1 SUBROUTINE ZLABRD( 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 D( * ), E( * )
14 COMPLEX*16 A( LDA, * ), TAUP( * ), TAUQ( * ), X( LDX, * ),
15 $ Y( LDY, * )
16 * ..
17 *
18 * Purpose
19 * =======
20 *
21 * ZLABRD reduces the first NB rows and columns of a complex general
22 * m by n matrix A to upper or lower real bidiagonal form by a unitary
23 * transformation Q**H * A * P, and returns the matrices X and Y which
24 * are needed to apply the transformation to the unreduced part of A.
25 *
26 * If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
27 * bidiagonal form.
28 *
29 * This is an auxiliary routine called by ZGEBRD
30 *
31 * Arguments
32 * =========
33 *
34 * M (input) INTEGER
35 * The number of rows in the matrix A.
36 *
37 * N (input) INTEGER
38 * The number of columns in the matrix A.
39 *
40 * NB (input) INTEGER
41 * The number of leading rows and columns of A to be reduced.
42 *
43 * A (input/output) COMPLEX*16 array, dimension (LDA,N)
44 * On entry, the m by n general matrix to be reduced.
45 * On exit, the first NB rows and columns of the matrix are
46 * overwritten; the rest of the array is unchanged.
47 * If m >= n, elements on and below the diagonal in the first NB
48 * columns, with the array TAUQ, represent the unitary
49 * matrix Q as a product of elementary reflectors; and
50 * elements above the diagonal in the first NB rows, with the
51 * array TAUP, represent the unitary matrix P as a product
52 * of elementary reflectors.
53 * If m < n, elements below the diagonal in the first NB
54 * columns, with the array TAUQ, represent the unitary
55 * matrix Q as a product of elementary reflectors, and
56 * elements on and above the diagonal in the first NB rows,
57 * with the array TAUP, represent the unitary matrix P as
58 * a product of elementary reflectors.
59 * See Further Details.
60 *
61 * LDA (input) INTEGER
62 * The leading dimension of the array A. LDA >= max(1,M).
63 *
64 * D (output) DOUBLE PRECISION array, dimension (NB)
65 * The diagonal elements of the first NB rows and columns of
66 * the reduced matrix. D(i) = A(i,i).
67 *
68 * E (output) DOUBLE PRECISION array, dimension (NB)
69 * The off-diagonal elements of the first NB rows and columns of
70 * the reduced matrix.
71 *
72 * TAUQ (output) COMPLEX*16 array dimension (NB)
73 * The scalar factors of the elementary reflectors which
74 * represent the unitary matrix Q. See Further Details.
75 *
76 * TAUP (output) COMPLEX*16 array, dimension (NB)
77 * The scalar factors of the elementary reflectors which
78 * represent the unitary matrix P. See Further Details.
79 *
80 * X (output) COMPLEX*16 array, dimension (LDX,NB)
81 * The m-by-nb matrix X required to update the unreduced part
82 * of A.
83 *
84 * LDX (input) INTEGER
85 * The leading dimension of the array X. LDX >= max(1,M).
86 *
87 * Y (output) COMPLEX*16 array, dimension (LDY,NB)
88 * The n-by-nb matrix Y required to update the unreduced part
89 * of A.
90 *
91 * LDY (input) INTEGER
92 * The leading dimension of the array Y. LDY >= max(1,N).
93 *
94 * Further Details
95 * ===============
96 *
97 * The matrices Q and P are represented as products of elementary
98 * reflectors:
99 *
100 * Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb)
101 *
102 * Each H(i) and G(i) has the form:
103 *
104 * H(i) = I - tauq * v * v**H and G(i) = I - taup * u * u**H
105 *
106 * where tauq and taup are complex scalars, and v and u are complex
107 * vectors.
108 *
109 * If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
110 * A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in
111 * A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
112 *
113 * If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
114 * A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
115 * A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
116 *
117 * The elements of the vectors v and u together form the m-by-nb matrix
118 * V and the nb-by-n matrix U**H which are needed, with X and Y, to apply
119 * the transformation to the unreduced part of the matrix, using a block
120 * update of the form: A := A - V*Y**H - X*U**H.
