|
2329
|
1 SUBROUTINE ZGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, |
|
|
2 $ INFO ) |
|
|
3 * |
|
7034
|
4 * -- LAPACK routine (version 3.1) -- |
|
|
5 * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. |
|
|
6 * November 2006 |
|
2329
|
7 * |
|
|
8 * .. Scalar Arguments .. |
|
|
9 INTEGER INFO, LDA, LWORK, M, N |
|
|
10 * .. |
|
|
11 * .. Array Arguments .. |
|
|
12 DOUBLE PRECISION D( * ), E( * ) |
|
3333
|
13 COMPLEX*16 A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * ) |
|
2329
|
14 * .. |
|
|
15 * |
|
|
16 * Purpose |
|
|
17 * ======= |
|
|
18 * |
|
|
19 * ZGEBRD reduces a general complex M-by-N matrix A to upper or lower |
|
|
20 * bidiagonal form B by a unitary transformation: Q**H * 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) COMPLEX*16 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 unitary matrix Q as a product of elementary |
|
|
40 * reflectors, and the elements above the first superdiagonal, |
|
|
41 * with the array TAUP, represent the unitary 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 unitary matrix Q as a product of |
|
|
47 * elementary reflectors, and the elements above the diagonal, |
|
|
48 * with the array TAUP, represent the unitary 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) COMPLEX*16 array dimension (min(M,N)) |
|
|
65 * The scalar factors of the elementary reflectors which |
|
|
66 * represent the unitary matrix Q. See Further Details. |
|
|
67 * |
|
|
68 * TAUP (output) COMPLEX*16 array, dimension (min(M,N)) |
|
|
69 * The scalar factors of the elementary reflectors which |
|
|
70 * represent the unitary matrix P. See Further Details. |
|
|
71 * |
|
7034
|
72 * WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK)) |
|
2329
|
73 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. |
|
|
74 * |
|
|
75 * LWORK (input) INTEGER |
|
|
76 * The length of the array WORK. LWORK >= max(1,M,N). |
|
|
77 * For optimum performance LWORK >= (M+N)*NB, where NB |
|
|
78 * is the optimal blocksize. |
|
|
79 * |
|
3333
|
80 * If LWORK = -1, then a workspace query is assumed; the routine |
|
|
81 * only calculates the optimal size of the WORK array, returns |
|
|
82 * this value as the first entry of the WORK array, and no error |
|
|
83 * message related to LWORK is issued by XERBLA. |
|
|
84 * |
|
2329
|
85 * INFO (output) INTEGER |
|
|
86 * = 0: successful exit. |
|
|
87 * < 0: if INFO = -i, the i-th argument had an illegal value. |
|
|
88 * |
|
|
89 * Further Details |
|
|
90 * =============== |
|
|
91 * |
|
|
92 * The matrices Q and P are represented as products of elementary |
|
|
93 * reflectors: |
|
|
94 * |
|
|
95 * If m >= n, |
|
|
96 * |
|
|
97 * Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1) |
|
|
98 * |
|
|
99 * Each H(i) and G(i) has the form: |
|
|
100 * |
|
|
101 * H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' |
|
|
102 * |
|
|
103 * where tauq and taup are complex scalars, and v and u are complex |
|
|
104 * vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in |
|
|
105 * A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in |
|
|
106 * A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i). |
|
|
107 * |
|
|
108 * If m < n, |
|
|
109 * |
|
|
110 * Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m) |
|
|
111 * |
|
|
112 * Each H(i) and G(i) has the form: |
|
|
113 * |
|
|
114 * H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' |
|
|
115 * |
|
|
116 * where tauq and taup are complex scalars, and v and u are complex |
|
|
117 * vectors; v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in |
|
|
118 * A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in |
|
|
119 * A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i). |
|
|
120 * |
|
|
121 * The contents of A on exit are illustrated by the following examples: |
|
|
122 * |
|
|
123 * m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): |
|
|
124 * |
|
|
125 * ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 ) |
|
|
126 * ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 ) |
|
|
127 * ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 ) |
|
|
128 * ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 ) |
|
|
129 * ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 ) |
|
|
130 * ( v1 v2 v3 v4 v5 ) |
|
|
131 * |
|
|
132 * where d and e denote diagonal and off-diagonal elements of B, vi |
|
|
