External Subroutines ..
85 EXTERNAL DGELQ2, DLARFB, DLARFT, XERBLA
86 * ..
87 * .. Intrinsic Functions ..
88 INTRINSIC MAX, MIN
89 * ..
90 * .. External Functions ..
91 INTEGER ILAENV
92 EXTERNAL ILAENV
93 * ..
94 * .. Executable Statements ..
95 *
96 * Test the input arguments
97 *
98 INFO = 0
99 NB = ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
100 LWKOPT = M*NB
101 WORK( 1 ) = LWKOPT
102 LQUERY = ( LWORK.EQ.-1 )
103 IF( M.LT.0 ) THEN
104 INFO = -1
105 ELSE IF( N.LT.0 ) THEN
106 INFO = -2
107 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
108 INFO = -4
109 ELSE IF( LWORK.LT.MAX( 1, M ) .AND. .NOT.LQUERY ) THEN
110 INFO = -7
111 END IF
112 IF( INFO.NE.0 ) THEN
113 CALL XERBLA( 'DGELQF', -INFO )
114 RETURN
115 ELSE IF( LQUERY ) THEN
116 RETURN
117 END IF
118 *
119 * Quick return if possible
120 *
121 K = MIN( M, N )
122 IF( K.EQ.0 ) THEN
123 WORK( 1 ) = 1
124 RETURN
125 END IF
126 *
127 NBMIN = 2
128 NX = 0
129 IWS = M
130 IF( NB.GT.1 .AND. NB.LT.K ) THEN
131 *
132 * Determine when to cross over from blocked to unblocked code.
133 *
134 NX = MAX( 0, ILAENV( 3, 'DGELQF', ' ', M, N, -1, -1 ) )
135 IF( NX.LT.K ) THEN
136 *
137 * Determine if workspace is large enough for blocked code.
138 *
139 LDWORK = M
140 IWS = LDWORK*NB
141 IF( LWORK.LT.IWS ) THEN
142 *
143 * Not enough workspace to use optimal NB: reduce NB and
144 * determine the minimum value of NB.
145 *
146 NB = LWORK / LDWORK
147 NBMIN = MAX( 2, ILAENV( 2, 'DGELQF', ' ', M, N, -1,
148 $ -1 ) )
149 END IF
150 END IF
151 END IF
152 *
153 IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
154 *
155 * Use blocked code initially
156 *
157 DO 10 I = 1, K - NX, NB
158 IB = MIN( K-I+1, NB )
159 *
160 * Compute the LQ factorization of the current block
161 * A(i:i+ib-1,i:n)
162 *
163 CALL DGELQ2( IB, N-I+1, A( I, I ), LDA, TAU( I ), WORK,
164 $ IINFO )
165 IF( I+IB.LE.M ) THEN
166 *
167 * Form the triangular factor of the block reflector
168 * H = H(i) H(i+1) . . . H(i+ib-1)
169 *
170 CALL DLARFT( 'Forward', 'Rowwise', N-I+1, IB, A( I, I ),
171 $ LDA, TAU( I ), WORK, LDWORK )
172 *
173 * Apply H to A(i+ib:m,i:n) from the right
174 *
175 CALL DLARFB( 'Right', 'No transpose', 'Forward',
176 $ 'Rowwise', M-I-IB+1, N-I+1, IB, A( I, I ),
177 $ LDA, WORK, LDWORK, A( I+IB, I ), LDA,
178 $ WORK( IB+1 ), LDWORK )
179 END IF
180 10 CONTINUE
181 ELSE
182 I = 1
183 END IF
184 *
185 * Use unblocked code to factor the last or only block.
186 *
187 IF( I.LE.K )
188 $ CALL DGELQ2( M-I+1, N-I+1, A( I, I ), LDA, TAU( I ), WORK,
189 $ IINFO )
190 *
191 WORK( 1 ) = IWS
192 RETURN
193 *
194 * End of DGELQF
195 *
196 END
2 *
3 * -- LAPACK routine (version 3.3.1) --
4 * -- LAPACK is a software package provided by Univ. of Tennessee, --
5 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
6 * -- April 2011 --
7 *
8 * .. Scalar Arguments ..
9 INTEGER INFO, LDA, LWORK, M, N
10 * ..
11 * .. Array Arguments ..
12 DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
13 * ..
14 *
15 * Purpose
16 * =======
17 *
18 * DGELQF computes an LQ factorization of a real M-by-N matrix A:
19 * A = L * Q.
20 *
21 * Arguments
22 * =========
23 *
24 * M (input) INTEGER
25 * The number of rows of the matrix A. M >= 0.
26 *
27 * N (input) INTEGER
28 * The number of columns of the matrix A. N >= 0.
29 *
30 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
31 * On entry, the M-by-N matrix A.
32 * On exit, the elements on and below the diagonal of the array
33 * contain the m-by-min(m,n) lower trapezoidal matrix L (L is
34 * lower triangular if m <= n); the elements above the diagonal,
35 * with the array TAU, represent the orthogonal matrix Q as a
36 * product of elementary reflectors (see Further Details).
37 *
38 * LDA (input) INTEGER
39 * The leading dimension of the array A. LDA >= max(1,M).
40 *
41 * TAU (output) DOUBLE PRECISION array, dimension (min(M,N))
42 * The scalar factors of the elementary reflectors (see Further
43 * Details).