121 *
122 * The contents of A on exit are illustrated by the following examples
123 * with nb = 2:
124 *
125 * m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
126 *
127 * ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 )
128 * ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 )
129 * ( v1 v2 a a a ) ( v1 1 a a a a )
130 * ( v1 v2 a a a ) ( v1 v2 a a a a )
131 * ( v1 v2 a a a ) ( v1 v2 a a a a )
132 * ( v1 v2 a a a )
133 *
134 * where a denotes an element of the original matrix which is unchanged,
135 * vi denotes an element of the vector defining H(i), and ui an element
136 * of the vector defining G(i).
137 *
138 * =====================================================================
139 *
140 * .. Parameters ..
141 COMPLEX*16 ZERO, ONE
142 PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ),
143 $ ONE = ( 1.0D+0, 0.0D+0 ) )
144 * ..
145 * .. Local Scalars ..
146 INTEGER I
147 COMPLEX*16 ALPHA
148 * ..
149 * .. External Subroutines ..
150 EXTERNAL ZGEMV, ZLACGV, ZLARFG, ZSCAL
151 * ..
152 * .. Intrinsic Functions ..
153 INTRINSIC MIN
154 * ..
155 * .. Executable Statements ..
156 *
157 * Quick return if possible
158 *
159 IF( M.LE.0 .OR. N.LE.0 )
160 $ RETURN
161 *
162 IF( M.GE.N ) THEN
163 *
164 * Reduce to upper bidiagonal form
165 *
166 DO 10 I = 1, NB
167 *
168 * Update A(i:m,i)
169 *
170 CALL ZLACGV( I-1, Y( I, 1 ), LDY )
171 CALL ZGEMV( 'No transpose', M-I+1, I-1, -ONE, A( I, 1 ),
172 $ LDA, Y( I, 1 ), LDY, ONE, A( I, I ), 1 )
173 CALL ZLACGV( I-1, Y( I, 1 ), LDY )
174 CALL ZGEMV( 'No transpose', M-I+1, I-1, -ONE, X( I, 1 ),
175 $ LDX, A( 1, I ), 1, ONE, A( I, I ), 1 )
176 *
177 * Generate reflection Q(i) to annihilate A(i+1:m,i)
178 *
179 ALPHA = A( I, I )
180 CALL ZLARFG( M-I+1, ALPHA, A( MIN( I+1, M ), I ), 1,
181 $ TAUQ( I ) )
182 D( I ) = ALPHA
183 IF( I.LT.N ) THEN
184 A( I, I ) = ONE
185 *
186 * Compute Y(i+1:n,i)
187 *
188 CALL ZGEMV( 'Conjugate transpose', M-I+1, N-I, ONE,
189 $ A( I, I+1 ), LDA, A( I, I ), 1, ZERO,
190 $ Y( I+1, I ), 1 )
191 CALL ZGEMV( 'Conjugate transpose', M-I+1, I-1, ONE,
192 $ A( I, 1 ), LDA, A( I, I ), 1, ZERO,
193 $ Y( 1, I ), 1 )
194 CALL ZGEMV( 'No transpose', N-I, I-1, -ONE, Y( I+1, 1 ),
195 $ LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
196 CALL ZGEMV( 'Conjugate transpose', M-I+1, I-1, ONE,
197 $ X( I, 1 ), LDX, A( I, I ), 1, ZERO,
198 $ Y( 1, I ), 1 )
199 CALL ZGEMV( 'Conjugate transpose', I-1, N-I, -ONE,
200 $ A( 1, I+1 ), LDA, Y( 1, I ), 1, ONE,
201 $ Y( I+1, I ), 1 )
202 CALL ZSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 )
203 *
204 * Update A(i,i+1:n)
205 *
206 CALL ZLACGV( N-I, A( I, I+1 ), LDA )
207 CALL ZLACGV( I, A( I, 1 ), LDA )
208 CALL ZGEMV( 'No transpose', N-I, I, -ONE, Y( I+1, 1 ),
209 $ LDY, A( I, 1 ), LDA, ONE, A( I, I+1 ), LDA )
210 CALL ZLACGV( I, A( I, 1 ), LDA )
211 CALL ZLACGV( I-1, X( I, 1 ), LDX )
212 CALL ZGEMV( 'Conjugate transpose', I-1, N-I, -ONE,
213 $ A( 1, I+1 ), LDA, X( I, 1 ), LDX, ONE,
214 $ A( I, I+1 ), LDA )
