1 SUBROUTINE DGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, 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, M, N
10 * ..
11 * .. Array Arguments ..
12 DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
13 $ TAUQ( * ), WORK( * )
14 * ..
15 *
16 * Purpose
17 * =======
18 *
19 * DGEBD2 reduces a real general m by n matrix A to upper or lower
20 * bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
21 *
22 * If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
23 *
24 * Arguments
25 * =========
26 *
27 * M (input) INTEGER
28 * The number of rows in the matrix A. M >= 0.
29 *
30 * N (input) INTEGER
31 * The number of columns in the matrix A. N >= 0.
32 *
33 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
34 * On entry, the m by n general matrix to be reduced.
35 * On exit,
36 * if m >= n, the diagonal and the first superdiagonal are
37 * overwritten with the upper bidiagonal matrix B; the
38 * elements below the diagonal, with the array TAUQ, represent
39 * the orthogonal matrix Q as a product of elementary
40 * reflectors, and the elements above the first superdiagonal,
41 * with the array TAUP, represent the orthogonal matrix P as
42 * a product of elementary reflectors;
43 * if m < n, the diagonal and the first subdiagonal are
44 * overwritten with the lower bidiagonal matrix B; the
45 * elements below the first subdiagonal, with the array TAUQ,
46 * represent the orthogonal matrix Q as a product of
47 * elementary reflectors, and the elements above the diagonal,
48 * with the array TAUP, represent the orthogonal matrix P as
49 * a product of elementary reflectors.
50 * See Further Details.
51 *
52 * LDA (input) INTEGER
53 * The leading dimension of the array A. LDA >= max(1,M).
54 *
55 * D (output) DOUBLE PRECISION array, dimension (min(M,N))
56 * The diagonal elements of the bidiagonal matrix B:
57 * D(i) = A(i,i).
58 *
59 * E (output) DOUBLE PRECISION array, dimension (min(M,N)-1)
60 * The off-diagonal elements of the bidiagonal matrix B:
61 * if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
62 * if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
63 *
64 * TAUQ (output) DOUBLE PRECISION array dimension (min(M,N))
65 * The scalar factors of the elementary reflectors which
66 * represent the orthogonal matrix Q. See Further Details.
67 *
68 * TAUP (output) DOUBLE PRECISION array, dimension (min(M,N))
69 * The scalar factors of the elementary reflectors which
70 * represent the orthogonal matrix P. See Further Details.
71 *
72 * WORK (workspace) DOUBLE PRECISION array, dimension (max(M,N))
73 *
74 * INFO (output) INTEGER
75 * = 0: successful exit.
76 * < 0: if INFO = -i, the i-th argument had an illegal value.
77 *
78 * Further Details
79 * ===============
80 *
81 * The matrices Q and P are represented as products of elementary
82 * reflectors:
83 *
84 * If m >= n,
85 *
86 * Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)
87 *
88 * Each H(i) and G(i) has the form:
89 *
90 * H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
91 *
92 * where tauq and taup are real scalars, and v and u are real vectors;
93 * v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
94 * u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
95 * tauq is stored in TAUQ(i) and taup in TAUP(i).
96 *
97 * If m < n,
98 *
99 * Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)
100 *
101 * Each H(i) and G(i) has the form:
102 *
103 * H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
104 *
105 * where tauq and taup are real scalars, and v and u are real vectors;
106 * v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
107 * u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
108 * tauq is stored in TAUQ(i) and taup in TAUP(i).
109 *
110 * The contents of A on exit are illustrated by the following examples:
111 *
112 * m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
113 *
114 * ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
115 * ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
116 * ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
117 * ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
118 * ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
119 * ( v1 v2 v3 v4 v5 )
120 *
121 * where d and e denote diagonal and off-diagonal elements of B, vi
122 * denotes an element of the vector defining H(i), and ui an element of
123 * the vector defining G(i).
124 *
125 * =====================================================================
126 *
127 * .. Parameters ..
128 DOUBLE PRECISION ZERO, ONE
129 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
130 * ..
