1*> \brief \b SSYTRD
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 SSYTRD( 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               A( LDA, * ), D( * ), E( * ), TAU( * ),
29*      $                   WORK( * )
30*       ..
31*
32*
33*> \par Purpose:
34*  =============
35*>
36*> \verbatim
37*>
38*> SSYTRD 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 REAL 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 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 REAL 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 REAL 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*> \ingroup realSYcomputational
143*
144*> \par Further Details:
145*  =====================
146*>
147*> \verbatim
148*>
149*>  If UPLO = 'U', the matrix Q is represented as a product of elementary
150*>  reflectors
151*>
152*>     Q = H(n-1) . . . H(2) H(1).
153*>
154*>  Each H(i) has the form
155*>
156*>     H(i) = I - tau * v * v**T
157*>
158*>  where tau is a real scalar, and v is a real vector with
159*>  v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
160*>  A(1:i-1,i+1), and tau in TAU(i).
161*>
162*>  If UPLO = 'L', the matrix Q is represented as a product of elementary
163*>  reflectors
164*>
165*>     Q = H(1) H(2) . . . H(n-1).
166*>
167*>  Each H(i) has the form
168*>
169*>     H(i) = I - tau * v * v**T
170*>
171*>  where tau is a real scalar, and v is a real vector with
172*>  v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
173*>  and tau in TAU(i).
174*>
175*>  The contents of A on exit are illustrated by the following examples
176*>  with n = 5:
177*>
178*>  if UPLO = 'U':                       if UPLO = 'L':
179*>
180*>    (  d   e   v2  v3  v4 )              (  d                  )
181*>    (      d   e   v3  v4 )              (  e   d              )
182*>    (          d   e   v4 )              (  v1  e   d          )
183*>    (              d   e  )              (  v1  v2  e   d      )
184*>    (                  d  )              (  v1  v2  v3  e   d  )
185*>
186*>  where d and e denote diagonal and off-diagonal elements of T, and vi
187*>  denotes an element of the vector defining H(i).
188*> \endverbatim
189*>
190*  =====================================================================
191      SUBROUTINE SSYTRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
192*
193*  -- LAPACK computational routine --
194*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
195*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
196*
197*     .. Scalar Arguments ..
198      CHARACTER          UPLO
199      INTEGER            INFO, LDA, LWORK, N
200*     ..
201*     .. Array Arguments ..
202      REAL               A( LDA, * ), D( * ), E( * ), TAU( * ),
203     $                   WORK( * )
204*     ..
205*
206*  =====================================================================
207*
208*     .. Parameters ..
209      REAL               ONE
210      PARAMETER          ( ONE = 1.0E+0 )
211*     ..
212*     .. Local Scalars ..
213      LOGICAL            LQUERY, UPPER
214      INTEGER            I, IINFO, IWS, J, KK, LDWORK, LWKOPT, NB,
215     $                   NBMIN, NX
216*     ..
217*     .. External Subroutines ..
218      EXTERNAL           SLATRD, SSYR2K, SSYTD2, XERBLA
219*     ..
220*     .. Intrinsic Functions ..
221      INTRINSIC          MAX
222*     ..
223*     .. External Functions ..
224      LOGICAL            LSAME
225      INTEGER            ILAENV
226      EXTERNAL           LSAME, ILAENV
227*     ..
228*     .. Executable Statements ..
229*
230*     Test the input parameters
231*
232      INFO = 0
233      UPPER = LSAME( UPLO, 'U' )
234      LQUERY = ( LWORK.EQ.-1 )
235      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
236         INFO = -1
237      ELSE IF( N.LT.0 ) THEN
238         INFO = -2
239      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
240         INFO = -4
241      ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
242         INFO = -9
243      END IF
244*
245      IF( INFO.EQ.0 ) THEN
246*
247*        Determine the block size.
248*
249         NB = ILAENV( 1, 'SSYTRD', UPLO, N, -1, -1, -1 )
250         LWKOPT = N*NB
251         WORK( 1 ) = LWKOPT
252      END IF
253*
254      IF( INFO.NE.0 ) THEN
255         CALL XERBLA( 'SSYTRD', -INFO )
256         RETURN
257      ELSE IF( LQUERY ) THEN
258         RETURN
259      END IF
260*
261*     Quick return if possible
262*
263      IF( N.EQ.0 ) THEN
264         WORK( 1 ) = 1
265         RETURN
266      END IF
267*
268      NX = N
269      IWS = 1
270      IF( NB.GT.1 .AND. NB.LT.N ) THEN
271*
272*        Determine when to cross over from blocked to unblocked code
273*        (last block is always handled by unblocked code).
