LAPACK 3.12.1
LAPACK: Linear Algebra PACKage
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zgebrd.f
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1*> \brief \b ZGEBRD
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> Download ZGEBRD + dependencies
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10*> [TGZ]</a>
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12*> [ZIP]</a>
13*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgebrd.f">
14*> [TXT]</a>
15*
16* Definition:
17* ===========
18*
19* SUBROUTINE ZGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK,
20* INFO )
21*
22* .. Scalar Arguments ..
23* INTEGER INFO, LDA, LWORK, M, N
24* ..
25* .. Array Arguments ..
26* DOUBLE PRECISION D( * ), E( * )
27* COMPLEX*16 A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * )
28* ..
29*
30*
31*> \par Purpose:
32* =============
33*>
34*> \verbatim
35*>
36*> ZGEBRD reduces a general complex M-by-N matrix A to upper or lower
37*> bidiagonal form B by a unitary transformation: Q**H * A * P = B.
38*>
39*> If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
40*> \endverbatim
41*
42* Arguments:
43* ==========
44*
45*> \param[in] M
46*> \verbatim
47*> M is INTEGER
48*> The number of rows in the matrix A. M >= 0.
49*> \endverbatim
50*>
51*> \param[in] N
52*> \verbatim
53*> N is INTEGER
54*> The number of columns in the matrix A. N >= 0.
55*> \endverbatim
56*>
57*> \param[in,out] A
58*> \verbatim
59*> A is COMPLEX*16 array, dimension (LDA,N)
60*> On entry, the M-by-N general matrix to be reduced.
61*> On exit,
62*> if m >= n, the diagonal and the first superdiagonal are
63*> overwritten with the upper bidiagonal matrix B; the
64*> elements below the diagonal, with the array TAUQ, represent
65*> the unitary matrix Q as a product of elementary
66*> reflectors, and the elements above the first superdiagonal,
67*> with the array TAUP, represent the unitary matrix P as
68*> a product of elementary reflectors;
69*> if m < n, the diagonal and the first subdiagonal are
70*> overwritten with the lower bidiagonal matrix B; the
71*> elements below the first subdiagonal, with the array TAUQ,
72*> represent the unitary matrix Q as a product of
73*> elementary reflectors, and the elements above the diagonal,
74*> with the array TAUP, represent the unitary matrix P as
75*> a product of elementary reflectors.
76*> See Further Details.
77*> \endverbatim
78*>
79*> \param[in] LDA
80*> \verbatim
81*> LDA is INTEGER
82*> The leading dimension of the array A. LDA >= max(1,M).
83*> \endverbatim
84*>
85*> \param[out] D
86*> \verbatim
87*> D is DOUBLE PRECISION array, dimension (min(M,N))
88*> The diagonal elements of the bidiagonal matrix B:
89*> D(i) = A(i,i).
90*> \endverbatim
91*>
92*> \param[out] E
93*> \verbatim
94*> E is DOUBLE PRECISION array, dimension (min(M,N)-1)
95*> The off-diagonal elements of the bidiagonal matrix B:
96*> if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
97*> if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
98*> \endverbatim
99*>
100*> \param[out] TAUQ
101*> \verbatim
102*> TAUQ is COMPLEX*16 array, dimension (min(M,N))
103*> The scalar factors of the elementary reflectors which
104*> represent the unitary matrix Q. See Further Details.
105*> \endverbatim
106*>
107*> \param[out] TAUP
108*> \verbatim
109*> TAUP is COMPLEX*16 array, dimension (min(M,N))
110*> The scalar factors of the elementary reflectors which
111*> represent the unitary matrix P. See Further Details.
112*> \endverbatim
113*>
114*> \param[out] WORK
115*> \verbatim
116*> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
117*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
118*> \endverbatim
119*>
120*> \param[in] LWORK
121*> \verbatim
122*> LWORK is INTEGER
123*> The length of the array WORK.
124*> LWORK >= 1, if MIN(M,N) = 0, and LWORK >= MAX(M,N), otherwise.
125*> For optimum performance LWORK >= (M+N)*NB, where NB
126*> is the optimal blocksize.
