1 SUBROUTINE DGEQLF( 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 * DGEQLF computes a QL factorization of a real M-by-N matrix A:
19 * A = Q * L.
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,
33 * if m >= n, the lower triangle of the subarray
34 * A(m-n+1:m,1:n) contains the N-by-N lower triangular matrix L;
35 * if m <= n, the elements on and below the (n-m)-th
36 * superdiagonal contain the M-by-N lower trapezoidal matrix L;
37 * the remaining elements, with the array TAU, represent the
38 * orthogonal matrix Q as a product of elementary reflectors
39 * (see Further Details).
40 *
41 * LDA (input) INTEGER
42 * The leading dimension of the array A. LDA >= max(1,M).
43 *
44 * TAU (output) DOUBLE PRECISION array, dimension (min(M,N))
45 * The scalar factors of the elementary reflectors (see Further
46 * Details).
47 *
48 * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
49 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
50 *
51 * LWORK (input) INTEGER
52 * The dimension of the array WORK. LWORK >= max(1,N).
53 * For optimum performance LWORK >= N*NB, where NB is the
54 * optimal blocksize.
55 *
56 * If LWORK = -1, then a workspace query is assumed; the routine
57 * only calculates the optimal size of the WORK array, returns
58 * this value as the first entry of the WORK array, and no error
59 * message related to LWORK is issued by XERBLA.
60 *
61 * INFO (output) INTEGER
62 * = 0: successful exit
63 * < 0: if INFO = -i, the i-th argument had an illegal value
64 *
65 * Further Details
66 * ===============
67 *
68 * The matrix Q is represented as a product of elementary reflectors
69 *
70 * Q = H(k) . . . H(2) H(1), where k = min(m,n).
71 *
72 * Each H(i) has the form
73 *
74 * H(i) = I - tau * v * v**T
75 *
76 * where tau is a real scalar, and v is a real vector with
77 * v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in
78 * A(1:m-k+i-1,n-k+i), and tau in TAU(i).
79 *
80 * =====================================================================
81 *
82 * .. Local Scalars ..
83 LOGICAL LQUERY
84 INTEGER I, IB, IINFO, IWS, K, KI, KK, LDWORK, LWKOPT,
85 $ MU, NB, NBMIN, NU, NX
86 * ..
87 * .. External Subroutines ..
88 EXTERNAL DGEQL2, DLARFB, DLARFT, XERBLA
89 * ..
90 * .. Intrinsic Functions ..
91 INTRINSIC MAX, MIN
92 * ..
93 * .. External Functions ..
94 INTEGER ILAENV
95 EXTERNAL ILAENV
96 * ..
97 * .. Executable Statements ..
98 *
99 * Test the input arguments
100 *
101 INFO = 0
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 END IF
110 *
111 IF( INFO.EQ.0 ) THEN
112 K = MIN( M, N )
113 IF( K.EQ.0 ) THEN
114 LWKOPT = 1
115 ELSE
116 NB = ILAENV( 1, 'DGEQLF', ' ', M, N, -1, -1 )
117 LWKOPT = N*NB
118 END IF
119 WORK( 1 ) = LWKOPT
120 *
121 IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
122 INFO = -7
123 END IF
124 END IF
125 *
126 IF( INFO.NE.0 ) THEN
127 CALL XERBLA( 'DGEQLF', -INFO )
128 RETURN
129 ELSE IF( LQUERY ) THEN
130 RETURN
131 END IF
132 *
133 * Quick return if possible
134 *
135 IF( K.EQ.0 ) THEN
136 RETURN
137 END IF
138 *
139 NBMIN = 2
140 NX = 1
141 IWS = N
142 IF( NB.GT.1 .AND. NB.LT.K ) THEN
143 *
144 * Determine when to cross over from blocked to unblocked code.
145 *
146 NX = MAX( 0, ILAENV( 3, 'DGEQLF', ' ', M, N, -1, -1 ) )
147 IF( NX.LT.K ) THEN
148 *
149 * Determine if workspace is large enough for blocked code.
150 *
151 LDWORK = N
152 IWS = LDWORK*NB
153 IF( LWORK.LT.IWS ) THEN
154 *
155 * Not enough workspace to use optimal NB: reduce NB and
156 * determine the minimum value of NB.