215 CALL ZLACGV( I-1, X( I, 1 ), LDX )
216 *
217 * Generate reflection P(i) to annihilate A(i,i+2:n)
218 *
219 ALPHA = A( I, I+1 )
220 CALL ZLARFG( N-I, ALPHA, A( I, MIN( I+2, N ) ), LDA,
221 $ TAUP( I ) )
222 E( I ) = ALPHA
223 A( I, I+1 ) = ONE
224 *
225 * Compute X(i+1:m,i)
226 *
227 CALL ZGEMV( 'No transpose', M-I, N-I, ONE, A( I+1, I+1 ),
228 $ LDA, A( I, I+1 ), LDA, ZERO, X( I+1, I ), 1 )
229 CALL ZGEMV( 'Conjugate transpose', N-I, I, ONE,
230 $ Y( I+1, 1 ), LDY, A( I, I+1 ), LDA, ZERO,
231 $ X( 1, I ), 1 )
232 CALL ZGEMV( 'No transpose', M-I, I, -ONE, A( I+1, 1 ),
233 $ LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
234 CALL ZGEMV( 'No transpose', I-1, N-I, ONE, A( 1, I+1 ),
235 $ LDA, A( I, I+1 ), LDA, ZERO, X( 1, I ), 1 )
236 CALL ZGEMV( 'No transpose', M-I, I-1, -ONE, X( I+1, 1 ),
237 $ LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
238 CALL ZSCAL( M-I, TAUP( I ), X( I+1, I ), 1 )
239 CALL ZLACGV( N-I, A( I, I+1 ), LDA )
240 END IF
241 10 CONTINUE
242 ELSE
243 *
244 * Reduce to lower bidiagonal form
245 *
246 DO 20 I = 1, NB
247 *
248 * Update A(i,i:n)
249 *
250 CALL ZLACGV( N-I+1, A( I, I ), LDA )
251 CALL ZLACGV( I-1, A( I, 1 ), LDA )
252 CALL ZGEMV( 'No transpose', N-I+1, I-1, -ONE, Y( I, 1 ),
253 $ LDY, A( I, 1 ), LDA, ONE, A( I, I ), LDA )
254 CALL ZLACGV( I-1, A( I, 1 ), LDA )
255 CALL ZLACGV( I-1, X( I, 1 ), LDX )
256 CALL ZGEMV( 'Conjugate transpose', I-1, N-I+1, -ONE,
257 $ A( 1, I ), LDA, X( I, 1 ), LDX, ONE, A( I, I ),
258 $ LDA )
259 CALL ZLACGV( I-1, X( I, 1 ), LDX )
260 *
261 * Generate reflection P(i) to annihilate A(i,i+1:n)
262 *
263 ALPHA = A( I, I )
264 CALL ZLARFG( N-I+1, ALPHA, A( I, MIN( I+1, N ) ), LDA,
265 $ TAUP( I ) )
266 D( I ) = ALPHA
267 IF( I.LT.M ) THEN
268 A( I, I ) = ONE
269 *
270 * Compute X(i+1:m,i)
271 *
272 CALL ZGEMV( 'No transpose', M-I, N-I+1, ONE, A( I+1, I ),
273 $ LDA, A( I, I ), LDA, ZERO, X( I+1, I ), 1 )
274 CALL ZGEMV( 'Conjugate transpose', N-I+1, I-1, ONE,
275 $ Y( I, 1 ), LDY, A( I, I ), LDA, ZERO,
276 $ X( 1, I ), 1 )
277 CALL ZGEMV( 'No transpose', M-I, I-1, -ONE, A( I+1, 1 ),
278 $ LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
279 CALL ZGEMV( 'No transpose', I-1, N-I+1, ONE, A( 1, I ),
280 $ LDA, A( I, I ), LDA, ZERO, X( 1, I ), 1 )
281 CALL ZGEMV( 'No transpose', M-I, I-1, -ONE, X( I+1, 1 ),
282 $ LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
283 CALL ZSCAL( M-I, TAUP( I ), X( I+1, I ), 1 )
284 CALL ZLACGV( N-I+1, A( I, I ), LDA )
285 *
286 * Update A(i+1:m,i)
287 *
288 CALL ZLACGV( I-1, Y( I, 1 ), LDY )
289 CALL ZGEMV( 'No transpose', M-I, I-1, -ONE, A( I+1, 1 ),
290 $ LDA, Y( I, 1 ), LDY, ONE, A( I+1, I ), 1 )
291 CALL ZLACGV( I-1, Y( I, 1 ), LDY )
292 CALL ZGEMV( 'No transpose', M-I, I, -ONE, X( I+1, 1 ),
293 $ LDX, A( 1, I ), 1, ONE, A( I+1, I ), 1 )
294 *
295 * Generate reflection Q(i) to annihilate A(i+2:m,i)
296 *
297 ALPHA = A( I+1, I )
298 CALL ZLARFG( M-I, ALPHA, A( MIN( I+2, M ), I ), 1,
299 $ TAUQ( I ) )
