1*> \brief <b> DSBEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
2*
3*  =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6*            http://www.netlib.org/lapack/explore-html/
7*
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16*> \endhtmlonly
17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE DSBEVD( JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK,
22*                          LWORK, IWORK, LIWORK, INFO )
23*
24*       .. Scalar Arguments ..
25*       CHARACTER          JOBZ, UPLO
26*       INTEGER            INFO, KD, LDAB, LDZ, LIWORK, LWORK, N
27*       ..
28*       .. Array Arguments ..
29*       INTEGER            IWORK( * )
30*       DOUBLE PRECISION   AB( LDAB, * ), W( * ), WORK( * ), Z( LDZ, * )
31*       ..
32*
33*
34*> \par Purpose:
35*  =============
36*>
37*> \verbatim
38*>
39*> DSBEVD computes all the eigenvalues and, optionally, eigenvectors of
40*> a real symmetric band matrix A. If eigenvectors are desired, it uses
41*> a divide and conquer algorithm.
42*>
43*> The divide and conquer algorithm makes very mild assumptions about
44*> floating point arithmetic. It will work on machines with a guard
45*> digit in add/subtract, or on those binary machines without guard
46*> digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
47*> Cray-2. It could conceivably fail on hexadecimal or decimal machines
48*> without guard digits, but we know of none.
49*> \endverbatim
50*
51*  Arguments:
52*  ==========
53*
54*> \param[in] JOBZ
55*> \verbatim
56*>          JOBZ is CHARACTER*1
57*>          = 'N':  Compute eigenvalues only;
58*>          = 'V':  Compute eigenvalues and eigenvectors.
59*> \endverbatim
60*>
61*> \param[in] UPLO
62*> \verbatim
63*>          UPLO is CHARACTER*1
64*>          = 'U':  Upper triangle of A is stored;
65*>          = 'L':  Lower triangle of A is stored.
66*> \endverbatim
67*>
68*> \param[in] N
69*> \verbatim
70*>          N is INTEGER
71*>          The order of the matrix A.  N >= 0.
72*> \endverbatim
73*>
74*> \param[in] KD
75*> \verbatim
76*>          KD is INTEGER
77*>          The number of superdiagonals of the matrix A if UPLO = 'U',
78*>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
79*> \endverbatim
80*>
81*> \param[in,out] AB
82*> \verbatim
83*>          AB is DOUBLE PRECISION array, dimension (LDAB, N)
84*>          On entry, the upper or lower triangle of the symmetric band
85*>          matrix A, stored in the first KD+1 rows of the array.  The
86*>          j-th column of A is stored in the j-th column of the array AB
87*>          as follows:
88*>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
89*>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
90*>
91*>          On exit, AB is overwritten by values generated during the
92*>          reduction to tridiagonal form.  If UPLO = 'U', the first
93*>          superdiagonal and the diagonal of the tridiagonal matrix T
94*>          are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
95*>          the diagonal and first subdiagonal of T are returned in the
96*>          first two rows of AB.
97*> \endverbatim
98*>
99*> \param[in] LDAB
100*> \verbatim
101*>          LDAB is INTEGER
102*>          The leading dimension of the array AB.  LDAB >= KD + 1.
103*> \endverbatim
104*>
105*> \param[out] W
106*> \verbatim
107*>          W is DOUBLE PRECISION array, dimension (N)
108*>          If INFO = 0, the eigenvalues in ascending order.
109*> \endverbatim
110*>
111*> \param[out] Z
112*> \verbatim
113*>          Z is DOUBLE PRECISION array, dimension (LDZ, N)
114*>          If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
115*>          eigenvectors of the matrix A, with the i-th column of Z
116*>          holding the eigenvector associated with W(i).
117*>          If JOBZ = 'N', then Z is not referenced.
118*> \endverbatim
119*>
120*> \param[in] LDZ
121*> \verbatim
122*>          LDZ is INTEGER
123*>          The leading dimension of the array Z.  LDZ >= 1, and if
124*>          JOBZ = 'V', LDZ >= max(1,N).
