1 SUBROUTINE DLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y,
2 $ LDY )
3 *
4 * -- LAPACK auxiliary 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 LDA, LDX, LDY, M, N, NB
11 * ..
12 * .. Array Arguments ..
13 DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
14 $ TAUQ( * ), X( LDX, * ), Y( LDY, * )
15 * ..
16 *
17 * Purpose
18 * =======
19 *
20 * DLABRD reduces the first NB rows and columns of a real general
21 * m by n matrix A to upper or lower bidiagonal form by an orthogonal
22 * transformation Q**T * A * P, and returns the matrices X and Y which
23 * are needed to apply the transformation to the unreduced part of A.
24 *
25 * If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
26 * bidiagonal form.
27 *
28 * This is an auxiliary routine called by DGEBRD
29 *
30 * Arguments
31 * =========
32 *
33 * M (input) INTEGER
34 * The number of rows in the matrix A.
35 *
36 * N (input) INTEGER
37 * The number of columns in the matrix A.
38 *
39 * NB (input) INTEGER
40 * The number of leading rows and columns of A to be reduced.
41 *
42 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
43 * On entry, the m by n general matrix to be reduced.
44 * On exit, the first NB rows and columns of the matrix are
45 * overwritten; the rest of the array is unchanged.
46 * If m >= n, elements on and below the diagonal in the first NB
47 * columns, with the array TAUQ, represent the orthogonal
48 * matrix Q as a product of elementary reflectors; and
49 * elements above the diagonal in the first NB rows, with the
50 * array TAUP, represent the orthogonal matrix P as a product
51 * of elementary reflectors.
52 * If m < n, elements below the diagonal in the first NB
53 * columns, with the array TAUQ, represent the orthogonal
54 * matrix Q as a product of elementary reflectors, and
55 * elements on and above the diagonal in the first NB rows,
56 * with the array TAUP, represent the orthogonal matrix P as
57 * a product of elementary reflectors.
58 * See Further Details.
59 *
60 * LDA (input) INTEGER
61 * The leading dimension of the array A. LDA >= max(1,M).
62 *
63 * D (output) DOUBLE PRECISION array, dimension (NB)
64 * The diagonal elements of the first NB rows and columns of
65 * the reduced matrix. D(i) = A(i,i).
66 *
67 * E (output) DOUBLE PRECISION array, dimension (NB)
68 * The off-diagonal elements of the first NB rows and columns of
69 * the reduced matrix.
70 *
71 * TAUQ (output) DOUBLE PRECISION array dimension (NB)
72 * The scalar factors of the elementary reflectors which
73 * represent the orthogonal matrix Q. See Further Details.
74 *
75 * TAUP (output) DOUBLE PRECISION array, dimension (NB)
76 * The scalar factors of the elementary reflectors which
77 * represent the orthogonal matrix P. See Further Details.
78 *
79 * X (output) DOUBLE PRECISION array, dimension (LDX,NB)
80 * The m-by-nb matrix X required to update the unreduced part
81 * of A.
82 *
83 * LDX (input) INTEGER
84 * The leading dimension of the array X. LDX >= max(1,M).
85 *
86 * Y (output) DOUBLE PRECISION array, dimension (LDY,NB)
87 * The n-by-nb matrix Y required to update the unreduced part
88 * of A.
89 *
90 * LDY (input) INTEGER
91 * The leading dimension of the array Y. LDY >= max(1,N).
92 *
93 * Further Details
94 * ===============
95 *
96 * The matrices Q and P are represented as products of elementary
97 * reflectors:
98 *
99 * Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb)
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 *
107 * If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
108 * A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in
109 * A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
110 *
111 * If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
112 * A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
113 * A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
114 *
115 * The elements of the vectors v and u together form the m-by-nb matrix
116 * V and the nb-by-n matrix U**T which are needed, with X and Y, to apply
117 * the transformation to the unreduced part of the matrix, using a block
118 * update of the form: A := A - V*Y**T - X*U**T.