274*
275         NX = MAX( NB, ILAENV( 3, 'SSYTRD', UPLO, N, -1, -1, -1 ) )
276         IF( NX.LT.N ) THEN
277*
278*           Determine if workspace is large enough for blocked code.
279*
280            LDWORK = N
281            IWS = LDWORK*NB
282            IF( LWORK.LT.IWS ) THEN
283*
284*              Not enough workspace to use optimal NB:  determine the
285*              minimum value of NB, and reduce NB or force use of
286*              unblocked code by setting NX = N.
287*
288               NB = MAX( LWORK / LDWORK, 1 )
289               NBMIN = ILAENV( 2, 'SSYTRD', UPLO, N, -1, -1, -1 )
290               IF( NB.LT.NBMIN )
291     $            NX = N
292            END IF
293         ELSE
294            NX = N
295         END IF
296      ELSE
297         NB = 1
298      END IF
299*
300      IF( UPPER ) THEN
301*
302*        Reduce the upper triangle of A.
303*        Columns 1:kk are handled by the unblocked method.
304*
305         KK = N - ( ( N-NX+NB-1 ) / NB )*NB
306         DO 20 I = N - NB + 1, KK + 1, -NB
307*
308*           Reduce columns i:i+nb-1 to tridiagonal form and form the
309*           matrix W which is needed to update the unreduced part of
310*           the matrix
311*
312            CALL SLATRD( UPLO, I+NB-1, NB, A, LDA, E, TAU, WORK,
313     $                   LDWORK )
314*
315*           Update the unreduced submatrix A(1:i-1,1:i-1), using an
316*           update of the form:  A := A - V*W**T - W*V**T
317*
318            CALL SSYR2K( UPLO, 'No transpose', I-1, NB, -ONE, A( 1, I ),
319     $                   LDA, WORK, LDWORK, ONE, A, LDA )
320*
321*           Copy superdiagonal elements back into A, and diagonal
322*           elements into D
323*
324            DO 10 J = I, I + NB - 1
325               A( J-1, J ) = E( J-1 )
326               D( J ) = A( J, J )
327   10       CONTINUE
328   20    CONTINUE
329*
330*        Use unblocked code to reduce the last or only block
331*
332         CALL SSYTD2( UPLO, KK, A, LDA, D, E, TAU, IINFO )
333      ELSE
334*
335*        Reduce the lower triangle of A
336*
337         DO 40 I = 1, N - NX, NB
338*
339*           Reduce columns i:i+nb-1 to tridiagonal form and form the
340*           matrix W which is needed to update the unreduced part of
341*           the matrix
342*
343            CALL SLATRD( UPLO, N-I+1, NB, A( I, I ), LDA, E( I ),
344     $                   TAU( I ), WORK, LDWORK )
345*
346*           Update the unreduced submatrix A(i+ib:n,i+ib:n), using
347*           an update of the form:  A := A - V*W**T - W*V**T
348*
349            CALL SSYR2K( UPLO, 'No transpose', N-I-NB+1, NB, -ONE,
350     $                   A( I+NB, I ), LDA, WORK( NB+1 ), LDWORK, ONE,
351     $                   A( I+NB, I+NB ), LDA )
352*
353*           Copy subdiagonal elements back into A, and diagonal
354*           elements into D
355*
356            DO 30 J = I, I + NB - 1
357               A( J+1, J ) = E( J )
358               D( J ) = A( J, J )
359   30       CONTINUE
360   40    CONTINUE
361*
362*        Use unblocked code to reduce the last or only block
363*
364         CALL SSYTD2( UPLO, N-I+1, A( I, I ), LDA, D( I ), E( I ),
365     $                TAU( I ), IINFO )
366      END IF
367*
368      WORK( 1 ) = LWKOPT
369      RETURN
370*
371*     End of SSYTRD
372*
373      END
374