44 *
45 * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
46 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
47 *
48 * LWORK (input) INTEGER
49 * The dimension of the array WORK. LWORK >= max(1,M).
50 * For optimum performance LWORK >= M*NB, where NB is the
51 * optimal blocksize.
52 *
53 * If LWORK = -1, then a workspace query is assumed; the routine
54 * only calculates the optimal size of the WORK array, returns
55 * this value as the first entry of the WORK array, and no error
56 * message related to LWORK is issued by XERBLA.
57 *
58 * INFO (output) INTEGER
59 * = 0: successful exit
60 * < 0: if INFO = -i, the i-th argument had an illegal value
61 *
62 * Further Details
63 * ===============
64 *
65 * The matrix Q is represented as a product of elementary reflectors
66 *
67 * Q = H(k) . . . H(2) H(1), where k = min(m,n).
68 *
69 * Each H(i) has the form
70 *
71 * H(i) = I - tau * v * v**T
72 *
73 * where tau is a real scalar, and v is a real vector with
74 * v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i,i+1:n),
75 * and tau in TAU(i).
76 *
77 * =====================================================================
78 *
79 * .. Local Scalars ..
80 LOGICAL LQUERY
81 INTEGER I, IB, IINFO, IWS, K, LDWORK, LWKOPT, NB,
82 $ NBMIN, NX
83 * ..
84 * .. External Subroutines ..
85 EXTERNAL DGELQ2, DLARFB, DLARFT, XERBLA
86 * ..
87 * .. Intrinsic Functions ..
88 INTRINSIC MAX, MIN
89 * ..
90 * .. External Functions ..
91 INTEGER ILAENV
92 EXTERNAL ILAENV
93 * ..
94 * .. Executable Statements ..
95 *
96 * Test the input arguments
97 *
98 INFO = 0
99 NB = ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
100 LWKOPT = M*NB
101 WORK( 1 ) = LWKOPT
102 LQUERY = ( LWORK.EQ.-1 )
103 IF( M.LT.0 ) THEN
104 INFO = -1
105 ELSE IF( N.LT.0 ) THEN
106 INFO = -2
107 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
108 INFO = -4
109 ELSE IF( LWORK.LT.MAX( 1, M ) .AND. .NOT.LQUERY ) THEN
110 INFO = -7
111 END IF
112 IF( INFO.NE.0 ) THEN
113 CALL XERBLA( 'DGELQF', -INFO )
114 RETURN
115 ELSE IF( LQUERY ) THEN
116 RETURN
117 END IF
118 *
119 * Quick return if possible
120 *
121 K = MIN( M, N )
122 IF( K.EQ.0 ) THEN
123 WORK( 1 ) = 1
124 RETURN
125 END IF
126 *
127 NBMIN = 2
128 NX = 0
129 IWS = M
130 IF( NB.GT.1 .AND. NB.LT.K ) THEN
131 *
132 * Determine when to cross over from blocked to unblocked code.
133 *
134 NX = MAX( 0, ILAENV( 3, 'DGELQF', ' ', M, N, -1, -1 ) )
135 IF( NX.LT.K ) THEN
136 *
137 * Determine if workspace is large enough for blocked code.
138 *
139 LDWORK = M
140 IWS = LDWORK*NB
141 IF( LWORK.LT.IWS ) THEN
142 *
143 * Not enough workspace to use optimal NB: reduce NB and
144 * determine the minimum value of NB.
145 *
146 NB = LWORK / LDWORK
147 NBMIN = MAX( 2, ILAENV( 2, 'DGELQF', ' ', M, N, -1,
148 $ -1 ) )
149 END IF
150 END IF
151 END IF
152 *
153 IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
154 *
155 * Use blocked code initially
156 *
157 DO 10 I = 1, K - NX, NB
158 IB = MIN( K-I+1, NB )
159 *
160 * Compute the LQ factorization of the current block
161 * A(i:i+ib-1,i:n)
162 *
163 CALL DGELQ2( IB, N-I+1, A( I, I ), LDA, TAU( I ), WORK,
164 $ IINFO )
165 IF( I+IB.LE.M ) THEN
166 *
167 * Form the triangular factor of the block reflector
168 * H = H(i) H(i+1) . . . H(i+ib-1)
169 *
170 CALL DLARFT( 'Forward', 'Rowwise', N-I+1, IB, A( I, I ),
171 $ LDA, TAU( I ), WORK, LDWORK )
172 *
173 * Apply H to A(i+ib:m,i:n) from the right
174 *
175 CALL DLARFB( 'Right', 'No transpose', 'Forward',
176 $ 'Rowwise', M-I-IB+1, N-I+1, IB, A( I, I ),
177 $ LDA, WORK, LDWORK, A( I+IB, I ), LDA,
178 $ WORK( IB+1 ), LDWORK )
179 END IF
180 10 CONTINUE
181 ELSE
182 I = 1
183 END IF
184 *
185 * Use unblocked code to factor the last or only block.
186 *
187 IF( I.LE.K )
188 $ CALL DGELQ2( M-I+1, N-I+1, A( I, I ), LDA, TAU( I ), WORK,
189 $ IINFO )
190 *
191 WORK( 1 ) = IWS
192 RETURN
193 *
194 * End of DGELQF
195 *
196 END