119 *
120 * The contents of A on exit are illustrated by the following examples
121 * with nb = 2:
122 *
123 * m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
124 *
125 * ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 )
126 * ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 )
127 * ( v1 v2 a a a ) ( v1 1 a a a a )
128 * ( v1 v2 a a a ) ( v1 v2 a a a a )
129 * ( v1 v2 a a a ) ( v1 v2 a a a a )
130 * ( v1 v2 a a a )
131 *
132 * where a denotes an element of the original matrix which is unchanged,
133 * vi denotes an element of the vector defining H(i), and ui an element
134 * of the vector defining G(i).
135 *
136 * =====================================================================
137 *
138 * .. Parameters ..
139 DOUBLE PRECISION ZERO, ONE
140 PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
141 * ..
142 * .. Local Scalars ..
143 INTEGER I
144 * ..
145 * .. External Subroutines ..
146 EXTERNAL DGEMV, DLARFG, DSCAL
147 * ..
148 * .. Intrinsic Functions ..
149 INTRINSIC MIN
150 * ..
151 * .. Executable Statements ..
152 *
153 * Quick return if possible
154 *
155 IF( M.LE.0 .OR. N.LE.0 )
156 $ RETURN
157 *
158 IF( M.GE.N ) THEN
159 *
160 * Reduce to upper bidiagonal form
161 *
162 DO 10 I = 1, NB
163 *
164 * Update A(i:m,i)
165 *
166 CALL DGEMV( 'No transpose', M-I+1, I-1, -ONE, A( I, 1 ),
167 $ LDA, Y( I, 1 ), LDY, ONE, A( I, I ), 1 )
168 CALL DGEMV( 'No transpose', M-I+1, I-1, -ONE, X( I, 1 ),
169 $ LDX, A( 1, I ), 1, ONE, A( I, I ), 1 )
170 *
171 * Generate reflection Q(i) to annihilate A(i+1:m,i)
172 *
173 CALL DLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1,
174 $ TAUQ( I ) )
175 D( I ) = A( I, I )
176 IF( I.LT.N ) THEN
177 A( I, I ) = ONE
178 *
179 * Compute Y(i+1:n,i)
180 *
181 CALL DGEMV( 'Transpose', M-I+1, N-I, ONE, A( I, I+1 ),
182 $ LDA, A( I, I ), 1, ZERO, Y( I+1, I ), 1 )
183 CALL DGEMV( 'Transpose', M-I+1, I-1, ONE, A( I, 1 ), LDA,
184 $ A( I, I ), 1, ZERO, Y( 1, I ), 1 )
185 CALL DGEMV( 'No transpose', N-I, I-1, -ONE, Y( I+1, 1 ),
186 $ LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
187 CALL DGEMV( 'Transpose', M-I+1, I-1, ONE, X( I, 1 ), LDX,
188 $ A( I, I ), 1, ZERO, Y( 1, I ), 1 )
189 CALL DGEMV( 'Transpose', I-1, N-I, -ONE, A( 1, I+1 ),
190 $ LDA, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
191 CALL DSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 )
192 *
193 * Update A(i,i+1:n)
194 *
195 CALL DGEMV( 'No transpose', N-I, I, -ONE, Y( I+1, 1 ),
196 $ LDY, A( I, 1 ), LDA, ONE, A( I, I+1 ), LDA )
197 CALL DGEMV( 'Transpose', I-1, N-I, -ONE, A( 1, I+1 ),
198 $ LDA, X( I, 1 ), LDX, ONE, A( I, I+1 ), LDA )
199 *
200 * Generate reflection P(i) to annihilate A(i,i+2:n)
201 *
202 CALL DLARFG( N-I, A( I, I+1 ), A( I, MIN( I+2, N ) ),
203 $ LDA, TAUP( I ) )
204 E( I ) = A( I, I+1 )
205 A( I, I+1 ) = ONE
206 *
207 * Compute X(i+1:m,i)
208 *
209 CALL DGEMV( 'No transpose', M-I, N-I, ONE, A( I+1, I+1 ),
210 $ LDA, A( I, I+1 ), LDA, ZERO, X( I+1, I ), 1 )
211 CALL DGEMV( 'Transpose', N-I, I, ONE, Y( I+1, 1 ), LDY,