131 * .. Local Scalars ..
132 INTEGER I
133 * ..
134 * .. External Subroutines ..
135 EXTERNAL DLARF, DLARFG, XERBLA
136 * ..
137 * .. Intrinsic Functions ..
138 INTRINSIC MAX, MIN
139 * ..
140 * .. Executable Statements ..
141 *
142 * Test the input parameters
143 *
144 INFO = 0
145 IF( M.LT.0 ) THEN
146 INFO = -1
147 ELSE IF( N.LT.0 ) THEN
148 INFO = -2
149 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
150 INFO = -4
151 END IF
152 IF( INFO.LT.0 ) THEN
153 CALL XERBLA( 'DGEBD2', -INFO )
154 RETURN
155 END IF
156 *
157 IF( M.GE.N ) THEN
158 *
159 * Reduce to upper bidiagonal form
160 *
161 DO 10 I = 1, N
162 *
163 * Generate elementary reflector H(i) to annihilate A(i+1:m,i)
164 *
165 CALL DLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1,
166 $ TAUQ( I ) )
167 D( I ) = A( I, I )
168 A( I, I ) = ONE
169 *
170 * Apply H(i) to A(i:m,i+1:n) from the left
171 *
172 IF( I.LT.N )
173 $ CALL DLARF( 'Left', M-I+1, N-I, A( I, I ), 1, TAUQ( I ),
174 $ A( I, I+1 ), LDA, WORK )
175 A( I, I ) = D( I )
176 *
177 IF( I.LT.N ) THEN
178 *
179 * Generate elementary reflector G(i) to annihilate
180 * A(i,i+2:n)
181 *
182 CALL DLARFG( N-I, A( I, I+1 ), A( I, MIN( I+2, N ) ),
183 $ LDA, TAUP( I ) )
184 E( I ) = A( I, I+1 )
185 A( I, I+1 ) = ONE
186 *
187 * Apply G(i) to A(i+1:m,i+1:n) from the right
188 *
189 CALL DLARF( 'Right', M-I, N-I, A( I, I+1 ), LDA,
190 $ TAUP( I ), A( I+1, I+1 ), LDA, WORK )
191 A( I, I+1 ) = E( I )
192 ELSE
193 TAUP( I ) = ZERO
194 END IF
195 10 CONTINUE
196 ELSE
197 *
198 * Reduce to lower bidiagonal form
199 *
200 DO 20 I = 1, M
201 *
202 * Generate elementary reflector G(i) to annihilate A(i,i+1:n)
203 *
204 CALL DLARFG( N-I+1, A( I, I ), A( I, MIN( I+1, N ) ), LDA,
205 $ TAUP( I ) )
206 D( I ) = A( I, I )
207 A( I, I ) = ONE
208 *
209 * Apply G(i) to A(i+1:m,i:n) from the right
210 *
211 IF( I.LT.M )
212 $ CALL DLARF( 'Right', M-I, N-I+1, A( I, I ), LDA,
213 $ TAUP( I ), A( I+1, I ), LDA, WORK )
214 A( I, I ) = D( I )
215 *
216 IF( I.LT.M ) THEN
217 *
218 * Generate elementary reflector H(i) to annihilate
219 * A(i+2:m,i)
220 *
221 CALL DLARFG( M-I, A( I+1, I ), A( MIN( I+2, M ), I ), 1,
222 $ TAUQ( I ) )
223 E( I ) = A( I+1, I )
224 A( I+1, I ) = ONE
225 *
226 * Apply H(i) to A(i+1:m,i+1:n) from the left
227 *
228 CALL DLARF( 'Left', M-I, N-I, A( I+1, I ), 1, TAUQ( I ),
229 $ A( I+1, I+1 ), LDA, WORK )
230 A( I+1, I ) = E( I )
231 ELSE
232 TAUQ( I ) = ZERO
233 END IF
234 20 CONTINUE
235 END IF
236 RETURN
237 *
238 * End of DGEBD2
239 *
240 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, M, N
10 * ..