300 E( I ) = ALPHA
301 A( I+1, I ) = ONE
302 *
303 * Compute Y(i+1:n,i)
304 *
305 CALL ZGEMV( 'Conjugate transpose', M-I, N-I, ONE,
306 $ A( I+1, I+1 ), LDA, A( I+1, I ), 1, ZERO,
307 $ Y( I+1, I ), 1 )
308 CALL ZGEMV( 'Conjugate transpose', M-I, I-1, ONE,
309 $ A( I+1, 1 ), LDA, A( I+1, I ), 1, ZERO,
310 $ Y( 1, I ), 1 )
311 CALL ZGEMV( 'No transpose', N-I, I-1, -ONE, Y( I+1, 1 ),
312 $ LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
313 CALL ZGEMV( 'Conjugate transpose', M-I, I, ONE,
314 $ X( I+1, 1 ), LDX, A( I+1, I ), 1, ZERO,
315 $ Y( 1, I ), 1 )
316 CALL ZGEMV( 'Conjugate transpose', I, N-I, -ONE,
317 $ A( 1, I+1 ), LDA, Y( 1, I ), 1, ONE,
318 $ Y( I+1, I ), 1 )
319 CALL ZSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 )
320 ELSE
321 CALL ZLACGV( N-I+1, A( I, I ), LDA )
322 END IF
323 20 CONTINUE
324 END IF
325 RETURN
326 *
327 * End of ZLABRD
328 *
329 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 D( * ), E( * )
14 COMPLEX*16 A( LDA, * ), TAUP( * ), TAUQ( * ), X( LDX, * ),
15 $ Y( LDY, * )
16 * ..
17 *
18 * Purpose
19 * =======
20 *
21 * ZLABRD reduces the first NB rows and columns of a complex general
22 * m by n matrix A to upper or lower real bidiagonal form by a unitary
23 * transformation Q**H * A * P, and returns the matrices X and Y which
24 * are needed to apply the transformation to the unreduced part of A.
25 *
26 * If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
27 * bidiagonal form.
28 *
29 * This is an auxiliary routine called by ZGEBRD
30 *
31 * Arguments
32 * =========
33 *
34 * M (input) INTEGER
35 * The number of rows in the matrix A.
36 *
37 * N (input) INTEGER
38 * The number of columns in the matrix A.
39 *
40 * NB (input) INTEGER
41 * The number of leading rows and columns of A to be reduced.
42 *
43 * A (input/output) COMPLEX*16 array, dimension (LDA,N)
44 * On entry, the m by n general matrix to be reduced.
45 * On exit, the first NB rows and columns of the matrix are
46 * overwritten; the rest of the array is unchanged.
47 * If m >= n, elements on and below the diagonal in the first NB
48 * columns, with the array TAUQ, represent the unitary
49 * matrix Q as a product of elementary reflectors; and
50 * elements above the diagonal in the first NB rows, with the
51 * array TAUP, represent the unitary matrix P as a product
52 * of elementary reflectors.
53 * If m < n, elements below the diagonal in the first NB
54 * columns, with the array TAUQ, represent the unitary
55 * matrix Q as a product of elementary reflectors, and
56 * elements on and above the diagonal in the first NB rows,
57 * with the array TAUP, represent the unitary matrix P as
58 * a product of elementary reflectors.
59 * See Further Details.
60 *
61 * LDA (input) INTEGER
62 * The leading dimension of the array A. LDA >= max(1,M).
63 *
64 * D (output) DOUBLE PRECISION array, dimension (NB)
65 * The diagonal elements of the first NB rows and columns of
66 * the reduced matrix. D(i) = A(i,i).
