1*> \brief \b DGEBRD
2*
3*  =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6*            http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
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16*> \endhtmlonly
17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE DGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK,
22*                          INFO )
23*
24*       .. Scalar Arguments ..
25*       INTEGER            INFO, LDA, LWORK, M, N
26*       ..
27*       .. Array Arguments ..
28*       DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), TAUP( * ),
29*      $                   TAUQ( * ), WORK( * )
30*       ..
31*
32*
33*> \par Purpose:
34*  =============
35*>
36*> \verbatim
37*>
38*> DGEBRD reduces a general real M-by-N matrix A to upper or lower
39*> bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
40*>
41*> If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
42*> \endverbatim
43*
44*  Arguments:
45*  ==========
46*
47*> \param[in] M
48*> \verbatim
49*>          M is INTEGER
50*>          The number of rows in the matrix A.  M >= 0.
51*> \endverbatim
52*>
53*> \param[in] N
54*> \verbatim
55*>          N is INTEGER
56*>          The number of columns in the matrix A.  N >= 0.
57*> \endverbatim
58*>
59*> \param[in,out] A
60*> \verbatim
61*>          A is DOUBLE PRECISION array, dimension (LDA,N)
62*>          On entry, the M-by-N general matrix to be reduced.
63*>          On exit,
64*>          if m >= n, the diagonal and the first superdiagonal are
65*>            overwritten with the upper bidiagonal matrix B; the
66*>            elements below the diagonal, with the array TAUQ, represent
67*>            the orthogonal matrix Q as a product of elementary
68*>            reflectors, and the elements above the first superdiagonal,
69*>            with the array TAUP, represent the orthogonal matrix P as
70*>            a product of elementary reflectors;
71*>          if m < n, the diagonal and the first subdiagonal are
72*>            overwritten with the lower bidiagonal matrix B; the
73*>            elements below the first subdiagonal, with the array TAUQ,
74*>            represent the orthogonal matrix Q as a product of
75*>            elementary reflectors, and the elements above the diagonal,
76*>            with the array TAUP, represent the orthogonal matrix P as
77*>            a product of elementary reflectors.
78*>          See Further Details.
79*> \endverbatim
80*>
81*> \param[in] LDA
82*> \verbatim
83*>          LDA is INTEGER
84*>          The leading dimension of the array A.  LDA >= max(1,M).
85*> \endverbatim
86*>
87*> \param[out] D
88*> \verbatim
89*>          D is DOUBLE PRECISION array, dimension (min(M,N))
90*>          The diagonal elements of the bidiagonal matrix B:
91*>          D(i) = A(i,i).
92*> \endverbatim
93*>
94*> \param[out] E
95*> \verbatim
96*>          E is DOUBLE PRECISION array, dimension (min(M,N)-1)
97*>          The off-diagonal elements of the bidiagonal matrix B:
98*>          if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
99*>          if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
100*> \endverbatim
101*>
102*> \param[out] TAUQ
103*> \verbatim
104*>          TAUQ is DOUBLE PRECISION array dimension (min(M,N))
105*>          The scalar factors of the elementary reflectors which
106*>          represent the orthogonal matrix Q. See Further Details.
107*> \endverbatim
108*>
109*> \param[out] TAUP
110*> \verbatim
111*>          TAUP is DOUBLE PRECISION array, dimension (min(M,N))
112*>          The scalar factors of the elementary reflectors which
113*>          represent the orthogonal matrix P. See Further Details.
114*> \endverbatim
115*>
116*> \param[out] WORK
117*> \verbatim
118*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
119*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
120*> \endverbatim
121*>
122*> \param[in] LWORK
123*> \verbatim
124*>          LWORK is INTEGER
125*>          The length of the array WORK.  LWORK >= max(1,M,N).
126*>          For optimum performance LWORK >= (M+N)*NB, where NB
127*>          is the optimal blocksize.