11 * .. Array Arguments ..
12 DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
13 $ TAUQ( * ), WORK( * )
14 * ..
15 *
16 * Purpose
17 * =======
18 *
19 * DGEBD2 reduces a real general m by n matrix A to upper or lower
20 * bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
21 *
22 * If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
23 *
24 * Arguments
25 * =========
26 *
27 * M (input) INTEGER
28 * The number of rows in the matrix A. M >= 0.
29 *
30 * N (input) INTEGER
31 * The number of columns in the matrix A. N >= 0.
32 *
33 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
34 * On entry, the m by n general matrix to be reduced.
35 * On exit,
36 * if m >= n, the diagonal and the first superdiagonal are
37 * overwritten with the upper bidiagonal matrix B; the
38 * elements below the diagonal, with the array TAUQ, represent
39 * the orthogonal matrix Q as a product of elementary
40 * reflectors, and the elements above the first superdiagonal,
41 * with the array TAUP, represent the orthogonal matrix P as
42 * a product of elementary reflectors;
43 * if m < n, the diagonal and the first subdiagonal are
44 * overwritten with the lower bidiagonal matrix B; the
45 * elements below the first subdiagonal, with the array TAUQ,
46 * represent the orthogonal matrix Q as a product of
47 * elementary reflectors, and the elements above the diagonal,
48 * with the array TAUP, represent the orthogonal matrix P as
49 * a product of elementary reflectors.
50 * See Further Details.
51 *
52 * LDA (input) INTEGER
53 * The leading dimension of the array A. LDA >= max(1,M).
54 *
55 * D (output) DOUBLE PRECISION array, dimension (min(M,N))
56 * The diagonal elements of the bidiagonal matrix B:
57 * D(i) = A(i,i).
58 *
59 * E (output) DOUBLE PRECISION array, dimension (min(M,N)-1)
60 * The off-diagonal elements of the bidiagonal matrix B:
61 * if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
62 * if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
63 *
64 * TAUQ (output) DOUBLE PRECISION array dimension (min(M,N))
65 * The scalar factors of the elementary reflectors which
66 * represent the orthogonal matrix Q. See Further Details.
67 *
68 * TAUP (output) DOUBLE PRECISION array, dimension (min(M,N))
69 * The scalar factors of the elementary reflectors which
70 * represent the orthogonal matrix P. See Further Details.
71 *
72 * WORK (workspace) DOUBLE PRECISION array, dimension (max(M,N))
73 *
74 * INFO (output) INTEGER
75 * = 0: successful exit.
76 * < 0: if INFO = -i, the i-th argument had an illegal value.
77 *
78 * Further Details
79 * ===============
80 *
81 * The matrices Q and P are represented as products of elementary
82 * reflectors:
83 *
84 * If m >= n,
85 *
86 * Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)
87 *
88 * Each H(i) and G(i) has the form:
89 *
90 * H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
91 *
92 * where tauq and taup are real scalars, and v and u are real vectors;
93 * v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
94 * u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
95 * tauq is stored in TAUQ(i) and taup in TAUP(i).
96 *
97 * If m < n,
98 *
99 * Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)
100 *
101 * Each H(i) and G(i) has the form:
102 *
103 * H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
104 *
105 * where tauq and taup are real scalars, and v and u are real vectors;
106 * v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
107 * u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
108 * tauq is stored in TAUQ(i) and taup in TAUP(i).
109 *
110 * The contents of A on exit are illustrated by the following examples:
111 *
112 * m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
113 *
114 * ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
115 * ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
116 * ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
117 * ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
118 * ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
119 * ( v1 v2 v3 v4 v5 )
120 *
121 * where d and e denote diagonal and off-diagonal elements of B, vi
122 * denotes an element of the vector defining H(i), and ui an element of
123 * the vector defining G(i).
124 *
125 * =====================================================================
126 *
127 * .. Parameters ..
128 DOUBLE PRECISION ZERO, ONE
129 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
130 * ..
