001:       SUBROUTINE DSBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB,
002:      $                   LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z,
003:      $                   LDZ, WORK, IWORK, IFAIL, INFO )
004: *
005: *  -- LAPACK driver routine (version 3.2) --
006: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
007: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
008: *     November 2006
009: *
010: *     .. Scalar Arguments ..
011:       CHARACTER          JOBZ, RANGE, UPLO
012:       INTEGER            IL, INFO, IU, KA, KB, LDAB, LDBB, LDQ, LDZ, M,
013:      $                   N
014:       DOUBLE PRECISION   ABSTOL, VL, VU
015: *     ..
016: *     .. Array Arguments ..
017:       INTEGER            IFAIL( * ), IWORK( * )
018:       DOUBLE PRECISION   AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ),
019:      $                   W( * ), WORK( * ), Z( LDZ, * )
020: *     ..
021: *
022: *  Purpose
023: *  =======
024: *
025: *  DSBGVX computes selected eigenvalues, and optionally, eigenvectors
026: *  of a real generalized symmetric-definite banded eigenproblem, of
027: *  the form A*x=(lambda)*B*x.  Here A and B are assumed to be symmetric
028: *  and banded, and B is also positive definite.  Eigenvalues and
029: *  eigenvectors can be selected by specifying either all eigenvalues,
030: *  a range of values or a range of indices for the desired eigenvalues.
031: *
032: *  Arguments
033: *  =========
034: *
035: *  JOBZ    (input) CHARACTER*1
036: *          = 'N':  Compute eigenvalues only;
037: *          = 'V':  Compute eigenvalues and eigenvectors.
038: *
039: *  RANGE   (input) CHARACTER*1
040: *          = 'A': all eigenvalues will be found.
041: *          = 'V': all eigenvalues in the half-open interval (VL,VU]
042: *                 will be found.
043: *          = 'I': the IL-th through IU-th eigenvalues will be found.
044: *
045: *  UPLO    (input) CHARACTER*1
046: *          = 'U':  Upper triangles of A and B are stored;
047: *          = 'L':  Lower triangles of A and B are stored.
048: *
049: *  N       (input) INTEGER
050: *          The order of the matrices A and B.  N >= 0.
051: *
052: *  KA      (input) INTEGER
053: *          The number of superdiagonals of the matrix A if UPLO = 'U',
054: *          or the number of subdiagonals if UPLO = 'L'.  KA >= 0.
055: *
056: *  KB      (input) INTEGER
057: *          The number of superdiagonals of the matrix B if UPLO = 'U',
058: *          or the number of subdiagonals if UPLO = 'L'.  KB >= 0.
059: *
060: *  AB      (input/output) DOUBLE PRECISION array, dimension (LDAB, N)
061: *          On entry, the upper or lower triangle of the symmetric band
062: *          matrix A, stored in the first ka+1 rows of the array.  The
063: *          j-th column of A is stored in the j-th column of the array AB
064: *          as follows:
065: *          if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
066: *          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).
067: *
068: *          On exit, the contents of AB are destroyed.
069: *
070: *  LDAB    (input) INTEGER
071: *          The leading dimension of the array AB.  LDAB >= KA+1.
072: *
073: *  BB      (input/output) DOUBLE PRECISION array, dimension (LDBB, N)
074: *          On entry, the upper or lower triangle of the symmetric band
075: *          matrix B, stored in the first kb+1 rows of the array.  The
076: *          j-th column of B is stored in the j-th column of the array BB
077: *          as follows:
078: *          if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
079: *          if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).
080: *
081: *          On exit, the factor S from the split Cholesky factorization
082: *          B = S**T*S, as returned by DPBSTF.
083: *
084: *  LDBB    (input) INTEGER
085: *          The leading dimension of the array BB.  LDBB >= KB+1.
086: *
087: *  Q       (output) DOUBLE PRECISION array, dimension (LDQ, N)
088: *          If JOBZ = 'V', the n-by-n matrix used in the reduction of
089: *          A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x,
090: *          and consequently C to tridiagonal form.