128*>
129*>          If LWORK = -1, then a workspace query is assumed; the routine
130*>          only calculates the optimal size of the WORK array, returns
131*>          this value as the first entry of the WORK array, and no error
132*>          message related to LWORK is issued by XERBLA.
133*> \endverbatim
134*>
135*> \param[out] INFO
136*> \verbatim
137*>          INFO is INTEGER
138*>          = 0:  successful exit
139*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
140*> \endverbatim
141*
142*  Authors:
143*  ========
144*
145*> \author Univ. of Tennessee
146*> \author Univ. of California Berkeley
147*> \author Univ. of Colorado Denver
148*> \author NAG Ltd.
149*
150*> \date November 2011
151*
152*> \ingroup doubleGEcomputational
153*
154*> \par Further Details:
155*  =====================
156*>
157*> \verbatim
158*>
159*>  The matrices Q and P are represented as products of elementary
160*>  reflectors:
161*>
162*>  If m >= n,
163*>
164*>     Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)
165*>
166*>  Each H(i) and G(i) has the form:
167*>
168*>     H(i) = I - tauq * v * v**T  and G(i) = I - taup * u * u**T
169*>
170*>  where tauq and taup are real scalars, and v and u are real vectors;
171*>  v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
172*>  u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
173*>  tauq is stored in TAUQ(i) and taup in TAUP(i).
174*>
175*>  If m < n,
176*>
177*>     Q = H(1) H(2) . . . H(m-1)  and  P = G(1) G(2) . . . G(m)
178*>
179*>  Each H(i) and G(i) has the form:
180*>
181*>     H(i) = I - tauq * v * v**T  and G(i) = I - taup * u * u**T
182*>
183*>  where tauq and taup are real scalars, and v and u are real vectors;
184*>  v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
185*>  u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
186*>  tauq is stored in TAUQ(i) and taup in TAUP(i).
187*>
188*>  The contents of A on exit are illustrated by the following examples:
189*>
190*>  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):
191*>
192*>    (  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )
193*>    (  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )
194*>    (  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )
195*>    (  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )
196*>    (  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )
197*>    (  v1  v2  v3  v4  v5 )
198*>
199*>  where d and e denote diagonal and off-diagonal elements of B, vi
200*>  denotes an element of the vector defining H(i), and ui an element of
201*>  the vector defining G(i).
202*> \endverbatim
203*>
204*  =====================================================================
205      SUBROUTINE DGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK,
206     $                   INFO )
207*
208*  -- LAPACK computational routine (version 3.4.0) --
209*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
210*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
211*     November 2011
212*
213*     .. Scalar Arguments ..
214      INTEGER            INFO, LDA, LWORK, M, N
215*     ..
216*     .. Array Arguments ..
217      DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), TAUP( * ),
218     $                   TAUQ( * ), WORK( * )
219*     ..
220*
221*  =====================================================================
222*
223*     .. Parameters ..
224      DOUBLE PRECISION   ONE
225      PARAMETER          ( ONE = 1.0D+0 )
226*     ..
227*     .. Local Scalars ..
228      LOGICAL            LQUERY
229      INTEGER            I, IINFO, J, LDWRKX, LDWRKY, LWKOPT, MINMN, NB,
230     $                   NBMIN, NX
231      DOUBLE PRECISION   WS
232*     ..
233*     .. External Subroutines ..
234      EXTERNAL           DGEBD2, DGEMM, DLABRD, XERBLA
235*     ..
236*     .. Intrinsic Functions ..
237      INTRINSIC          DBLE, MAX, MIN
238*     ..
239*     .. External Functions ..
240      INTEGER            ILAENV
241      EXTERNAL           ILAENV
242*     ..
243*     .. Executable Statements ..