131 * .. Local Scalars ..
132 INTEGER I
133 * ..
134 * .. External Subroutines ..
135 EXTERNAL DLARF, DLARFG, XERBLA
136 * ..
137 * .. Intrinsic Functions ..
138 INTRINSIC MAX, MIN
139 * ..
140 * .. Executable Statements ..
141 *
142 * Test the input parameters
143 *
144 INFO = 0
145 IF( M.LT.0 ) THEN
146 INFO = -1
147 ELSE IF( N.LT.0 ) THEN
148 INFO = -2
149 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
150 INFO = -4
151 END IF
152 IF( INFO.LT.0 ) THEN
153 CALL XERBLA( 'DGEBD2', -INFO )
154 RETURN
155 END IF
156 *
157 IF( M.GE.N ) THEN
158 *
159 * Reduce to upper bidiagonal form
160 *
161 DO 10 I = 1, N
162 *
163 * Generate elementary reflector H(i) to annihilate A(i+1:m,i)
164 *
165 CALL DLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1,
166 $ TAUQ( I ) )
167 D( I ) = A( I, I )
168 A( I, I ) = ONE
169 *
170 * Apply H(i) to A(i:m,i+1:n) from the left
171 *
172 IF( I.LT.N )
173 $ CALL DLARF( 'Left', M-I+1, N-I, A( I, I ), 1, TAUQ( I ),
174 $ A( I, I+1 ), LDA, WORK )
175 A( I, I ) = D( I )
176 *
177 IF( I.LT.N ) THEN
178 *
179 * Generate elementary reflector G(i) to annihilate
180 * A(i,i+2:n)
181 *
182 CALL DLARFG( N-I, A( I, I+1 ), A( I, MIN( I+2, N ) ),
183 $ LDA, TAUP( I ) )
184 E( I ) = A( I, I+1 )
185 A( I, I+1 ) = ONE
186 *
187 * Apply G(i) to A(i+1:m,i+1:n) from the right
188 *
189 CALL DLARF( 'Right', M-I, N-I, A( I, I+1 ), LDA,
190 $ TAUP( I ), A( I+1, I+1 ), LDA, WORK )
191 A( I, I+1 ) = E( I )
192 ELSE
193 TAUP( I ) = ZERO
194 END IF
195 10 CONTINUE
196 ELSE
197 *
198 * Reduce to lower bidiagonal form
199 *
200 DO 20 I = 1, M
201 *
202 * Generate elementary reflector G(i) to annihilate A(i,i+1:n)
203 *
204 CALL DLARFG( N-I+1, A( I, I ), A( I, MIN( I+1, N ) ), LDA,
205 $ TAUP( I ) )
206 D( I ) = A( I, I )
207 A( I, I ) = ONE
208 *
209 * Apply G(i) to A(i+1:m,i:n) from the right
210 *
211 IF( I.LT.M )
212 $ CALL DLARF( 'Right', M-I, N-I+1, A( I, I ), LDA,
213 $ TAUP( I ), A( I+1, I ), LDA, WORK )
214 A( I, I ) = D( I )
215 *
216 IF( I.LT.M ) THEN
217 *
218 * Generate elementary reflector H(i) to annihilate
219 * A(i+2:m,i)
220 *
221 CALL DLARFG( M-I, A( I+1, I ), A( MIN( I+2, M ), I ), 1,
222 $ TAUQ( I ) )
223 E( I ) = A( I+1, I )
224 A( I+1, I ) = ONE
225 *
226 * Apply H(i) to A(i+1:m,i+1:n) from the left
227 *
228 CALL DLARF( 'Left', M-I, N-I, A( I+1, I ), 1, TAUQ( I ),
229 $ A( I+1, I+1 ), LDA, WORK )
230 A( I+1, I ) = E( I )
231 ELSE
232 TAUQ( I ) = ZERO
233 END IF
234 20 CONTINUE
235 END IF
236 RETURN
237 *
238 * End of DGEBD2
239 *
240 END