091: *          If JOBZ = 'N', the array Q is not referenced.
092: *
093: *  LDQ     (input) INTEGER
094: *          The leading dimension of the array Q.  If JOBZ = 'N',
095: *          LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N).
096: *
097: *  VL      (input) DOUBLE PRECISION
098: *  VU      (input) DOUBLE PRECISION
099: *          If RANGE='V', the lower and upper bounds of the interval to
100: *          be searched for eigenvalues. VL < VU.
101: *          Not referenced if RANGE = 'A' or 'I'.
102: *
103: *  IL      (input) INTEGER
104: *  IU      (input) INTEGER
105: *          If RANGE='I', the indices (in ascending order) of the
106: *          smallest and largest eigenvalues to be returned.
107: *          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
108: *          Not referenced if RANGE = 'A' or 'V'.
109: *
110: *  ABSTOL  (input) DOUBLE PRECISION
111: *          The absolute error tolerance for the eigenvalues.
112: *          An approximate eigenvalue is accepted as converged
113: *          when it is determined to lie in an interval [a,b]
114: *          of width less than or equal to
115: *
116: *                  ABSTOL + EPS *   max( |a|,|b| ) ,
117: *
118: *          where EPS is the machine precision.  If ABSTOL is less than
119: *          or equal to zero, then  EPS*|T|  will be used in its place,
120: *          where |T| is the 1-norm of the tridiagonal matrix obtained
121: *          by reducing A to tridiagonal form.
122: *
123: *          Eigenvalues will be computed most accurately when ABSTOL is
124: *          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
125: *          If this routine returns with INFO>0, indicating that some
126: *          eigenvectors did not converge, try setting ABSTOL to
127: *          2*DLAMCH('S').
128: *
129: *  M       (output) INTEGER
130: *          The total number of eigenvalues found.  0 <= M <= N.
131: *          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
132: *
133: *  W       (output) DOUBLE PRECISION array, dimension (N)
134: *          If INFO = 0, the eigenvalues in ascending order.
135: *
136: *  Z       (output) DOUBLE PRECISION array, dimension (LDZ, N)
137: *          If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
138: *          eigenvectors, with the i-th column of Z holding the
139: *          eigenvector associated with W(i).  The eigenvectors are
140: *          normalized so Z**T*B*Z = I.
141: *          If JOBZ = 'N', then Z is not referenced.
142: *
143: *  LDZ     (input) INTEGER
144: *          The leading dimension of the array Z.  LDZ >= 1, and if
145: *          JOBZ = 'V', LDZ >= max(1,N).
146: *
147: *  WORK    (workspace/output) DOUBLE PRECISION array, dimension (7*N)
148: *
149: *  IWORK   (workspace/output) INTEGER array, dimension (5*N)
150: *
151: *  IFAIL   (output) INTEGER array, dimension (M)
152: *          If JOBZ = 'V', then if INFO = 0, the first M elements of
153: *          IFAIL are zero.  If INFO > 0, then IFAIL contains the
154: *          indices of the eigenvalues that failed to converge.
155: *          If JOBZ = 'N', then IFAIL is not referenced.
156: *
157: *  INFO    (output) INTEGER
158: *          = 0 : successful exit
159: *          < 0 : if INFO = -i, the i-th argument had an illegal value
160: *          <= N: if INFO = i, then i eigenvectors failed to converge.
161: *                  Their indices are stored in IFAIL.
162: *          > N : DPBSTF returned an error code; i.e.,
163: *                if INFO = N + i, for 1 <= i <= N, then the leading
164: *                minor of order i of B is not positive definite.
165: *                The factorization of B could not be completed and
166: *                no eigenvalues or eigenvectors were computed.
167: *
168: *  Further Details
169: *  ===============
170: *
171: *  Based on contributions by
172: *     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
173: *
174: *  =====================================================================
175: *
176: *     .. Parameters ..
177:       DOUBLE PRECISION   ZERO, ONE
178:       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
179: *     ..