212 $ A( I, I+1 ), LDA, ZERO, X( 1, I ), 1 )
213 CALL DGEMV( 'No transpose', M-I, I, -ONE, A( I+1, 1 ),
214 $ LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
215 CALL DGEMV( 'No transpose', I-1, N-I, ONE, A( 1, I+1 ),
216 $ LDA, A( I, I+1 ), LDA, ZERO, X( 1, I ), 1 )
217 CALL DGEMV( 'No transpose', M-I, I-1, -ONE, X( I+1, 1 ),
218 $ LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
219 CALL DSCAL( M-I, TAUP( I ), X( I+1, I ), 1 )
220 END IF
221 10 CONTINUE
222 ELSE
223 *
224 * Reduce to lower bidiagonal form
225 *
226 DO 20 I = 1, NB
227 *
228 * Update A(i,i:n)
229 *
230 CALL DGEMV( 'No transpose', N-I+1, I-1, -ONE, Y( I, 1 ),
231 $ LDY, A( I, 1 ), LDA, ONE, A( I, I ), LDA )
232 CALL DGEMV( 'Transpose', I-1, N-I+1, -ONE, A( 1, I ), LDA,
233 $ X( I, 1 ), LDX, ONE, A( I, I ), LDA )
234 *
235 * Generate reflection P(i) to annihilate A(i,i+1:n)
236 *
237 CALL DLARFG( N-I+1, A( I, I ), A( I, MIN( I+1, N ) ), LDA,
238 $ TAUP( I ) )
239 D( I ) = A( I, I )
240 IF( I.LT.M ) THEN
241 A( I, I ) = ONE
242 *
243 * Compute X(i+1:m,i)
244 *
245 CALL DGEMV( 'No transpose', M-I, N-I+1, ONE, A( I+1, I ),
246 $ LDA, A( I, I ), LDA, ZERO, X( I+1, I ), 1 )
247 CALL DGEMV( 'Transpose', N-I+1, I-1, ONE, Y( I, 1 ), LDY,
248 $ A( I, I ), LDA, ZERO, X( 1, I ), 1 )
249 CALL DGEMV( 'No transpose', M-I, I-1, -ONE, A( I+1, 1 ),
250 $ LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
251 CALL DGEMV( 'No transpose', I-1, N-I+1, ONE, A( 1, I ),
252 $ LDA, A( I, I ), LDA, ZERO, X( 1, I ), 1 )
253 CALL DGEMV( 'No transpose', M-I, I-1, -ONE, X( I+1, 1 ),
254 $ LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
255 CALL DSCAL( M-I, TAUP( I ), X( I+1, I ), 1 )
256 *
257 * Update A(i+1:m,i)
258 *
259 CALL DGEMV( 'No transpose', M-I, I-1, -ONE, A( I+1, 1 ),
260 $ LDA, Y( I, 1 ), LDY, ONE, A( I+1, I ), 1 )
261 CALL DGEMV( 'No transpose', M-I, I, -ONE, X( I+1, 1 ),
262 $ LDX, A( 1, I ), 1, ONE, A( I+1, I ), 1 )
263 *
264 * Generate reflection Q(i) to annihilate A(i+2:m,i)
265 *
266 CALL DLARFG( M-I, A( I+1, I ), A( MIN( I+2, M ), I ), 1,
267 $ TAUQ( I ) )
268 E( I ) = A( I+1, I )
269 A( I+1, I ) = ONE
270 *
271 * Compute Y(i+1:n,i)
272 *
273 CALL DGEMV( 'Transpose', M-I, N-I, ONE, A( I+1, I+1 ),
274 $ LDA, A( I+1, I ), 1, ZERO, Y( I+1, I ), 1 )
275 CALL DGEMV( 'Transpose', M-I, I-1, ONE, A( I+1, 1 ), LDA,
276 $ A( I+1, I ), 1, ZERO, Y( 1, I ), 1 )
277 CALL DGEMV( 'No transpose', N-I, I-1, -ONE, Y( I+1, 1 ),
278 $ LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
279 CALL DGEMV( 'Transpose', M-I, I, ONE, X( I+1, 1 ), LDX,
280 $ A( I+1, I ), 1, ZERO, Y( 1, I ), 1 )
281 CALL DGEMV( 'Transpose', I, N-I, -ONE, A( 1, I+1 ), LDA,
282 $ Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
283 CALL DSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 )
284 END IF
285 20 CONTINUE
286 END IF
287 RETURN
288 *
289 * End of DLABRD
290 *
291 END
2 $ LDY )
3 *
4 * -- LAPACK auxiliary 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 LDA, LDX, LDY, M, N, NB
11 * ..
