1 SUBROUTINE DGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK,
2 $ INFO )
3 *
4 * -- LAPACK routine (version 3.3.1) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * -- April 2011 --
8 *
9 * .. Scalar Arguments ..
10 INTEGER INFO, LDA, LWORK, M, N
11 * ..
12 * .. Array Arguments ..
13 DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
14 $ TAUQ( * ), WORK( * )
15 * ..
16 *
17 * Purpose
18 * =======
19 *
20 * DGEBRD reduces a general real M-by-N matrix A to upper or lower
21 * bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
22 *
23 * If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
24 *
25 * Arguments
26 * =========
27 *
28 * M (input) INTEGER
29 * The number of rows in the matrix A. M >= 0.
30 *
31 * N (input) INTEGER
32 * The number of columns in the matrix A. N >= 0.
33 *
34 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
35 * On entry, the M-by-N general matrix to be reduced.
36 * On exit,
37 * if m >= n, the diagonal and the first superdiagonal are
38 * overwritten with the upper bidiagonal matrix B; the
39 * elements below the diagonal, with the array TAUQ, represent
40 * the orthogonal matrix Q as a product of elementary
41 * reflectors, and the elements above the first superdiagonal,
42 * with the array TAUP, represent the orthogonal matrix P as
43 * a product of elementary reflectors;
44 * if m < n, the diagonal and the first subdiagonal are
45 * overwritten with the lower bidiagonal matrix B; the
46 * elements below the first subdiagonal, with the array TAUQ,
47 * represent the orthogonal matrix Q as a product of
48 * elementary reflectors, and the elements above the diagonal,
49 * with the array TAUP, represent the orthogonal matrix P as
50 * a product of elementary reflectors.
51 * See Further Details.
52 *
53 * LDA (input) INTEGER
54 * The leading dimension of the array A. LDA >= max(1,M).
55 *
56 * D (output) DOUBLE PRECISION array, dimension (min(M,N))
57 * The diagonal elements of the bidiagonal matrix B:
58 * D(i) = A(i,i).
59 *
60 * E (output) DOUBLE PRECISION array, dimension (min(M,N)-1)
61 * The off-diagonal elements of the bidiagonal matrix B:
62 * if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
63 * if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
64 *
65 * TAUQ (output) DOUBLE PRECISION array dimension (min(M,N))
66 * The scalar factors of the elementary reflectors which
67 * represent the orthogonal matrix Q. See Further Details.
68 *
69 * TAUP (output) DOUBLE PRECISION array, dimension (min(M,N))
70 * The scalar factors of the elementary reflectors which
71 * represent the orthogonal matrix P. See Further Details.
72 *
73 * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
74 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
75 *
76 * LWORK (input) INTEGER
77 * The length of the array WORK. LWORK >= max(1,M,N).
78 * For optimum performance LWORK >= (M+N)*NB, where NB
79 * is the optimal blocksize.
80 *
81 * If LWORK = -1, then a workspace query is assumed; the routine
82 * only calculates the optimal size of the WORK array, returns
83 * this value as the first entry of the WORK array, and no error
84 * message related to LWORK is issued by XERBLA.
85 *
86 * INFO (output) INTEGER
87 * = 0: successful exit
88 * < 0: if INFO = -i, the i-th argument had an illegal value.
89 *
90 * Further Details
91 * ===============
92 *
93 * The matrices Q and P are represented as products of elementary
94 * reflectors:
95 *
96 * If m >= n,
97 *
98 * Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)
99 *
100 * Each H(i) and G(i) has the form:
101 *
102 * H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
103 *
104 * where tauq and taup are real scalars, and v and u are real vectors;
105 * v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
106 * u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
107 * tauq is stored in TAUQ(i) and taup in TAUP(i).