67 *
68 * E (output) DOUBLE PRECISION array, dimension (NB)
69 * The off-diagonal elements of the first NB rows and columns of
70 * the reduced matrix.
71 *
72 * TAUQ (output) COMPLEX*16 array dimension (NB)
73 * The scalar factors of the elementary reflectors which
74 * represent the unitary matrix Q. See Further Details.
75 *
76 * TAUP (output) COMPLEX*16 array, dimension (NB)
77 * The scalar factors of the elementary reflectors which
78 * represent the unitary matrix P. See Further Details.
79 *
80 * X (output) COMPLEX*16 array, dimension (LDX,NB)
81 * The m-by-nb matrix X required to update the unreduced part
82 * of A.
83 *
84 * LDX (input) INTEGER
85 * The leading dimension of the array X. LDX >= max(1,M).
86 *
87 * Y (output) COMPLEX*16 array, dimension (LDY,NB)
88 * The n-by-nb matrix Y required to update the unreduced part
89 * of A.
90 *
91 * LDY (input) INTEGER
92 * The leading dimension of the array Y. LDY >= max(1,N).
93 *
94 * Further Details
95 * ===============
96 *
97 * The matrices Q and P are represented as products of elementary
98 * reflectors:
99 *
100 * Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb)
101 *
102 * Each H(i) and G(i) has the form:
103 *
104 * H(i) = I - tauq * v * v**H and G(i) = I - taup * u * u**H
105 *
106 * where tauq and taup are complex scalars, and v and u are complex
107 * vectors.
108 *
109 * If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
110 * A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in
111 * A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
112 *
113 * If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
114 * A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
115 * A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
116 *
117 * The elements of the vectors v and u together form the m-by-nb matrix
118 * V and the nb-by-n matrix U**H which are needed, with X and Y, to apply
119 * the transformation to the unreduced part of the matrix, using a block
120 * update of the form: A := A - V*Y**H - X*U**H.
121 *
122 * The contents of A on exit are illustrated by the following examples
123 * with nb = 2:
124 *
125 * m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
126 *
127 * ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 )
128 * ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 )
129 * ( v1 v2 a a a ) ( v1 1 a a a a )
130 * ( v1 v2 a a a ) ( v1 v2 a a a a )
131 * ( v1 v2 a a a ) ( v1 v2 a a a a )
132 * ( v1 v2 a a a )
133 *
134 * where a denotes an element of the original matrix which is unchanged,
135 * vi denotes an element of the vector defining H(i), and ui an element
136 * of the vector defining G(i).
137 *
138 * =====================================================================
139 *
140 * .. Parameters ..
141 COMPLEX*16 ZERO, ONE
142 PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ),
143 $ ONE = ( 1.0D+0, 0.0D+0 ) )
144 * ..
145 * .. Local Scalars ..
146 INTEGER I
147 COMPLEX*16 ALPHA
148 * ..
149 * .. External Subroutines ..
150 EXTERNAL ZGEMV, ZLACGV, ZLARFG, ZSCAL
151 * ..
152 * .. Intrinsic Functions ..
153 INTRINSIC MIN
154 * ..
155 * .. Executable Statements ..