157 *
158 NB = LWORK / LDWORK
159 NBMIN = MAX( 2, ILAENV( 2, 'DGEQLF', ' ', M, N, -1,
160 $ -1 ) )
161 END IF
162 END IF
163 END IF
164 *
165 IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
166 *
167 * Use blocked code initially.
168 * The last kk columns are handled by the block method.
169 *
170 KI = ( ( K-NX-1 ) / NB )*NB
171 KK = MIN( K, KI+NB )
172 *
173 DO 10 I = K - KK + KI + 1, K - KK + 1, -NB
174 IB = MIN( K-I+1, NB )
175 *
176 * Compute the QL factorization of the current block
177 * A(1:m-k+i+ib-1,n-k+i:n-k+i+ib-1)
178 *
179 CALL DGEQL2( M-K+I+IB-1, IB, A( 1, N-K+I ), LDA, TAU( I ),
180 $ WORK, IINFO )
181 IF( N-K+I.GT.1 ) THEN
182 *
183 * Form the triangular factor of the block reflector
184 * H = H(i+ib-1) . . . H(i+1) H(i)
185 *
186 CALL DLARFT( 'Backward', 'Columnwise', M-K+I+IB-1, IB,
187 $ A( 1, N-K+I ), LDA, TAU( I ), WORK, LDWORK )
188 *
189 * Apply H**T to A(1:m-k+i+ib-1,1:n-k+i-1) from the left
190 *
191 CALL DLARFB( 'Left', 'Transpose', 'Backward',
192 $ 'Columnwise', M-K+I+IB-1, N-K+I-1, IB,
193 $ A( 1, N-K+I ), LDA, WORK, LDWORK, A, LDA,
194 $ WORK( IB+1 ), LDWORK )
195 END IF
196 10 CONTINUE
197 MU = M - K + I + NB - 1
198 NU = N - K + I + NB - 1
199 ELSE
200 MU = M
201 NU = N
202 END IF
203 *
204 * Use unblocked code to factor the last or only block
205 *
206 IF( MU.GT.0 .AND. NU.GT.0 )
207 $ CALL DGEQL2( MU, NU, A, LDA, TAU, WORK, IINFO )
208 *
209 WORK( 1 ) = IWS
210 RETURN
211 *
212 * End of DGEQLF
213 *
214 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 * DGEQLF computes a QL factorization of a real M-by-N matrix A:
19 * A = Q * L.
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,
33 * if m >= n, the lower triangle of the subarray
34 * A(m-n+1:m,1:n) contains the N-by-N lower triangular matrix L;
35 * if m <= n, the elements on and below the (n-m)-th
36 * superdiagonal contain the M-by-N lower trapezoidal matrix L;
37 * the remaining elements, with the array TAU, represent the
38 * orthogonal matrix Q as a product of elementary reflectors
39 * (see Further Details).
40 *
41 * LDA (input) INTEGER
42 * The leading dimension of the array A. LDA >= max(1,M).
43 *
44 * TAU (output) DOUBLE PRECISION array, dimension (min(M,N))
45 * The scalar factors of the elementary reflectors (see Further
46 * Details).
47 *
48 * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
49 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
50 *
51 * LWORK (input) INTEGER
52 * The dimension of the array WORK. LWORK >= max(1,N).
53 * For optimum performance LWORK >= N*NB, where NB is the
54 * optimal blocksize.
55 *
56 * If LWORK = -1, then a workspace query is assumed; the routine
57 * only calculates the optimal size of the WORK array, returns
58 * this value as the first entry of the WORK array, and no error
59 * message related to LWORK is issued by XERBLA.
60 *
61 * INFO (output) INTEGER
62 * = 0: successful exit
63 * < 0: if INFO = -i, the i-th argument had an illegal value
64 *
65 * Further Details
66 * ===============
67 *
68 * The matrix Q is represented as a product of elementary reflectors
69 *
70 * Q = H(k) . . . H(2) H(1), where k = min(m,n).
71 *
72 * Each H(i) has the form
73 *
74 * H(i) = I - tau * v * v**T
75 *
76 * where tau is a real scalar, and v is a real vector with
77 * v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in
78 * A(1:m-k+i-1,n-k+i), and tau in TAU(i).
79 *
80 * =====================================================================
81 *
82 * .. Local Scalars ..
83 LOGICAL LQUERY
84 INTEGER I, IB, IINFO, IWS, K, KI, KK, LDWORK, LWKOPT,
85 $ MU, NB, NBMIN, NU, NX
86 * ..