108 *
109 * If m < n,
110 *
111 * Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)
112 *
113 * Each H(i) and G(i) has the form:
114 *
115 * H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
116 *
117 * where tauq and taup are real scalars, and v and u are real vectors;
118 * v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
119 * u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
120 * tauq is stored in TAUQ(i) and taup in TAUP(i).
121 *
122 * The contents of A on exit are illustrated by the following examples:
123 *
124 * m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
125 *
126 * ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
127 * ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
128 * ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
129 * ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
130 * ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
131 * ( v1 v2 v3 v4 v5 )
132 *
133 * where d and e denote diagonal and off-diagonal elements of B, vi
134 * denotes an element of the vector defining H(i), and ui an element of
135 * the vector defining G(i).
136 *
137 * =====================================================================
138 *
139 * .. Parameters ..
140 DOUBLE PRECISION ONE
141 PARAMETER ( ONE = 1.0D+0 )
142 * ..
143 * .. Local Scalars ..
144 LOGICAL LQUERY
145 INTEGER I, IINFO, J, LDWRKX, LDWRKY, LWKOPT, MINMN, NB,
146 $ NBMIN, NX
147 DOUBLE PRECISION WS
148 * ..
149 * .. External Subroutines ..
150 EXTERNAL DGEBD2, DGEMM, DLABRD, XERBLA
151 * ..
152 * .. Intrinsic Functions ..
153 INTRINSIC DBLE, MAX, MIN
154 * ..
155 * .. External Functions ..
156 INTEGER ILAENV
157 EXTERNAL ILAENV
158 * ..
159 * .. Executable Statements ..
160 *
161 * Test the input parameters
162 *
163 INFO = 0
164 NB = MAX( 1, ILAENV( 1, 'DGEBRD', ' ', M, N, -1, -1 ) )
165 LWKOPT = ( M+N )*NB
166 WORK( 1 ) = DBLE( LWKOPT )
167 LQUERY = ( LWORK.EQ.-1 )
168 IF( M.LT.0 ) THEN
169 INFO = -1
170 ELSE IF( N.LT.0 ) THEN
171 INFO = -2
172 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
173 INFO = -4
174 ELSE IF( LWORK.LT.MAX( 1, M, N ) .AND. .NOT.LQUERY ) THEN
175 INFO = -10
176 END IF
177 IF( INFO.LT.0 ) THEN
178 CALL XERBLA( 'DGEBRD', -INFO )
179 RETURN
180 ELSE IF( LQUERY ) THEN
181 RETURN
182 END IF
183 *
184 * Quick return if possible
185 *
186 MINMN = MIN( M, N )
187 IF( MINMN.EQ.0 ) THEN
188 WORK( 1 ) = 1
189 RETURN
190 END IF
191 *
192 WS = MAX( M, N )
193 LDWRKX = M
194 LDWRKY = N
195 *
196 IF( NB.GT.1 .AND. NB.LT.MINMN ) THEN
197 *
198 * Set the crossover point NX.
199 *
200 NX = MAX( NB, ILAENV( 3, 'DGEBRD', ' ', M, N, -1, -1 ) )
201 *
202 * Determine when to switch from blocked to unblocked code.
203 *
204 IF( NX.LT.MINMN ) THEN
205 WS = ( M+N )*NB
206 IF( LWORK.LT.WS ) THEN
207 *
208 * Not enough work space for the optimal NB, consider using
209 * a smaller block size.