244*
245*     Test the input parameters
246*
247      INFO = 0
248      NB = MAX( 1, ILAENV( 1, 'DGEBRD', ' ', M, N, -1, -1 ) )
249      LWKOPT = ( M+N )*NB
250      WORK( 1 ) = DBLE( LWKOPT )
251      LQUERY = ( LWORK.EQ.-1 )
252      IF( M.LT.0 ) THEN
253         INFO = -1
254      ELSE IF( N.LT.0 ) THEN
255         INFO = -2
256      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
257         INFO = -4
258      ELSE IF( LWORK.LT.MAX( 1, M, N ) .AND. .NOT.LQUERY ) THEN
259         INFO = -10
260      END IF
261      IF( INFO.LT.0 ) THEN
262         CALL XERBLA( 'DGEBRD', -INFO )
263         RETURN
264      ELSE IF( LQUERY ) THEN
265         RETURN
266      END IF
267*
268*     Quick return if possible
269*
270      MINMN = MIN( M, N )
271      IF( MINMN.EQ.0 ) THEN
272         WORK( 1 ) = 1
273         RETURN
274      END IF
275*
276      WS = MAX( M, N )
277      LDWRKX = M
278      LDWRKY = N
279*
280      IF( NB.GT.1 .AND. NB.LT.MINMN ) THEN
281*
282*        Set the crossover point NX.
283*
284         NX = MAX( NB, ILAENV( 3, 'DGEBRD', ' ', M, N, -1, -1 ) )
285*
286*        Determine when to switch from blocked to unblocked code.
287*
288         IF( NX.LT.MINMN ) THEN
289            WS = ( M+N )*NB
290            IF( LWORK.LT.WS ) THEN
291*
292*              Not enough work space for the optimal NB, consider using
293*              a smaller block size.
294*
295               NBMIN = ILAENV( 2, 'DGEBRD', ' ', M, N, -1, -1 )
296               IF( LWORK.GE.( M+N )*NBMIN ) THEN
297                  NB = LWORK / ( M+N )
298               ELSE
299                  NB = 1
300                  NX = MINMN
301               END IF
302            END IF
303         END IF
304      ELSE
305         NX = MINMN
306      END IF
307*
308      DO 30 I = 1, MINMN - NX, NB
309*
310*        Reduce rows and columns i:i+nb-1 to bidiagonal form and return
311*        the matrices X and Y which are needed to update the unreduced
312*        part of the matrix
313*
314         CALL DLABRD( M-I+1, N-I+1, NB, A( I, I ), LDA, D( I ), E( I ),
315     $                TAUQ( I ), TAUP( I ), WORK, LDWRKX,
316     $                WORK( LDWRKX*NB+1 ), LDWRKY )
317*
318*        Update the trailing submatrix A(i+nb:m,i+nb:n), using an update
319*        of the form  A := A - V*Y**T - X*U**T
320*
321         CALL DGEMM( 'No transpose', 'Transpose', M-I-NB+1, N-I-NB+1,
322     $               NB, -ONE, A( I+NB, I ), LDA,
323     $               WORK( LDWRKX*NB+NB+1 ), LDWRKY, ONE,
324     $               A( I+NB, I+NB ), LDA )
325         CALL DGEMM( 'No transpose', 'No transpose', M-I-NB+1, N-I-NB+1,
326     $               NB, -ONE, WORK( NB+1 ), LDWRKX, A( I, I+NB ), LDA,
327     $               ONE, A( I+NB, I+NB ), LDA )
328*
329*        Copy diagonal and off-diagonal elements of B back into A
330*
331         IF( M.GE.N ) THEN
332            DO 10 J = I, I + NB - 1
333               A( J, J ) = D( J )
334               A( J, J+1 ) = E( J )
335   10       CONTINUE
336         ELSE
337            DO 20 J = I, I + NB - 1
338               A( J, J ) = D( J )
339               A( J+1, J ) = E( J )
340   20       CONTINUE
341         END IF
342   30 CONTINUE
343*
344*     Use unblocked code to reduce the remainder of the matrix
345*
346      CALL DGEBD2( M-I+1, N-I+1, A( I, I ), LDA, D( I ), E( I ),
347     $             TAUQ( I ), TAUP( I ), WORK, IINFO )
348      WORK( 1 ) = WS
349      RETURN
350*
351*     End of DGEBRD
352*
353      END
354c $Id$
355