127*>
128*> If LWORK = -1, then a workspace query is assumed; the routine
129*> only calculates the optimal size of the WORK array, returns
130*> this value as the first entry of the WORK array, and no error
131*> message related to LWORK is issued by XERBLA.
132*> \endverbatim
133*>
134*> \param[out] INFO
135*> \verbatim
136*> INFO is INTEGER
137*> = 0: successful exit.
138*> < 0: if INFO = -i, the i-th argument had an illegal value.
139*> \endverbatim
140*
141* Authors:
142* ========
143*
144*> \author Univ. of Tennessee
145*> \author Univ. of California Berkeley
146*> \author Univ. of Colorado Denver
147*> \author NAG Ltd.
148*
149*> \ingroup gebrd
150*
151*> \par Further Details:
152* =====================
153*>
154*> \verbatim
155*>
156*> The matrices Q and P are represented as products of elementary
157*> reflectors:
158*>
159*> If m >= n,
160*>
161*> Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)
162*>
163*> Each H(i) and G(i) has the form:
164*>
165*> H(i) = I - tauq * v * v**H and G(i) = I - taup * u * u**H
166*>
167*> where tauq and taup are complex scalars, and v and u are complex
168*> vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in
169*> A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in
170*> A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
171*>
172*> If m < n,
173*>
174*> Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)
175*>
176*> Each H(i) and G(i) has the form:
177*>
178*> H(i) = I - tauq * v * v**H and G(i) = I - taup * u * u**H
179*>
180*> where tauq and taup are complex scalars, and v and u are complex
181*> vectors; v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in
182*> A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in
183*> A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
184*>
185*> The contents of A on exit are illustrated by the following examples:
186*>
187*> m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
188*>
189*> ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
190*> ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
191*> ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
192*> ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
193*> ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
194*> ( v1 v2 v3 v4 v5 )
195*>
196*> where d and e denote diagonal and off-diagonal elements of B, vi
197*> denotes an element of the vector defining H(i), and ui an element of
198*> the vector defining G(i).
199*> \endverbatim
200*>
201* =====================================================================
202 SUBROUTINE zgebrd( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK,
203 $ INFO )
204*
205* -- LAPACK computational routine --
206* -- LAPACK is a software package provided by Univ. of Tennessee, --
207* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
208*
209* .. Scalar Arguments ..
210 INTEGER INFO, LDA, LWORK, M, N
211* ..
212* .. Array Arguments ..
213 DOUBLE PRECISION D( * ), E( * )
214 COMPLEX*16 A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * )
215* ..
216*
217* =====================================================================
218*
219* .. Parameters ..
220 COMPLEX*16 ONE
221 parameter( one = ( 1.0d+0, 0.0d+0 ) )
222* ..
223* .. Local Scalars ..
224 LOGICAL LQUERY
225 INTEGER I, IINFO, J, LDWRKX, LDWRKY, LWKMIN, LWKOPT,
226 $ minmn, nb, nbmin, nx, ws
227* ..
228* .. External Subroutines ..
229 EXTERNAL xerbla, zgebd2, zgemm, zlabrd
230* ..
231* .. Intrinsic Functions ..
232 INTRINSIC dble, max, min
233* ..
234* .. External Functions ..
235 INTEGER ILAENV
236 EXTERNAL ilaenv
237* ..
238* .. Executable Statements ..
239*
240* Test the input parameters
241*
242 info = 0
243 minmn = min( m, n )
244 IF( minmn.EQ.0 ) THEN
245 lwkmin = 1
246 lwkopt = 1
247 ELSE
248 lwkmin = max( m, n )
249 nb = max( 1, ilaenv( 1, 'ZGEBRD', ' ', m, n, -1, -1 ) )
250 lwkopt = ( m+n )*nb
251 END IF
252 work( 1 ) = dble( lwkopt )
253*
254 lquery = ( lwork.EQ.-1 )
255 IF( m.LT.0 ) THEN
256 info = -1
257 ELSE IF( n.LT.0 ) THEN
258 info = -2
259 ELSE IF( lda.LT.max( 1, m ) ) THEN
260 info = -4
261 ELSE IF( lwork.LT.lwkmin .AND. .NOT.lquery ) THEN
262 info = -10
263 END IF
264 IF( info.LT.0 ) THEN
265 CALL xerbla( 'ZGEBRD', -info )
266 RETURN
267 ELSE IF( lquery ) THEN
268 RETURN
269 END IF
270*
271* Quick return if possible
272*
273 IF( minmn.EQ.0 ) THEN
274 work( 1 ) = 1
275 RETURN
276 END IF
277*
278 ws = max( m, n )
279 ldwrkx = m
280 ldwrky = n
281*
282 IF( nb.GT.1 .AND. nb.LT.minmn ) THEN
283*
284* Set the crossover point NX.