156 *
157 * Quick return if possible
158 *
159 IF( M.LE.0 .OR. N.LE.0 )
160 $ RETURN
161 *
162 IF( M.GE.N ) THEN
163 *
164 * Reduce to upper bidiagonal form
165 *
166 DO 10 I = 1, NB
167 *
168 * Update A(i:m,i)
169 *
170 CALL ZLACGV( I-1, Y( I, 1 ), LDY )
171 CALL ZGEMV( 'No transpose', M-I+1, I-1, -ONE, A( I, 1 ),
172 $ LDA, Y( I, 1 ), LDY, ONE, A( I, I ), 1 )
173 CALL ZLACGV( I-1, Y( I, 1 ), LDY )
174 CALL ZGEMV( 'No transpose', M-I+1, I-1, -ONE, X( I, 1 ),
175 $ LDX, A( 1, I ), 1, ONE, A( I, I ), 1 )
176 *
177 * Generate reflection Q(i) to annihilate A(i+1:m,i)
178 *
179 ALPHA = A( I, I )
180 CALL ZLARFG( M-I+1, ALPHA, A( MIN( I+1, M ), I ), 1,
181 $ TAUQ( I ) )
182 D( I ) = ALPHA
183 IF( I.LT.N ) THEN
184 A( I, I ) = ONE
185 *
186 * Compute Y(i+1:n,i)
187 *
188 CALL ZGEMV( 'Conjugate transpose', M-I+1, N-I, ONE,
189 $ A( I, I+1 ), LDA, A( I, I ), 1, ZERO,
190 $ Y( I+1, I ), 1 )
191 CALL ZGEMV( 'Conjugate transpose', M-I+1, I-1, ONE,
192 $ A( I, 1 ), LDA, A( I, I ), 1, ZERO,
193 $ Y( 1, I ), 1 )
194 CALL ZGEMV( 'No transpose', N-I, I-1, -ONE, Y( I+1, 1 ),
195 $ LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
196 CALL ZGEMV( 'Conjugate transpose', M-I+1, I-1, ONE,
197 $ X( I, 1 ), LDX, A( I, I ), 1, ZERO,
198 $ Y( 1, I ), 1 )
199 CALL ZGEMV( 'Conjugate transpose', I-1, N-I, -ONE,
200 $ A( 1, I+1 ), LDA, Y( 1, I ), 1, ONE,
201 $ Y( I+1, I ), 1 )
202 CALL ZSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 )
203 *
204 * Update A(i,i+1:n)
205 *
206 CALL ZLACGV( N-I, A( I, I+1 ), LDA )
207 CALL ZLACGV( I, A( I, 1 ), LDA )
208 CALL ZGEMV( 'No transpose', N-I, I, -ONE, Y( I+1, 1 ),
209 $ LDY, A( I, 1 ), LDA, ONE, A( I, I+1 ), LDA )
210 CALL ZLACGV( I, A( I, 1 ), LDA )
211 CALL ZLACGV( I-1, X( I, 1 ), LDX )
212 CALL ZGEMV( 'Conjugate transpose', I-1, N-I, -ONE,
213 $ A( 1, I+1 ), LDA, X( I, 1 ), LDX, ONE,
214 $ A( I, I+1 ), LDA )
215 CALL ZLACGV( I-1, X( I, 1 ), LDX )
216 *
217 * Generate reflection P(i) to annihilate A(i,i+2:n)
218 *
219 ALPHA = A( I, I+1 )
220 CALL ZLARFG( N-I, ALPHA, A( I, MIN( I+2, N ) ), LDA,
221 $ TAUP( I ) )
222 E( I ) = ALPHA
223 A( I, I+1 ) = ONE
224 *
225 * Compute X(i+1:m,i)
226 *
227 CALL ZGEMV( 'No transpose', M-I, N-I, ONE, A( I+1, I+1 ),
228 $ LDA, A( I, I+1 ), LDA, ZERO, X( I+1, I ), 1 )
229 CALL ZGEMV( 'Conjugate transpose', N-I, I, ONE,
230 $ Y( I+1, 1 ), LDY, A( I, I+1 ), LDA, ZERO,
231 $ X( 1, I ), 1 )
232 CALL ZGEMV( 'No transpose', M-I, I, -ONE, A( I+1, 1 ),
233 $ LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
234 CALL ZGEMV( 'No transpose', I-1, N-I, ONE, A( 1, I+1 ),
235 $ LDA, A( I, I+1 ), LDA, ZERO, X( 1, I ), 1 )
236 CALL ZGEMV( 'No transpose', M-I, I-1, -ONE, X( I+1, 1 ),
237 $ LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
238 CALL ZSCAL( M-I, TAUP( I ), X( I+1, I ), 1 )
239 CALL ZLACGV( N-I, A( I, I+1 ), LDA )
240 END IF
241 10 CONTINUE
242 ELSE
243 *
244 * Reduce to lower bidiagonal form
245 *
246 DO 20 I = 1, NB
247 *
248 * Update A(i,i:n)
249 *
250 CALL ZLACGV( N-I+1, A( I, I ), LDA )
251 CALL ZLACGV( I-1, A( I, 1 ), LDA )
252 CALL ZGEMV( 'No transpose', N-I+1, I-1, -ONE, Y( I, 1 ),
253 $ LDY, A( I, 1 ), LDA, ONE, A( I, I ), LDA )
254 CALL ZLACGV( I-1, A( I, 1 ), LDA )
255 CALL ZLACGV( I-1, X( I, 1 ), LDX )
256 CALL ZGEMV( 'Conjugate transpose', I-1, N-I+1, -ONE,
257 $ A( 1, I ), LDA, X( I, 1 ), LDX, ONE, A( I, I ),
258 $ LDA )
259 CALL ZLACGV( I-1, X( I, 1 ), LDX )
260 *
261 * Generate reflection P(i) to annihilate A(i,i+1:n)
262 *
263 ALPHA = A( I, I )
264 CALL ZLARFG( N-I+1, ALPHA, A( I, MIN( I+1, N ) ), LDA,
265 $ TAUP( I ) )
266 D( I ) = ALPHA
267 IF( I.LT.M ) THEN
268 A( I, I ) = ONE
269 *
270 * Compute X(i+1:m,i)
271 *
272 CALL ZGEMV( 'No transpose', M-I, N-I+1, ONE, A( I+1, I ),
273 $ LDA, A( I, I ), LDA, ZERO, X( I+1, I ), 1 )
274 CALL ZGEMV( 'Conjugate transpose', N-I+1, I-1, ONE,
275 $ Y( I, 1 ), LDY, A( I, I ), LDA, ZERO,
276 $ X( 1, I ), 1 )
277 CALL ZGEMV( 'No transpose', M-I, I-1, -ONE, A( I+1, 1 ),
278 $ LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
279 CALL ZGEMV( 'No transpose', I-1, N-I+1, ONE, A( 1, I ),
280 $ LDA, A( I, I ), LDA, ZERO, X( 1, I ), 1 )
281 CALL ZGEMV( 'No transpose', M-I, I-1, -ONE, X( I+1, 1 ),
282 $ LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
283 CALL ZSCAL( M-I, TAUP( I ), X( I+1, I ), 1 )
284 CALL ZLACGV( N-I+1, A( I, I ), LDA )
285 *
286 * Update A(i+1:m,i)
287 *
288 CALL ZLACGV( I-1, Y( I, 1 ), LDY )
289 CALL ZGEMV( 'No transpose', M-I, I-1, -ONE, A( I+1, 1 ),
290 $ LDA, Y( I, 1 ), LDY, ONE, A( I+1, I ), 1 )
291 CALL ZLACGV( I-1, Y( I, 1 ), LDY )
292 CALL ZGEMV( 'No transpose', M-I, I, -ONE, X( I+1, 1 ),
293 $ LDX, A( 1, I ), 1, ONE, A( I+1, I ), 1 )
294 *
295 * Generate reflection Q(i) to annihilate A(i+2:m,i)
296 *
297 ALPHA = A( I+1, I )
298 CALL ZLARFG( M-I, ALPHA, A( MIN( I+2, M ), I ), 1,
299 $ TAUQ( I ) )
300 E( I ) = ALPHA
301 A( I+1, I ) = ONE
302 *
303 * Compute Y(i+1:n,i)
304 *
305 CALL ZGEMV( 'Conjugate transpose', M-I, N-I, ONE,
306 $ A( I+1, I+1 ), LDA, A( I+1, I ), 1, ZERO,
307 $ Y( I+1, I ), 1 )
308 CALL ZGEMV( 'Conjugate transpose', M-I, I-1, ONE,
309 $ A( I+1, 1 ), LDA, A( I+1, I ), 1, ZERO,
310 $ Y( 1, I ), 1 )
311 CALL ZGEMV( 'No transpose', N-I, I-1, -ONE, Y( I+1, 1 ),
312 $ LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
313 CALL ZGEMV( 'Conjugate transpose', M-I, I, ONE,
314 $ X( I+1, 1 ), LDX, A( I+1, I ), 1, ZERO,
315 $ Y( 1, I ), 1 )
316 CALL ZGEMV( 'Conjugate transpose', I, N-I, -ONE,
317 $ A( 1, I+1 ), LDA, Y( 1, I ), 1, ONE,
318 $ Y( I+1, I ), 1 )
319 CALL ZSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 )
320 ELSE
321 CALL ZLACGV( N-I+1, A( I, I ), LDA )
322 END IF
323 20 CONTINUE
324 END IF
325 RETURN
326 *
327 * End of ZLABRD
328 *
329 END