1*> \brief \b CHETRD
2*
3*  =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6*            http://www.netlib.org/lapack/explore-html/
7*
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17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE CHETRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
22*
23*       .. Scalar Arguments ..
24*       CHARACTER          UPLO
25*       INTEGER            INFO, LDA, LWORK, N
26*       ..
27*       .. Array Arguments ..
28*       REAL               D( * ), E( * )
29*       COMPLEX            A( LDA, * ), TAU( * ), WORK( * )
30*       ..
31*
32*
33*> \par Purpose:
34*  =============
35*>
36*> \verbatim
37*>
38*> CHETRD reduces a complex Hermitian matrix A to real symmetric
39*> tridiagonal form T by a unitary similarity transformation:
40*> Q**H * A * Q = T.
41*> \endverbatim
42*
43*  Arguments:
44*  ==========
45*
46*> \param[in] UPLO
47*> \verbatim
48*>          UPLO is CHARACTER*1
49*>          = 'U':  Upper triangle of A is stored;
50*>          = 'L':  Lower triangle of A is stored.
51*> \endverbatim
52*>
53*> \param[in] N
54*> \verbatim
55*>          N is INTEGER
56*>          The order of the matrix A.  N >= 0.
57*> \endverbatim
58*>
59*> \param[in,out] A
60*> \verbatim
61*>          A is COMPLEX array, dimension (LDA,N)
62*>          On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
63*>          N-by-N upper triangular part of A contains the upper
64*>          triangular part of the matrix A, and the strictly lower
65*>          triangular part of A is not referenced.  If UPLO = 'L', the
66*>          leading N-by-N lower triangular part of A contains the lower
67*>          triangular part of the matrix A, and the strictly upper
68*>          triangular part of A is not referenced.
69*>          On exit, if UPLO = 'U', the diagonal and first superdiagonal
70*>          of A are overwritten by the corresponding elements of the
71*>          tridiagonal matrix T, and the elements above the first
72*>          superdiagonal, with the array TAU, represent the unitary
73*>          matrix Q as a product of elementary reflectors; if UPLO
74*>          = 'L', the diagonal and first subdiagonal of A are over-
75*>          written by the corresponding elements of the tridiagonal
76*>          matrix T, and the elements below the first subdiagonal, with
77*>          the array TAU, represent the unitary matrix Q as a product
78*>          of elementary reflectors. 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,N).
85*> \endverbatim
86*>
87*> \param[out] D
88*> \verbatim
89*>          D is REAL array, dimension (N)
90*>          The diagonal elements of the tridiagonal matrix T:
91*>          D(i) = A(i,i).
92*> \endverbatim
93*>
94*> \param[out] E
95*> \verbatim
96*>          E is REAL array, dimension (N-1)
97*>          The off-diagonal elements of the tridiagonal matrix T:
98*>          E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
99*> \endverbatim
100*>
101*> \param[out] TAU
102*> \verbatim
103*>          TAU is COMPLEX array, dimension (N-1)
104*>          The scalar factors of the elementary reflectors (see Further
105*>          Details).
106*> \endverbatim
107*>
108*> \param[out] WORK
109*> \verbatim
110*>          WORK is COMPLEX array, dimension (MAX(1,LWORK))
111*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
112*> \endverbatim
113*>
114*> \param[in] LWORK
115*> \verbatim
116*>          LWORK is INTEGER
117*>          The dimension of the array WORK.  LWORK >= 1.
118*>          For optimum performance LWORK >= N*NB, where NB is the
119*>          optimal blocksize.
120*>
121*>          If LWORK = -1, then a workspace query is assumed; the routine
122*>          only calculates the optimal size of the WORK array, returns
123*>          this value as the first entry of the WORK array, and no error
124*>          message related to LWORK is issued by XERBLA.
125*> \endverbatim
126*>
127*> \param[out] INFO
128*> \verbatim
129*>          INFO is INTEGER
130*>          = 0:  successful exit
131*>          < 0:  if INFO = -i, the i-th argument had an illegal value
132*> \endverbatim
133*
134*  Authors:
135*  ========
136*
137*> \author Univ. of Tennessee
138*> \author Univ. of California Berkeley
139*> \author Univ. of Colorado Denver
140*> \author NAG Ltd.
141*
142*> \date November 2011
143*
144*> \ingroup complexHEcomputational
145*
146*> \par Further Details:
147*  =====================
148*>
149*> \verbatim
150*>
151*>  If UPLO = 'U', the matrix Q is represented as a product of elementary
152*>  reflectors
153*>
154*>     Q = H(n-1) . . . H(2) H(1).
155*>
156*>  Each H(i) has the form
157*>
158*>     H(i) = I - tau * v * v**H
159*>
160*>  where tau is a complex scalar, and v is a complex vector with
161*>  v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
162*>  A(1:i-1,i+1), and tau in TAU(i).
163*>
164*>  If UPLO = 'L', the matrix Q is represented as a product of elementary
165*>  reflectors
166*>
167*>     Q = H(1) H(2) . . . H(n-1).
168*>
169*>  Each H(i) has the form
170*>
171*>     H(i) = I - tau * v * v**H
172*>
173*>  where tau is a complex scalar, and v is a complex vector with
174*>  v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
175*>  and tau in TAU(i).
176*>
177*>  The contents of A on exit are illustrated by the following examples
178*>  with n = 5:
179*>
180*>  if UPLO = 'U':                       if UPLO = 'L':
181*>
182*>    (  d   e   v2  v3  v4 )              (  d                  )
183*>    (      d   e   v3  v4 )              (  e   d              )
184*>    (          d   e   v4 )              (  v1  e   d          )
185*>    (              d   e  )              (  v1  v2  e   d      )
186*>    (                  d  )              (  v1  v2  v3  e   d  )
187*>
188*>  where d and e denote diagonal and off-diagonal elements of T, and vi
189*>  denotes an element of the vector defining H(i).
190*> \endverbatim
191*>
192*  =====================================================================
193      SUBROUTINE CHETRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
194*
195*  -- LAPACK computational routine (version 3.4.0) --
196*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
197*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
198*     November 2011
199*
200*     .. Scalar Arguments ..
201      CHARACTER          UPLO
202      INTEGER            INFO, LDA, LWORK, N
203*     ..
204*     .. Array Arguments ..
205      REAL               D( * ), E( * )
206      COMPLEX            A( LDA, * ), TAU( * ), WORK( * )
207*     ..
208*
209*  =====================================================================
210*
211*     .. Parameters ..
212      REAL               ONE
213      PARAMETER          ( ONE = 1.0E+0 )
214      COMPLEX            CONE
215      PARAMETER          ( CONE = ( 1.0E+0, 0.0E+0 ) )
216*     ..
217*     .. Local Scalars ..
218      LOGICAL            LQUERY, UPPER
219      INTEGER            I, IINFO, IWS, J, KK, LDWORK, LWKOPT, NB,
220     $                   NBMIN, NX
221*     ..
222*     .. External Subroutines ..
223      EXTERNAL           CHER2K, CHETD2, CLATRD, XERBLA
224*     ..
225*     .. Intrinsic Functions ..
226      INTRINSIC          MAX
227*     ..
228*     .. External Functions ..
229      LOGICAL            LSAME
230      INTEGER            ILAENV
231      EXTERNAL           LSAME, ILAENV
232*     ..
233*     .. Executable Statements ..
234*
235*     Test the input parameters
236*
237      INFO = 0
238      UPPER = LSAME( UPLO, 'U' )
239      LQUERY = ( LWORK.EQ.-1 )
240      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
241         INFO = -1
242      ELSE IF( N.LT.0 ) THEN
243         INFO = -2
244      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
245         INFO = -4
246      ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
247         INFO = -9
248      END IF
249*
250      IF( INFO.EQ.0 ) THEN
251*
252*        Determine the block size.
253*
254         NB = ILAENV( 1, 'CHETRD', UPLO, N, -1, -1, -1 )
255         LWKOPT = N*NB
256         WORK( 1 ) = LWKOPT
257      END IF
258*
259      IF( INFO.NE.0 ) THEN
260         CALL XERBLA( 'CHETRD', -INFO )
261         RETURN
262      ELSE IF( LQUERY ) THEN
263         RETURN
264      END IF
265*
266*     Quick return if possible
267*
268      IF( N.EQ.0 ) THEN
269         WORK( 1 ) = 1
270         RETURN
271      END IF
272*
273      NX = N
274      IWS = 1
275      IF( NB.GT.1 .AND. NB.LT.N ) THEN
276*
277*        Determine when to cross over from blocked to unblocked code
278*        (last block is always handled by unblocked code).
279*
280         NX = MAX( NB, ILAENV( 3, 'CHETRD', UPLO, N, -1, -1, -1 ) )
281         IF( NX.LT.N ) THEN
282*
283*           Determine if workspace is large enough for blocked code.
284*
285            LDWORK = N
286            IWS = LDWORK*NB
287            IF( LWORK.LT.IWS ) THEN
288*
289*              Not enough workspace to use optimal NB:  determine the
290*              minimum value of NB, and reduce NB or force use of
291*              unblocked code by setting NX = N.
292*
293               NB = MAX( LWORK / LDWORK, 1 )
294               NBMIN = ILAENV( 2, 'CHETRD', UPLO, N, -1, -1, -1 )
295               IF( NB.LT.NBMIN )
296     $            NX = N
297            END IF
298         ELSE
299            NX = N
300         END IF
301      ELSE
302         NB = 1
303      END IF
304*
305      IF( UPPER ) THEN
306*
307*        Reduce the upper triangle of A.
308*        Columns 1:kk are handled by the unblocked method.
309*
310         KK = N - ( ( N-NX+NB-1 ) / NB )*NB
311         DO 20 I = N - NB + 1, KK + 1, -NB
312*
313*           Reduce columns i:i+nb-1 to tridiagonal form and form the
314*           matrix W which is needed to update the unreduced part of
315*           the matrix
316*
317            CALL CLATRD( UPLO, I+NB-1, NB, A, LDA, E, TAU, WORK,
318     $                   LDWORK )
319*
320*           Update the unreduced submatrix A(1:i-1,1:i-1), using an
321*           update of the form:  A := A - V*W**H - W*V**H
322*
323            CALL CHER2K( UPLO, 'No transpose', I-1, NB, -CONE,
324     $                   A( 1, I ), LDA, WORK, LDWORK, ONE, A, LDA )
325*
326*           Copy superdiagonal elements back into A, and diagonal
327*           elements into D
328*
329            DO 10 J = I, I + NB - 1
330               A( J-1, J ) = E( J-1 )
331               D( J ) = A( J, J )
332   10       CONTINUE
333   20    CONTINUE
334*
335*        Use unblocked code to reduce the last or only block
336*
337         CALL CHETD2( UPLO, KK, A, LDA, D, E, TAU, IINFO )
338      ELSE
339*
340*        Reduce the lower triangle of A
341*
342         DO 40 I = 1, N - NX, NB
343*
344*           Reduce columns i:i+nb-1 to tridiagonal form and form the
345*           matrix W which is needed to update the unreduced part of
346*           the matrix
347*
348            CALL CLATRD( UPLO, N-I+1, NB, A( I, I ), LDA, E( I ),
349     $                   TAU( I ), WORK, LDWORK )
350*
351*           Update the unreduced submatrix A(i+nb:n,i+nb:n), using
352*           an update of the form:  A := A - V*W**H - W*V**H
353*
354            CALL CHER2K( UPLO, 'No transpose', N-I-NB+1, NB, -CONE,
355     $                   A( I+NB, I ), LDA, WORK( NB+1 ), LDWORK, ONE,
356     $                   A( I+NB, I+NB ), LDA )
357*
358*           Copy subdiagonal elements back into A, and diagonal
359*           elements into D
360*
361            DO 30 J = I, I + NB - 1
362               A( J+1, J ) = E( J )
363               D( J ) = A( J, J )
364   30       CONTINUE
365   40    CONTINUE
366*
367*        Use unblocked code to reduce the last or only block
368*
369         CALL CHETD2( UPLO, N-I+1, A( I, I ), LDA, D( I ), E( I ),
370     $                TAU( I ), IINFO )
371      END IF
372*
373      WORK( 1 ) = LWKOPT
374      RETURN
375*
376*     End of CHETRD
377*
378      END
379