285*
286 nx = max( nb, ilaenv( 3, 'ZGEBRD', ' ', m, n, -1, -1 ) )
287*
288* Determine when to switch from blocked to unblocked code.
289*
290 IF( nx.LT.minmn ) THEN
291 ws = lwkopt
292 IF( lwork.LT.ws ) THEN
293*
294* Not enough work space for the optimal NB, consider using
295* a smaller block size.
296*
297 nbmin = ilaenv( 2, 'ZGEBRD', ' ', m, n, -1, -1 )
298 IF( lwork.GE.( m+n )*nbmin ) THEN
299 nb = lwork / ( m+n )
300 ELSE
301 nb = 1
302 nx = minmn
303 END IF
304 END IF
305 END IF
306 ELSE
307 nx = minmn
308 END IF
309*
310 DO 30 i = 1, minmn - nx, nb
311*
312* Reduce rows and columns i:i+ib-1 to bidiagonal form and return
313* the matrices X and Y which are needed to update the unreduced
314* part of the matrix
315*
316 CALL zlabrd( m-i+1, n-i+1, nb, a( i, i ), lda, d( i ),
317 $ e( i ),
318 $ tauq( i ), taup( i ), work, ldwrkx,
319 $ work( ldwrkx*nb+1 ), ldwrky )
320*
321* Update the trailing submatrix A(i+ib:m,i+ib:n), using
322* an update of the form A := A - V*Y**H - X*U**H
323*
324 CALL zgemm( 'No transpose', 'Conjugate transpose', m-i-nb+1,
325 $ n-i-nb+1, nb, -one, a( i+nb, i ), lda,
326 $ work( ldwrkx*nb+nb+1 ), ldwrky, one,
327 $ a( i+nb, i+nb ), lda )
328 CALL zgemm( 'No transpose', 'No transpose', m-i-nb+1,
329 $ n-i-nb+1,
330 $ nb, -one, work( nb+1 ), ldwrkx, a( i, i+nb ), lda,
331 $ one, a( i+nb, i+nb ), lda )
332*
333* Copy diagonal and off-diagonal elements of B back into A
334*
335 IF( m.GE.n ) THEN
336 DO 10 j = i, i + nb - 1
337 a( j, j ) = d( j )
338 a( j, j+1 ) = e( j )
339 10 CONTINUE
340 ELSE
341 DO 20 j = i, i + nb - 1
342 a( j, j ) = d( j )
343 a( j+1, j ) = e( j )
344 20 CONTINUE
345 END IF
346 30 CONTINUE
347*
348* Use unblocked code to reduce the remainder of the matrix
349*
350 CALL zgebd2( m-i+1, n-i+1, a( i, i ), lda, d( i ), e( i ),
351 $ tauq( i ), taup( i ), work, iinfo )
352 work( 1 ) = ws
353 RETURN
354*
355* End of ZGEBRD
356*
357 END
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine zgebd2(m, n, a, lda, d, e, tauq, taup, work, info)
ZGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.
Definition zgebd2.f:187
subroutine zgebrd(m, n, a, lda, d, e, tauq, taup, work, lwork, info)
ZGEBRD
Definition zgebrd.f:204
subroutine zgemm(transa, transb, m, n, k, alpha, a, lda, b, ldb, beta, c, ldc)
ZGEMM
Definition zgemm.f:188
subroutine zlabrd(m, n, nb, a, lda, d, e, tauq, taup, x, ldx, y, ldy)
ZLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.
Definition zlabrd.f:211