87 * .. External Subroutines ..
88 EXTERNAL DGEQL2, DLARFB, DLARFT, XERBLA
89 * ..
90 * .. Intrinsic Functions ..
91 INTRINSIC MAX, MIN
92 * ..
93 * .. External Functions ..
94 INTEGER ILAENV
95 EXTERNAL ILAENV
96 * ..
97 * .. Executable Statements ..
98 *
99 * Test the input arguments
100 *
101 INFO = 0
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 END IF
110 *
111 IF( INFO.EQ.0 ) THEN
112 K = MIN( M, N )
113 IF( K.EQ.0 ) THEN
114 LWKOPT = 1
115 ELSE
116 NB = ILAENV( 1, 'DGEQLF', ' ', M, N, -1, -1 )
117 LWKOPT = N*NB
118 END IF
119 WORK( 1 ) = LWKOPT
120 *
121 IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
122 INFO = -7
123 END IF
124 END IF
125 *
126 IF( INFO.NE.0 ) THEN
127 CALL XERBLA( 'DGEQLF', -INFO )
128 RETURN
129 ELSE IF( LQUERY ) THEN
130 RETURN
131 END IF
132 *
133 * Quick return if possible
134 *
135 IF( K.EQ.0 ) THEN
136 RETURN
137 END IF
138 *
139 NBMIN = 2
140 NX = 1
141 IWS = N
142 IF( NB.GT.1 .AND. NB.LT.K ) THEN
143 *
144 * Determine when to cross over from blocked to unblocked code.
145 *
146 NX = MAX( 0, ILAENV( 3, 'DGEQLF', ' ', M, N, -1, -1 ) )
147 IF( NX.LT.K ) THEN
148 *
149 * Determine if workspace is large enough for blocked code.
150 *
151 LDWORK = N
152 IWS = LDWORK*NB
153 IF( LWORK.LT.IWS ) THEN
154 *
155 * Not enough workspace to use optimal NB: reduce NB and
156 * determine the minimum value of NB.
157 *
158 NB = LWORK / LDWORK
159 NBMIN = MAX( 2, ILAENV( 2, 'DGEQLF', ' ', M, N, -1,
160 $ -1 ) )
161 END IF
162 END IF
163 END IF
164 *
165 IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
166 *
167 * Use blocked code initially.
168 * The last kk columns are handled by the block method.
169 *
170 KI = ( ( K-NX-1 ) / NB )*NB
171 KK = MIN( K, KI+NB )
172 *
173 DO 10 I = K - KK + KI + 1, K - KK + 1, -NB
174 IB = MIN( K-I+1, NB )
175 *
176 * Compute the QL factorization of the current block
177 * A(1:m-k+i+ib-1,n-k+i:n-k+i+ib-1)
178 *
179 CALL DGEQL2( M-K+I+IB-1, IB, A( 1, N-K+I ), LDA, TAU( I ),
180 $ WORK, IINFO )
181 IF( N-K+I.GT.1 ) THEN
182 *
183 * Form the triangular factor of the block reflector
184 * H = H(i+ib-1) . . . H(i+1) H(i)
185 *
186 CALL DLARFT( 'Backward', 'Columnwise', M-K+I+IB-1, IB,
187 $ A( 1, N-K+I ), LDA, TAU( I ), WORK, LDWORK )
188 *
189 * Apply H**T to A(1:m-k+i+ib-1,1:n-k+i-1) from the left
190 *
191 CALL DLARFB( 'Left', 'Transpose', 'Backward',
192 $ 'Columnwise', M-K+I+IB-1, N-K+I-1, IB,
193 $ A( 1, N-K+I ), LDA, WORK, LDWORK, A, LDA,
194 $ WORK( IB+1 ), LDWORK )
195 END IF
196 10 CONTINUE
197 MU = M - K + I + NB - 1
198 NU = N - K + I + NB - 1
199 ELSE
200 MU = M
201 NU = N
202 END IF
203 *
204 * Use unblocked code to factor the last or only block
205 *
206 IF( MU.GT.0 .AND. NU.GT.0 )
207 $ CALL DGEQL2( MU, NU, A, LDA, TAU, WORK, IINFO )
208 *
209 WORK( 1 ) = IWS
210 RETURN
211 *
212 * End of DGEQLF
213 *
214 END