12 * .. Array Arguments ..
13 DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
14 $ TAUQ( * ), X( LDX, * ), Y( LDY, * )
15 * ..
16 *
17 * Purpose
18 * =======
19 *
20 * DLABRD reduces the first NB rows and columns of a real general
21 * m by n matrix A to upper or lower bidiagonal form by an orthogonal
22 * transformation Q**T * A * P, and returns the matrices X and Y which
23 * are needed to apply the transformation to the unreduced part of A.
24 *
25 * If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
26 * bidiagonal form.
27 *
28 * This is an auxiliary routine called by DGEBRD
29 *
30 * Arguments
31 * =========
32 *
33 * M (input) INTEGER
34 * The number of rows in the matrix A.
35 *
36 * N (input) INTEGER
37 * The number of columns in the matrix A.
38 *
39 * NB (input) INTEGER
40 * The number of leading rows and columns of A to be reduced.
41 *
42 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
43 * On entry, the m by n general matrix to be reduced.
44 * On exit, the first NB rows and columns of the matrix are
45 * overwritten; the rest of the array is unchanged.
46 * If m >= n, elements on and below the diagonal in the first NB
47 * columns, with the array TAUQ, represent the orthogonal
48 * matrix Q as a product of elementary reflectors; and
49 * elements above the diagonal in the first NB rows, with the
50 * array TAUP, represent the orthogonal matrix P as a product
51 * of elementary reflectors.
52 * If m < n, elements below the diagonal in the first NB
53 * columns, with the array TAUQ, represent the orthogonal
54 * matrix Q as a product of elementary reflectors, and
55 * elements on and above the diagonal in the first NB rows,
56 * with the array TAUP, represent the orthogonal matrix P as
57 * a product of elementary reflectors.
58 * See Further Details.
59 *
60 * LDA (input) INTEGER
61 * The leading dimension of the array A. LDA >= max(1,M).
62 *
63 * D (output) DOUBLE PRECISION array, dimension (NB)
64 * The diagonal elements of the first NB rows and columns of
65 * the reduced matrix. D(i) = A(i,i).
66 *
67 * E (output) DOUBLE PRECISION array, dimension (NB)
68 * The off-diagonal elements of the first NB rows and columns of
69 * the reduced matrix.
70 *
71 * TAUQ (output) DOUBLE PRECISION array dimension (NB)
72 * The scalar factors of the elementary reflectors which
73 * represent the orthogonal matrix Q. See Further Details.
74 *
75 * TAUP (output) DOUBLE PRECISION array, dimension (NB)
76 * The scalar factors of the elementary reflectors which
77 * represent the orthogonal matrix P. See Further Details.
78 *
79 * X (output) DOUBLE PRECISION array, dimension (LDX,NB)
80 * The m-by-nb matrix X required to update the unreduced part
81 * of A.
82 *
83 * LDX (input) INTEGER
84 * The leading dimension of the array X. LDX >= max(1,M).
85 *
86 * Y (output) DOUBLE PRECISION array, dimension (LDY,NB)
87 * The n-by-nb matrix Y required to update the unreduced part
88 * of A.
89 *
90 * LDY (input) INTEGER
91 * The leading dimension of the array Y. LDY >= max(1,N).
92 *
93 * Further Details
94 * ===============
95 *
96 * The matrices Q and P are represented as products of elementary
97 * reflectors:
98 *
99 * Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb)
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 *
107 * If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
108 * A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in
109 * A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
110 *
111 * If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
112 * A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
113 * A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
114 *
115 * The elements of the vectors v and u together form the m-by-nb matrix
116 * V and the nb-by-n matrix U**T which are needed, with X and Y, to apply
117 * the transformation to the unreduced part of the matrix, using a block
118 * update of the form: A := A - V*Y**T - X*U**T.