180: *     .. Local Scalars ..
181:       LOGICAL            ALLEIG, INDEIG, TEST, UPPER, VALEIG, WANTZ
182:       CHARACTER          ORDER, VECT
183:       INTEGER            I, IINFO, INDD, INDE, INDEE, INDIBL, INDISP,
184:      $                   INDIWO, INDWRK, ITMP1, J, JJ, NSPLIT
185:       DOUBLE PRECISION   TMP1
186: *     ..
187: *     .. External Functions ..
188:       LOGICAL            LSAME
189:       EXTERNAL           LSAME
190: *     ..
191: *     .. External Subroutines ..
192:       EXTERNAL           DCOPY, DGEMV, DLACPY, DPBSTF, DSBGST, DSBTRD,
193:      $                   DSTEBZ, DSTEIN, DSTEQR, DSTERF, DSWAP, XERBLA
194: *     ..
195: *     .. Intrinsic Functions ..
196:       INTRINSIC          MIN
197: *     ..
198: *     .. Executable Statements ..
199: *
200: *     Test the input parameters.
201: *
202:       WANTZ = LSAME( JOBZ, 'V' )
203:       UPPER = LSAME( UPLO, 'U' )
204:       ALLEIG = LSAME( RANGE, 'A' )
205:       VALEIG = LSAME( RANGE, 'V' )
206:       INDEIG = LSAME( RANGE, 'I' )
207: *
208:       INFO = 0
209:       IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
210:          INFO = -1
211:       ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
212:          INFO = -2
213:       ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
214:          INFO = -3
215:       ELSE IF( N.LT.0 ) THEN
216:          INFO = -4
217:       ELSE IF( KA.LT.0 ) THEN
218:          INFO = -5
219:       ELSE IF( KB.LT.0 .OR. KB.GT.KA ) THEN
220:          INFO = -6
221:       ELSE IF( LDAB.LT.KA+1 ) THEN
222:          INFO = -8
223:       ELSE IF( LDBB.LT.KB+1 ) THEN
224:          INFO = -10
225:       ELSE IF( LDQ.LT.1 .OR. ( WANTZ .AND. LDQ.LT.N ) ) THEN
226:          INFO = -12
227:       ELSE
228:          IF( VALEIG ) THEN
229:             IF( N.GT.0 .AND. VU.LE.VL )
230:      $         INFO = -14
231:          ELSE IF( INDEIG ) THEN
232:             IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
233:                INFO = -15
234:             ELSE IF ( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
235:                INFO = -16
236:             END IF
237:          END IF
238:       END IF
239:       IF( INFO.EQ.0) THEN
240:          IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
241:             INFO = -21
242:          END IF
243:       END IF
244: *
245:       IF( INFO.NE.0 ) THEN
246:          CALL XERBLA( 'DSBGVX', -INFO )
247:          RETURN
248:       END IF
249: *
250: *     Quick return if possible
251: *
252:       M = 0
253:       IF( N.EQ.0 )
254:      $   RETURN
255: *
256: *     Form a split Cholesky factorization of B.
257: *
258:       CALL DPBSTF( UPLO, N, KB, BB, LDBB, INFO )
259:       IF( INFO.NE.0 ) THEN
260:          INFO = N + INFO
261:          RETURN
262:       END IF
263: *
264: *     Transform problem to standard eigenvalue problem.
265: *
266:       CALL DSBGST( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Q, LDQ,
267:      $             WORK, IINFO )
268: *
269: *     Reduce symmetric band matrix to tridiagonal form.
270: *
271:       INDD = 1
272:       INDE = INDD + N
273:       INDWRK = INDE + N
274:       IF( WANTZ ) THEN
275:          VECT = 'U'
276:       ELSE
277:          VECT = 'N'
278:       END IF
279:       CALL DSBTRD( VECT, UPLO, N, KA, AB, LDAB, WORK( INDD ),
280:      $             WORK( INDE ), Q, LDQ, WORK( INDWRK ), IINFO )
281: *
282: *     If all eigenvalues are desired and ABSTOL is less than or equal
283: *     to zero, then call DSTERF or SSTEQR.  If this fails for some
284: *     eigenvalue, then try DSTEBZ.