125*> \endverbatim
126*>
127*> \param[out] WORK
128*> \verbatim
129*>          WORK is DOUBLE PRECISION array,
130*>                                         dimension (LWORK)
131*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
132*> \endverbatim
133*>
134*> \param[in] LWORK
135*> \verbatim
136*>          LWORK is INTEGER
137*>          The dimension of the array WORK.
138*>          IF N <= 1,                LWORK must be at least 1.
139*>          If JOBZ  = 'N' and N > 2, LWORK must be at least 2*N.
140*>          If JOBZ  = 'V' and N > 2, LWORK must be at least
141*>                         ( 1 + 5*N + 2*N**2 ).
142*>
143*>          If LWORK = -1, then a workspace query is assumed; the routine
144*>          only calculates the optimal sizes of the WORK and IWORK
145*>          arrays, returns these values as the first entries of the WORK
146*>          and IWORK arrays, and no error message related to LWORK or
147*>          LIWORK is issued by XERBLA.
148*> \endverbatim
149*>
150*> \param[out] IWORK
151*> \verbatim
152*>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
153*>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
154*> \endverbatim
155*>
156*> \param[in] LIWORK
157*> \verbatim
158*>          LIWORK is INTEGER
159*>          The dimension of the array IWORK.
160*>          If JOBZ  = 'N' or N <= 1, LIWORK must be at least 1.
161*>          If JOBZ  = 'V' and N > 2, LIWORK must be at least 3 + 5*N.
162*>
163*>          If LIWORK = -1, then a workspace query is assumed; the
164*>          routine only calculates the optimal sizes of the WORK and
165*>          IWORK arrays, returns these values as the first entries of
166*>          the WORK and IWORK arrays, and no error message related to
167*>          LWORK or LIWORK is issued by XERBLA.
168*> \endverbatim
169*>
170*> \param[out] INFO
171*> \verbatim
172*>          INFO is INTEGER
173*>          = 0:  successful exit
174*>          < 0:  if INFO = -i, the i-th argument had an illegal value
175*>          > 0:  if INFO = i, the algorithm failed to converge; i
176*>                off-diagonal elements of an intermediate tridiagonal
177*>                form did not converge to zero.
178*> \endverbatim
179*
180*  Authors:
181*  ========
182*
183*> \author Univ. of Tennessee
184*> \author Univ. of California Berkeley
185*> \author Univ. of Colorado Denver
186*> \author NAG Ltd.
187*
188*> \ingroup doubleOTHEReigen
189*
190*  =====================================================================
191      SUBROUTINE DSBEVD( JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK,
192     $                   LWORK, IWORK, LIWORK, INFO )
193*
194*  -- LAPACK driver routine --
195*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
196*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
197*
198*     .. Scalar Arguments ..
199      CHARACTER          JOBZ, UPLO
200      INTEGER            INFO, KD, LDAB, LDZ, LIWORK, LWORK, N
201*     ..
202*     .. Array Arguments ..
203      INTEGER            IWORK( * )
204      DOUBLE PRECISION   AB( LDAB, * ), W( * ), WORK( * ), Z( LDZ, * )
205*     ..
206*
207*  =====================================================================
208*
209*     .. Parameters ..
210      DOUBLE PRECISION   ZERO, ONE
211      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
212*     ..
213*     .. Local Scalars ..
214      LOGICAL            LOWER, LQUERY, WANTZ
215      INTEGER            IINFO, INDE, INDWK2, INDWRK, ISCALE, LIWMIN,
216     $                   LLWRK2, LWMIN
217      DOUBLE PRECISION   ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
218     $                   SMLNUM
219*     ..
220*     .. External Functions ..
221      LOGICAL            LSAME
222      DOUBLE PRECISION   DLAMCH, DLANSB
223      EXTERNAL           LSAME, DLAMCH, DLANSB
224*     ..
225*     .. External Subroutines ..
226      EXTERNAL           DGEMM, DLACPY, DLASCL, DSBTRD, DSCAL, DSTEDC,
227     $                   DSTERF, XERBLA
228*     ..
229*     .. Intrinsic Functions ..
230      INTRINSIC          SQRT
231*     ..
232*     .. Executable Statements ..
233*
234*     Test the input parameters.