119 *
120 * The contents of A on exit are illustrated by the following examples
121 * with nb = 2:
122 *
123 * m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
124 *
125 * ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 )
126 * ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 )
127 * ( v1 v2 a a a ) ( v1 1 a a a a )
128 * ( v1 v2 a a a ) ( v1 v2 a a a a )
129 * ( v1 v2 a a a ) ( v1 v2 a a a a )
130 * ( v1 v2 a a a )
131 *
132 * where a denotes an element of the original matrix which is unchanged,
133 * vi denotes an element of the vector defining H(i), and ui an element
134 * of the vector defining G(i).
135 *
136 * =====================================================================
137 *
138 * .. Parameters ..
139 DOUBLE PRECISION ZERO, ONE
140 PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
141 * ..
142 * .. Local Scalars ..
143 INTEGER I
144 * ..
145 * .. External Subroutines ..
146 EXTERNAL DGEMV, DLARFG, DSCAL
147 * ..
148 * .. Intrinsic Functions ..
149 INTRINSIC MIN
150 * ..
151 * .. Executable Statements ..
152 *
153 * Quick return if possible
154 *
155 IF( M.LE.0 .OR. N.LE.0 )
156 $ RETURN
157 *
158 IF( M.GE.N ) THEN
159 *
160 * Reduce to upper bidiagonal form
161 *
162 DO 10 I = 1, NB
163 *
164 * Update A(i:m,i)
165 *
166 CALL DGEMV( 'No transpose', M-I+1, I-1, -ONE, A( I, 1 ),
167 $ LDA, Y( I, 1 ), LDY, ONE, A( I, I ), 1 )
168 CALL DGEMV( 'No transpose', M-I+1, I-1, -ONE, X( I, 1 ),
169 $ LDX, A( 1, I ), 1, ONE, A( I, I ), 1 )
170 *
171 * Generate reflection Q(i) to annihilate A(i+1:m,i)
172 *
173 CALL DLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1,
174 $ TAUQ( I ) )
175 D( I ) = A( I, I )
176 IF( I.LT.N ) THEN
177 A( I, I ) = ONE
178 *
179 * Compute Y(i+1:n,i)
180 *
181 CALL DGEMV( 'Transpose', M-I+1, N-I, ONE, A( I, I+1 ),
182 $ LDA, A( I, I ), 1, ZERO, Y( I+1, I ), 1 )
183 CALL DGEMV( 'Transpose', M-I+1, I-1, ONE, A( I, 1 ), LDA,
184 $ A( I, I ), 1, ZERO, Y( 1, I ), 1 )
185 CALL DGEMV( 'No transpose', N-I, I-1, -ONE, Y( I+1, 1 ),
186 $ LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
187 CALL DGEMV( 'Transpose', M-I+1, I-1, ONE, X( I, 1 ), LDX,
188 $ A( I, I ), 1, ZERO, Y( 1, I ), 1 )
189 CALL DGEMV( 'Transpose', I-1, N-I, -ONE, A( 1, I+1 ),
190 $ LDA, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
191 CALL DSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 )
192 *
193 * Update A(i,i+1:n)
194 *
195 CALL DGEMV( 'No transpose', N-I, I, -ONE, Y( I+1, 1 ),
196 $ LDY, A( I, 1 ), LDA, ONE, A( I, I+1 ), LDA )
197 CALL DGEMV( 'Transpose', I-1, N-I, -ONE, A( 1, I+1 ),
198 $ LDA, X( I, 1 ), LDX, ONE, A( I, I+1 ), LDA )
199 *
200 * Generate reflection P(i) to annihilate A(i,i+2:n)
201 *
202 CALL DLARFG( N-I, A( I, I+1 ), A( I, MIN( I+2, N ) ),
203 $ LDA, TAUP( I ) )
204 E( I ) = A( I, I+1 )
205 A( I, I+1 ) = ONE
206 *
207 * Compute X(i+1:m,i)
208 *
209 CALL DGEMV( 'No transpose', M-I, N-I, ONE, A( I+1, I+1 ),
210 $ LDA, A( I, I+1 ), LDA, ZERO, X( I+1, I ), 1 )