285: *
286:       TEST = .FALSE.
287:       IF( INDEIG ) THEN
288:          IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
289:             TEST = .TRUE.
290:          END IF
291:       END IF
292:       IF( ( ALLEIG .OR. TEST ) .AND. ( ABSTOL.LE.ZERO ) ) THEN
293:          CALL DCOPY( N, WORK( INDD ), 1, W, 1 )
294:          INDEE = INDWRK + 2*N
295:          CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
296:          IF( .NOT.WANTZ ) THEN
297:             CALL DSTERF( N, W, WORK( INDEE ), INFO )
298:          ELSE
299:             CALL DLACPY( 'A', N, N, Q, LDQ, Z, LDZ )
300:             CALL DSTEQR( JOBZ, N, W, WORK( INDEE ), Z, LDZ,
301:      $                   WORK( INDWRK ), INFO )
302:             IF( INFO.EQ.0 ) THEN
303:                DO 10 I = 1, N
304:                   IFAIL( I ) = 0
305:    10          CONTINUE
306:             END IF
307:          END IF
308:          IF( INFO.EQ.0 ) THEN
309:             M = N
310:             GO TO 30
311:          END IF
312:          INFO = 0
313:       END IF
314: *
315: *     Otherwise, call DSTEBZ and, if eigenvectors are desired,
316: *     call DSTEIN.
317: *
318:       IF( WANTZ ) THEN
319:          ORDER = 'B'
320:       ELSE
321:          ORDER = 'E'
322:       END IF
323:       INDIBL = 1
324:       INDISP = INDIBL + N
325:       INDIWO = INDISP + N
326:       CALL DSTEBZ( RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL,
327:      $             WORK( INDD ), WORK( INDE ), M, NSPLIT, W,
328:      $             IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWRK ),
329:      $             IWORK( INDIWO ), INFO )
330: *
331:       IF( WANTZ ) THEN
332:          CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W,
333:      $                IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
334:      $                WORK( INDWRK ), IWORK( INDIWO ), IFAIL, INFO )
335: *
336: *        Apply transformation matrix used in reduction to tridiagonal
337: *        form to eigenvectors returned by DSTEIN.
338: *
339:          DO 20 J = 1, M
340:             CALL DCOPY( N, Z( 1, J ), 1, WORK( 1 ), 1 )
341:             CALL DGEMV( 'N', N, N, ONE, Q, LDQ, WORK, 1, ZERO,
342:      $                  Z( 1, J ), 1 )
343:    20    CONTINUE
344:       END IF
345: *
346:    30 CONTINUE
347: *
348: *     If eigenvalues are not in order, then sort them, along with
349: *     eigenvectors.
350: *
351:       IF( WANTZ ) THEN
352:          DO 50 J = 1, M - 1
353:             I = 0
354:             TMP1 = W( J )
355:             DO 40 JJ = J + 1, M
356:                IF( W( JJ ).LT.TMP1 ) THEN
357:                   I = JJ
358:                   TMP1 = W( JJ )
359:                END IF
360:    40       CONTINUE
361: *
362:             IF( I.NE.0 ) THEN
363:                ITMP1 = IWORK( INDIBL+I-1 )
364:                W( I ) = W( J )
365:                IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
366:                W( J ) = TMP1
367:                IWORK( INDIBL+J-1 ) = ITMP1
368:                CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
369:                IF( INFO.NE.0 ) THEN
370:                   ITMP1 = IFAIL( I )
371:                   IFAIL( I ) = IFAIL( J )
372:                   IFAIL( J ) = ITMP1
373:                END IF
374:             END IF
375:    50    CONTINUE
376:       END IF
377: *
378:       RETURN
379: *
380: *     End of DSBGVX
381: *
382:       END
383: