1*> \brief \b DSYTRD
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 DSYTRD( 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*       DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), TAU( * ),
29*      $                   WORK( * )
30*       ..
31*
32*
33*> \par Purpose:
34*  =============
35*>
36*> \verbatim
37*>
38*> DSYTRD reduces a real symmetric matrix A to real symmetric
39*> tridiagonal form T by an orthogonal similarity transformation:
40*> Q**T * 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 DOUBLE PRECISION array, dimension (LDA,N)
62*>          On entry, the symmetric 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 orthogonal
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 orthogonal 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 December 2016
143*
144*> \ingroup doubleSYcomputational
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**T
159*>
160*>  where tau is a real scalar, and v is a real 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**T
172*>
173*>  where tau is a real scalar, and v is a real 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 DSYTRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
194*
195*  -- LAPACK computational routine (version 3.7.0) --
196*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
197*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
198*     December 2016
199*
200*     .. Scalar Arguments ..
201      CHARACTER          UPLO
202      INTEGER            INFO, LDA, LWORK, N
203*     ..
204*     .. Array Arguments ..
205      DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), TAU( * ),
206     $                   WORK( * )
207*     ..
208*
209*  =====================================================================
210*
211*     .. Parameters ..
212      DOUBLE PRECISION   ONE
213      PARAMETER          ( ONE = 1.0D+0 )
214*     ..
215*     .. Local Scalars ..
216      LOGICAL            LQUERY, UPPER
217      INTEGER            I, IINFO, IWS, J, KK, LDWORK, LWKOPT, NB,
218     $                   NBMIN, NX
219*     ..
220*     .. External Subroutines ..
221      EXTERNAL           DLATRD, DSYR2K, DSYTD2, XERBLA
222*     ..
223*     .. Intrinsic Functions ..
224      INTRINSIC          MAX
225*     ..
226*     .. External Functions ..
227      LOGICAL            LSAME
228      INTEGER            ILAENV
229      EXTERNAL           LSAME, ILAENV
230*     ..
231*     .. Executable Statements ..
232*
233*     Test the input parameters
234*
235      INFO = 0
236      UPPER = LSAME( UPLO, 'U' )
237      LQUERY = ( LWORK.EQ.-1 )
238      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
239         INFO = -1
240      ELSE IF( N.LT.0 ) THEN
241         INFO = -2
242      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
243         INFO = -4
244      ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
245         INFO = -9
246      END IF
247*
248      IF( INFO.EQ.0 ) THEN
249*
250*        Determine the block size.
251*
252         NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 )
253         LWKOPT = N*NB
254         WORK( 1 ) = LWKOPT
255      END IF
256*
257      IF( INFO.NE.0 ) THEN
258         CALL XERBLA( 'DSYTRD', -INFO )
259         RETURN
260      ELSE IF( LQUERY ) THEN
261         RETURN
262      END IF
263*
264*     Quick return if possible
265*
266      IF( N.EQ.0 ) THEN
267         WORK( 1 ) = 1
268         RETURN
269      END IF
270*
271      NX = N
272      IWS = 1
273      IF( NB.GT.1 .AND. NB.LT.N ) THEN
274*
275*        Determine when to cross over from blocked to unblocked code
276*        (last block is always handled by unblocked code).
277*
278         NX = MAX( NB, ILAENV( 3, 'DSYTRD', UPLO, N, -1, -1, -1 ) )
279         IF( NX.LT.N ) THEN
280*
281*           Determine if workspace is large enough for blocked code.
282*
283            LDWORK = N
284            IWS = LDWORK*NB
285            IF( LWORK.LT.IWS ) THEN
286*
287*              Not enough workspace to use optimal NB:  determine the
288*              minimum value of NB, and reduce NB or force use of
289*              unblocked code by setting NX = N.
290*
291               NB = MAX( LWORK / LDWORK, 1 )
292               NBMIN = ILAENV( 2, 'DSYTRD', UPLO, N, -1, -1, -1 )
293               IF( NB.LT.NBMIN )
294     $            NX = N
295            END IF
296         ELSE
297            NX = N
298         END IF
299      ELSE
300         NB = 1
301      END IF
302*
303      IF( UPPER ) THEN
304*
305*        Reduce the upper triangle of A.
306*        Columns 1:kk are handled by the unblocked method.
307*
308         KK = N - ( ( N-NX+NB-1 ) / NB )*NB
309         DO 20 I = N - NB + 1, KK + 1, -NB
310*
311*           Reduce columns i:i+nb-1 to tridiagonal form and form the
312*           matrix W which is needed to update the unreduced part of
313*           the matrix
314*
315            CALL DLATRD( UPLO, I+NB-1, NB, A, LDA, E, TAU, WORK,
316     $                   LDWORK )
317*
318*           Update the unreduced submatrix A(1:i-1,1:i-1), using an
319*           update of the form:  A := A - V*W**T - W*V**T
320*
321            CALL DSYR2K( UPLO, 'No transpose', I-1, NB, -ONE, A( 1, I ),
322     $                   LDA, WORK, LDWORK, ONE, A, LDA )
323*
324*           Copy superdiagonal elements back into A, and diagonal
325*           elements into D
326*
327            DO 10 J = I, I + NB - 1
328               A( J-1, J ) = E( J-1 )
329               D( J ) = A( J, J )
330   10       CONTINUE
331   20    CONTINUE
332*
333*        Use unblocked code to reduce the last or only block
334*
335         CALL DSYTD2( UPLO, KK, A, LDA, D, E, TAU, IINFO )
336      ELSE
337*
338*        Reduce the lower triangle of A
339*
340         DO 40 I = 1, N - NX, NB
341*
342*           Reduce columns i:i+nb-1 to tridiagonal form and form the
343*           matrix W which is needed to update the unreduced part of
344*           the matrix
345*
346            CALL DLATRD( UPLO, N-I+1, NB, A( I, I ), LDA, E( I ),
347     $                   TAU( I ), WORK, LDWORK )
348*
349*           Update the unreduced submatrix A(i+ib:n,i+ib:n), using
350*           an update of the form:  A := A - V*W**T - W*V**T
351*
352            CALL DSYR2K( UPLO, 'No transpose', N-I-NB+1, NB, -ONE,
353     $                   A( I+NB, I ), LDA, WORK( NB+1 ), LDWORK, ONE,
354     $                   A( I+NB, I+NB ), LDA )
355*
356*           Copy subdiagonal elements back into A, and diagonal
357*           elements into D
358*
359            DO 30 J = I, I + NB - 1
360               A( J+1, J ) = E( J )
361               D( J ) = A( J, J )
362   30       CONTINUE
363   40    CONTINUE
364*
365*        Use unblocked code to reduce the last or only block
366*
367         CALL DSYTD2( UPLO, N-I+1, A( I, I ), LDA, D( I ), E( I ),
368     $                TAU( I ), IINFO )
369      END IF
370*
371      WORK( 1 ) = LWKOPT
372      RETURN
373*
374*     End of DSYTRD
375*
376      END
377