211 CALL DGEMV( 'Transpose', N-I, I, ONE, Y( I+1, 1 ), LDY,
212 $ A( I, I+1 ), LDA, ZERO, X( 1, I ), 1 )
213 CALL DGEMV( 'No transpose', M-I, I, -ONE, A( I+1, 1 ),
214 $ LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
215 CALL DGEMV( 'No transpose', I-1, N-I, ONE, A( 1, I+1 ),
216 $ LDA, A( I, I+1 ), LDA, ZERO, X( 1, I ), 1 )
217 CALL DGEMV( 'No transpose', M-I, I-1, -ONE, X( I+1, 1 ),
218 $ LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
219 CALL DSCAL( M-I, TAUP( I ), X( I+1, I ), 1 )
220 END IF
221 10 CONTINUE
222 ELSE
223 *
224 * Reduce to lower bidiagonal form
225 *
226 DO 20 I = 1, NB
227 *
228 * Update A(i,i:n)
229 *
230 CALL DGEMV( 'No transpose', N-I+1, I-1, -ONE, Y( I, 1 ),
231 $ LDY, A( I, 1 ), LDA, ONE, A( I, I ), LDA )
232 CALL DGEMV( 'Transpose', I-1, N-I+1, -ONE, A( 1, I ), LDA,
233 $ X( I, 1 ), LDX, ONE, A( I, I ), LDA )
234 *
235 * Generate reflection P(i) to annihilate A(i,i+1:n)
236 *
237 CALL DLARFG( N-I+1, A( I, I ), A( I, MIN( I+1, N ) ), LDA,
238 $ TAUP( I ) )
239 D( I ) = A( I, I )
240 IF( I.LT.M ) THEN
241 A( I, I ) = ONE
242 *
243 * Compute X(i+1:m,i)
244 *
245 CALL DGEMV( 'No transpose', M-I, N-I+1, ONE, A( I+1, I ),
246 $ LDA, A( I, I ), LDA, ZERO, X( I+1, I ), 1 )
247 CALL DGEMV( 'Transpose', N-I+1, I-1, ONE, Y( I, 1 ), LDY,
248 $ A( I, I ), LDA, ZERO, X( 1, I ), 1 )
249 CALL DGEMV( 'No transpose', M-I, I-1, -ONE, A( I+1, 1 ),
250 $ LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
251 CALL DGEMV( 'No transpose', I-1, N-I+1, ONE, A( 1, I ),
252 $ LDA, A( I, I ), LDA, ZERO, X( 1, I ), 1 )
253 CALL DGEMV( 'No transpose', M-I, I-1, -ONE, X( I+1, 1 ),
254 $ LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
255 CALL DSCAL( M-I, TAUP( I ), X( I+1, I ), 1 )
256 *
257 * Update A(i+1:m,i)
258 *
259 CALL DGEMV( 'No transpose', M-I, I-1, -ONE, A( I+1, 1 ),
260 $ LDA, Y( I, 1 ), LDY, ONE, A( I+1, I ), 1 )
261 CALL DGEMV( 'No transpose', M-I, I, -ONE, X( I+1, 1 ),
262 $ LDX, A( 1, I ), 1, ONE, A( I+1, I ), 1 )
263 *
264 * Generate reflection Q(i) to annihilate A(i+2:m,i)
265 *
266 CALL DLARFG( M-I, A( I+1, I ), A( MIN( I+2, M ), I ), 1,
267 $ TAUQ( I ) )
268 E( I ) = A( I+1, I )
269 A( I+1, I ) = ONE
270 *
271 * Compute Y(i+1:n,i)
272 *
273 CALL DGEMV( 'Transpose', M-I, N-I, ONE, A( I+1, I+1 ),
274 $ LDA, A( I+1, I ), 1, ZERO, Y( I+1, I ), 1 )
275 CALL DGEMV( 'Transpose', M-I, I-1, ONE, A( I+1, 1 ), LDA,
276 $ A( I+1, I ), 1, ZERO, Y( 1, I ), 1 )
277 CALL DGEMV( 'No transpose', N-I, I-1, -ONE, Y( I+1, 1 ),
278 $ LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
279 CALL DGEMV( 'Transpose', M-I, I, ONE, X( I+1, 1 ), LDX,
280 $ A( I+1, I ), 1, ZERO, Y( 1, I ), 1 )
281 CALL DGEMV( 'Transpose', I, N-I, -ONE, A( 1, I+1 ), LDA,
282 $ Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
283 CALL DSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 )
284 END IF
285 20 CONTINUE
286 END IF
287 RETURN
288 *
289 * End of DLABRD
290 *
291 END