235*
236      WANTZ = LSAME( JOBZ, 'V' )
237      LOWER = LSAME( UPLO, 'L' )
238      LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
239*
240      INFO = 0
241      IF( N.LE.1 ) THEN
242         LIWMIN = 1
243         LWMIN = 1
244      ELSE
245         IF( WANTZ ) THEN
246            LIWMIN = 3 + 5*N
247            LWMIN = 1 + 5*N + 2*N**2
248         ELSE
249            LIWMIN = 1
250            LWMIN = 2*N
251         END IF
252      END IF
253      IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
254         INFO = -1
255      ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
256         INFO = -2
257      ELSE IF( N.LT.0 ) THEN
258         INFO = -3
259      ELSE IF( KD.LT.0 ) THEN
260         INFO = -4
261      ELSE IF( LDAB.LT.KD+1 ) THEN
262         INFO = -6
263      ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
264         INFO = -9
265      END IF
266*
267      IF( INFO.EQ.0 ) THEN
268         WORK( 1 ) = LWMIN
269         IWORK( 1 ) = LIWMIN
270*
271         IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
272            INFO = -11
273         ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
274            INFO = -13
275         END IF
276      END IF
277*
278      IF( INFO.NE.0 ) THEN
279         CALL XERBLA( 'DSBEVD', -INFO )
280         RETURN
281      ELSE IF( LQUERY ) THEN
282         RETURN
283      END IF
284*
285*     Quick return if possible
286*
287      IF( N.EQ.0 )
288     $   RETURN
289*
290      IF( N.EQ.1 ) THEN
291         W( 1 ) = AB( 1, 1 )
292         IF( WANTZ )
293     $      Z( 1, 1 ) = ONE
294         RETURN
295      END IF
296*
297*     Get machine constants.
298*
299      SAFMIN = DLAMCH( 'Safe minimum' )
300      EPS = DLAMCH( 'Precision' )
301      SMLNUM = SAFMIN / EPS
302      BIGNUM = ONE / SMLNUM
303      RMIN = SQRT( SMLNUM )
304      RMAX = SQRT( BIGNUM )
305*
306*     Scale matrix to allowable range, if necessary.
307*
308      ANRM = DLANSB( 'M', UPLO, N, KD, AB, LDAB, WORK )
309      ISCALE = 0
310      IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
311         ISCALE = 1
312         SIGMA = RMIN / ANRM
313      ELSE IF( ANRM.GT.RMAX ) THEN
314         ISCALE = 1
315         SIGMA = RMAX / ANRM
316      END IF
317      IF( ISCALE.EQ.1 ) THEN
318         IF( LOWER ) THEN
319            CALL DLASCL( 'B', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
320         ELSE
321            CALL DLASCL( 'Q', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
322         END IF
323      END IF
324*
325*     Call DSBTRD to reduce symmetric band matrix to tridiagonal form.
326*
327      INDE = 1
328      INDWRK = INDE + N
329      INDWK2 = INDWRK + N*N
330      LLWRK2 = LWORK - INDWK2 + 1
331      CALL DSBTRD( JOBZ, UPLO, N, KD, AB, LDAB, W, WORK( INDE ), Z, LDZ,
332     $             WORK( INDWRK ), IINFO )
333*
334*     For eigenvalues only, call DSTERF.  For eigenvectors, call SSTEDC.
335*
336      IF( .NOT.WANTZ ) THEN
337         CALL DSTERF( N, W, WORK( INDE ), INFO )
338      ELSE
339         CALL DSTEDC( 'I', N, W, WORK( INDE ), WORK( INDWRK ), N,
340     $                WORK( INDWK2 ), LLWRK2, IWORK, LIWORK, INFO )
341         CALL DGEMM( 'N', 'N', N, N, N, ONE, Z, LDZ, WORK( INDWRK ), N,
342     $               ZERO, WORK( INDWK2 ), N )
343         CALL DLACPY( 'A', N, N, WORK( INDWK2 ), N, Z, LDZ )
344      END IF
345*
346*     If matrix was scaled, then rescale eigenvalues appropriately.
347*
348      IF( ISCALE.EQ.1 )
349     $   CALL DSCAL( N, ONE / SIGMA, W, 1 )
350*
351      WORK( 1 ) = LWMIN
352      IWORK( 1 ) = LIWMIN
353      RETURN
354*
355*     End of DSBEVD
356*
357      END
358