133 * denotes an element of the vector defining H(i), and ui an element of |
|
|
134 * the vector defining G(i). |
|
|
135 * |
|
|
136 * ===================================================================== |
|
|
137 * |
|
|
138 * .. Parameters .. |
|
|
139 COMPLEX*16 ONE |
|
|
140 PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) ) |
|
|
141 * .. |
|
|
142 * .. Local Scalars .. |
|
3333
|
143 LOGICAL LQUERY |
|
|
144 INTEGER I, IINFO, J, LDWRKX, LDWRKY, LWKOPT, MINMN, NB, |
|
|
145 $ NBMIN, NX |
|
2329
|
146 DOUBLE PRECISION WS |
|
|
147 * .. |
|
|
148 * .. External Subroutines .. |
|
|
149 EXTERNAL XERBLA, ZGEBD2, ZGEMM, ZLABRD |
|
|
150 * .. |
|
|
151 * .. Intrinsic Functions .. |
|
3333
|
152 INTRINSIC DBLE, MAX, MIN |
|
2329
|
153 * .. |
|
|
154 * .. External Functions .. |
|
|
155 INTEGER ILAENV |
|
|
156 EXTERNAL ILAENV |
|
|
157 * .. |
|
|
158 * .. Executable Statements .. |
|
|
159 * |
|
|
160 * Test the input parameters |
|
|
161 * |
|
|
162 INFO = 0 |
|
3333
|
163 NB = MAX( 1, ILAENV( 1, 'ZGEBRD', ' ', M, N, -1, -1 ) ) |
|
|
164 LWKOPT = ( M+N )*NB |
|
|
165 WORK( 1 ) = DBLE( LWKOPT ) |
|
|
166 LQUERY = ( LWORK.EQ.-1 ) |
|
2329
|
167 IF( M.LT.0 ) THEN |
|
|
168 INFO = -1 |
|
|
169 ELSE IF( N.LT.0 ) THEN |
|
|
170 INFO = -2 |
|
|
171 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN |
|
|
172 INFO = -4 |
|
3333
|
173 ELSE IF( LWORK.LT.MAX( 1, M, N ) .AND. .NOT.LQUERY ) THEN |
|
2329
|
174 INFO = -10 |
|
|
175 END IF |
|
|
176 IF( INFO.LT.0 ) THEN |
|
|
177 CALL XERBLA( 'ZGEBRD', -INFO ) |
|
|
178 RETURN |
|
3333
|
179 ELSE IF( LQUERY ) THEN |
|
|
180 RETURN |
|
2329
|
181 END IF |
|
|
182 * |
|
|
183 * Quick return if possible |
|
|
184 * |
|
|
185 MINMN = MIN( M, N ) |
|
|
186 IF( MINMN.EQ.0 ) THEN |
|
|
187 WORK( 1 ) = 1 |
|
|
188 RETURN |
|
|
189 END IF |
|
|
190 * |
|
|
191 WS = MAX( M, N ) |
|
|
192 LDWRKX = M |
|
|
193 LDWRKY = N |
|
|
194 * |
|
3333
|
195 IF( NB.GT.1 .AND. NB.LT.MINMN ) THEN |
|
2329
|
196 * |
|
3333
|
197 * Set the crossover point NX. |
|
2329
|
198 * |
|
3333
|
199 NX = MAX( NB, ILAENV( 3, 'ZGEBRD', ' ', M, N, -1, -1 ) ) |
|
2329
|
200 * |
|
|
201 * Determine when to switch from blocked to unblocked code. |
|
|
202 * |
|
|
203 IF( NX.LT.MINMN ) THEN |
|
|
204 WS = ( M+N )*NB |
|
|
205 IF( LWORK.LT.WS ) THEN |
|
|
206 * |
|
|
207 * Not enough work space for the optimal NB, consider using |
|
|
208 * a smaller block size. |
|
|
209 * |
|
|
210 NBMIN = ILAENV( 2, 'ZGEBRD', ' ', M, N, -1, -1 ) |
|
|
211 IF( LWORK.GE.( M+N )*NBMIN ) THEN |
|
|
212 NB = LWORK / ( M+N ) |
|
|
213 ELSE |
|
|
214 NB = 1 |
|
|
215 NX = MINMN |
|
|
216 END IF |
|
|
217 END IF |
|
|
218 END IF |
|
|
219 ELSE |
|
|
220 NX = MINMN |
|
|
221 END IF |
|
|
222 * |
|
|
223 DO 30 I = 1, MINMN - NX, NB |
|
|
224 * |
|
|
225 * Reduce rows and columns i:i+ib-1 to bidiagonal form and return |
|
|
226 * the matrices X and Y which are needed to update the unreduced |
|
|
227 * part of the matrix |
|
|
228 * |
|
|
229 CALL ZLABRD( M-I+1, N-I+1, NB, A( I, I ), LDA, D( I ), E( I ), |
|
|
230 $ TAUQ( I ), TAUP( I ), WORK, LDWRKX, |
|
|
231 $ WORK( LDWRKX*NB+1 ), LDWRKY ) |
|
|
232 * |
|
|
233 * Update the trailing submatrix A(i+ib:m,i+ib:n), using |
|
|
234 * an update of the form A := A - V*Y' - X*U' |
|
|
235 * |
|
|
236 CALL ZGEMM( 'No transpose', 'Conjugate transpose', M-I-NB+1, |
|
|
237 $ N-I-NB+1, NB, -ONE, A( I+NB, I ), LDA, |
|
|
238 $ WORK( LDWRKX*NB+NB+1 ), LDWRKY, ONE, |
|
|
239 $ A( I+NB, I+NB ), LDA ) |
|
|
240 CALL ZGEMM( 'No transpose', 'No transpose', M-I-NB+1, N-I-NB+1, |
|
|
241 $ NB, -ONE, WORK( NB+1 ), LDWRKX, A( I, I+NB ), LDA, |
|
|
242 $ ONE, A( I+NB, I+NB ), LDA ) |
|
|
243 * |
|
|
244 * Copy diagonal and off-diagonal elements of B back into A |
|
|
245 * |
|
|
246 IF( M.GE.N ) THEN |
|
|
247 DO 10 J = I, I + NB - 1 |
|
|
248 A( J, J ) = D( J ) |
|
|
249 A( J, J+1 ) = E( J ) |
|
|
250 10 CONTINUE |
|
|
251 ELSE |
|
|
252 DO 20 J = I, I + NB - 1 |
|
|
253 A( J, J ) = D( J ) |
|
|
254 A( J+1, J ) = E( J ) |
|
|
255 20 CONTINUE |
|
|
256 END IF |
|
|
257 30 CONTINUE |
|
|
258 * |
|
|
259 * Use unblocked code to reduce the remainder of the matrix |
|
|
260 * |
|
|
261 CALL ZGEBD2( M-I+1, N-I+1, A( I, I ), LDA, D( I ), E( I ), |
|
|
262 $ TAUQ( I ), TAUP( I ), WORK, IINFO ) |
|
|
263 WORK( 1 ) = WS |
|
|
264 RETURN |
|
|
265 * |
|
|
266 * End of ZGEBRD |
|
|
267 * |
|
|
268 END |