210 *
211 NBMIN = ILAENV( 2, 'DGEBRD', ' ', M, N, -1, -1 )
212 IF( LWORK.GE.( M+N )*NBMIN ) THEN
213 NB = LWORK / ( M+N )
214 ELSE
215 NB = 1
216 NX = MINMN
217 END IF
218 END IF
219 END IF
220 ELSE
221 NX = MINMN
222 END IF
223 *
224 DO 30 I = 1, MINMN - NX, NB
225 *
226 * Reduce rows and columns i:i+nb-1 to bidiagonal form and return
227 * the matrices X and Y which are needed to update the unreduced
228 * part of the matrix
229 *
230 CALL DLABRD( M-I+1, N-I+1, NB, A( I, I ), LDA, D( I ), E( I ),
231 $ TAUQ( I ), TAUP( I ), WORK, LDWRKX,
232 $ WORK( LDWRKX*NB+1 ), LDWRKY )
233 *
234 * Update the trailing submatrix A(i+nb:m,i+nb:n), using an update
235 * of the form A := A - V*Y**T - X*U**T
236 *
237 CALL DGEMM( 'No transpose', 'Transpose', M-I-NB+1, N-I-NB+1,
238 $ NB, -ONE, A( I+NB, I ), LDA,
239 $ WORK( LDWRKX*NB+NB+1 ), LDWRKY, ONE,
240 $ A( I+NB, I+NB ), LDA )
241 CALL DGEMM( 'No transpose', 'No transpose', M-I-NB+1, N-I-NB+1,
242 $ NB, -ONE, WORK( NB+1 ), LDWRKX, A( I, I+NB ), LDA,
243 $ ONE, A( I+NB, I+NB ), LDA )
244 *
245 * Copy diagonal and off-diagonal elements of B back into A
246 *
247 IF( M.GE.N ) THEN
248 DO 10 J = I, I + NB - 1
249 A( J, J ) = D( J )
250 A( J, J+1 ) = E( J )
251 10 CONTINUE
252 ELSE
253 DO 20 J = I, I + NB - 1
254 A( J, J ) = D( J )
255 A( J+1, J ) = E( J )
256 20 CONTINUE
257 END IF
258 30 CONTINUE
259 *
260 * Use unblocked code to reduce the remainder of the matrix
261 *
262 CALL DGEBD2( M-I+1, N-I+1, A( I, I ), LDA, D( I ), E( I ),
263 $ TAUQ( I ), TAUP( I ), WORK, IINFO )
264 WORK( 1 ) = WS
265 RETURN
266 *
267 * End of DGEBRD
268 *
269 END
2 $ INFO )
3 *
4 * -- LAPACK routine (version 3.3.1) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * -- April 2011 --
8 *
9 * .. Scalar Arguments ..
10 INTEGER INFO, LDA, LWORK, M, N
11 * ..
12 * .. Array Arguments ..
13 DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
14 $ TAUQ( * ), WORK( * )
15 * ..
16 *
17 * Purpose
18 * =======
19 *
20 * DGEBRD reduces a general real M-by-N matrix A to upper or lower
21 * bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
22 *
23 * If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
24 *
25 * Arguments
26 * =========
27 *
28 * M (input) INTEGER
29 * The number of rows in the matrix A. M >= 0.
30 *
31 * N (input) INTEGER
32 * The number of columns in the matrix A. N >= 0.
33 *
34 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
35 * On entry, the M-by-N general matrix to be reduced.
36 * On exit,
37 * if m >= n, the diagonal and the first superdiagonal are
38 * overwritten with the upper bidiagonal matrix B; the
39 * elements below the diagonal, with the array TAUQ, represent
40 * the orthogonal matrix Q as a product of elementary
41 * reflectors, and the elements above the first superdiagonal,
42 * with the array TAUP, represent the orthogonal matrix P as
43 * a product of elementary reflectors;
44 * if m < n, the diagonal and the first subdiagonal are
45 * overwritten with the lower bidiagonal matrix B; the
46 * elements below the first subdiagonal, with the array TAUQ,
47 * represent the orthogonal matrix Q as a product of
48 * elementary reflectors, and the elements above the diagonal,
49 * with the array TAUP, represent the orthogonal matrix P as
50 * a product of elementary reflectors.
51 * See Further Details.
52 *
53 * LDA (input) INTEGER
54 * The leading dimension of the array A. LDA >= max(1,M).
55 *
56 * D (output) DOUBLE PRECISION array, dimension (min(M,N))
57 * The diagonal elements of the bidiagonal matrix B:
58 * D(i) = A(i,i).
59 *
60 * E (output) DOUBLE PRECISION array, dimension (min(M,N)-1)
61 * The off-diagonal elements of the bidiagonal matrix B:
62 * if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
63 * if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
64 *
65 * TAUQ (output) DOUBLE PRECISION array dimension (min(M,N))
66 * The scalar factors of the elementary reflectors which
67 * represent the orthogonal matrix Q. See Further Details.
68 *
69 * TAUP (output) DOUBLE PRECISION array, dimension (min(M,N))
70 * The scalar factors of the elementary reflectors which
71 * represent the orthogonal matrix P. See Further Details.
72 *
73 * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
74 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
75 *
76 * LWORK (input) INTEGER
77 * The length of the array WORK. LWORK >= max(1,M,N).
78 * For optimum performance LWORK >= (M+N)*NB, where NB
79 * is the optimal blocksize.
80 *
81 * If LWORK = -1, then a workspace query is assumed; the routine
82 * only calculates the optimal size of the WORK array, returns
83 * this value as the first entry of the WORK array, and no error
84 * message related to LWORK is issued by XERBLA.
85 *
86 * INFO (output) INTEGER
87 * = 0: successful exit
88 * < 0: if INFO = -i, the i-th argument had an illegal value.
89 *
90 * Further Details
91 * ===============
92 *
93 * The matrices Q and P are represented as products of elementary
94 * reflectors:
95 *
96 * If m >= n,
97 *
98 * Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)
99 *
100 * Each H(i) and G(i) has the form:
101 *
102 * H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
103 *
104 * where tauq and taup are real scalars, and v and u are real vectors;
105 * v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
106 * u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
107 * tauq is stored in TAUQ(i) and taup in TAUP(i).
108 *
109 * If m < n,
110 *
111 * Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)
112 *
113 * Each H(i) and G(i) has the form:
114 *
115 * H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
116 *
117 * where tauq and taup are real scalars, and v and u are real vectors;
118 * v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
119 * u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
120 * tauq is stored in TAUQ(i) and taup in TAUP(i).
121 *
122 * The contents of A on exit are illustrated by the following examples:
123 *
124 * m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
125 *
126 * ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
127 * ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
128 * ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
129 * ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
130 * ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
131 * ( v1 v2 v3 v4 v5 )
132 *
133 * where d and e denote diagonal and off-diagonal elements of B, vi
134 * denotes an element of the vector defining H(i), and ui an element of
135 * the vector defining G(i).
136 *
137 * =====================================================================
138 *
139 * .. Parameters ..
140 DOUBLE PRECISION ONE
141 PARAMETER ( ONE = 1.0D+0 )
142 * ..
143 * .. Local Scalars ..
144 LOGICAL LQUERY
145 INTEGER I, IINFO, J, LDWRKX, LDWRKY, LWKOPT, MINMN, NB,
146 $ NBMIN, NX
147 DOUBLE PRECISION WS
148 * ..
149 * .. External Subroutines ..
150 EXTERNAL DGEBD2, DGEMM, DLABRD, XERBLA
151 * ..
152 * .. Intrinsic Functions ..
153 INTRINSIC DBLE, MAX, MIN
154 * ..
155 * .. External Functions ..
156 INTEGER ILAENV
157 EXTERNAL ILAENV
158 * ..
159 * .. Executable Statements ..
160 *
161 * Test the input parameters
162 *
163 INFO = 0
164 NB = MAX( 1, ILAENV( 1, 'DGEBRD', ' ', M, N, -1, -1 ) )
165 LWKOPT = ( M+N )*NB
166 WORK( 1 ) = DBLE( LWKOPT )
167 LQUERY = ( LWORK.EQ.-1 )
168 IF( M.LT.0 ) THEN
169 INFO = -1
170 ELSE IF( N.LT.0 ) THEN
171 INFO = -2
172 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
173 INFO = -4
174 ELSE IF( LWORK.LT.MAX( 1, M, N ) .AND. .NOT.LQUERY ) THEN
175 INFO = -10
176 END IF
177 IF( INFO.LT.0 ) THEN
178 CALL XERBLA( 'DGEBRD', -INFO )
179 RETURN
180 ELSE IF( LQUERY ) THEN
181 RETURN
182 END IF
183 *
184 * Quick return if possible
185 *
186 MINMN = MIN( M, N )
187 IF( MINMN.EQ.0 ) THEN
188 WORK( 1 ) = 1
189 RETURN
190 END IF
191 *
192 WS = MAX( M, N )
193 LDWRKX = M
194 LDWRKY = N
195 *
196 IF( NB.GT.1 .AND. NB.LT.MINMN ) THEN
197 *
198 * Set the crossover point NX.
199 *
200 NX = MAX( NB, ILAENV( 3, 'DGEBRD', ' ', M, N, -1, -1 ) )
201 *
202 * Determine when to switch from blocked to unblocked code.
203 *
204 IF( NX.LT.MINMN ) THEN
205 WS = ( M+N )*NB
206 IF( LWORK.LT.WS ) THEN
207 *
208 * Not enough work space for the optimal NB, consider using
209 * a smaller block size.
210 *
211 NBMIN = ILAENV( 2, 'DGEBRD', ' ', M, N, -1, -1 )
212 IF( LWORK.GE.( M+N )*NBMIN ) THEN
213 NB = LWORK / ( M+N )
214 ELSE
215 NB = 1
216 NX = MINMN
217 END IF
218 END IF
219 END IF
220 ELSE
221 NX = MINMN
222 END IF
223 *
224 DO 30 I = 1, MINMN - NX, NB
225 *
226 * Reduce rows and columns i:i+nb-1 to bidiagonal form and return
227 * the matrices X and Y which are needed to update the unreduced
228 * part of the matrix
229 *
230 CALL DLABRD( M-I+1, N-I+1, NB, A( I, I ), LDA, D( I ), E( I ),
231 $ TAUQ( I ), TAUP( I ), WORK, LDWRKX,
232 $ WORK( LDWRKX*NB+1 ), LDWRKY )
233 *
234 * Update the trailing submatrix A(i+nb:m,i+nb:n), using an update
235 * of the form A := A - V*Y**T - X*U**T
236 *
237 CALL DGEMM( 'No transpose', 'Transpose', M-I-NB+1, N-I-NB+1,
238 $ NB, -ONE, A( I+NB, I ), LDA,
239 $ WORK( LDWRKX*NB+NB+1 ), LDWRKY, ONE,
240 $ A( I+NB, I+NB ), LDA )
241 CALL DGEMM( 'No transpose', 'No transpose', M-I-NB+1, N-I-NB+1,
242 $ NB, -ONE, WORK( NB+1 ), LDWRKX, A( I, I+NB ), LDA,
243 $ ONE, A( I+NB, I+NB ), LDA )
244 *
245 * Copy diagonal and off-diagonal elements of B back into A
246 *
247 IF( M.GE.N ) THEN
248 DO 10 J = I, I + NB - 1
249 A( J, J ) = D( J )
250 A( J, J+1 ) = E( J )
251 10 CONTINUE
252 ELSE
253 DO 20 J = I, I + NB - 1
254 A( J, J ) = D( J )
255 A( J+1, J ) = E( J )
256 20 CONTINUE
257 END IF
258 30 CONTINUE
259 *
260 * Use unblocked code to reduce the remainder of the matrix
261 *
262 CALL DGEBD2( M-I+1, N-I+1, A( I, I ), LDA, D( I ), E( I ),
263 $ TAUQ( I ), TAUP( I ), WORK, IINFO )
264 WORK( 1 ) = WS
265 RETURN
266 *
267 * End of DGEBRD
268 *
269 END