001:       SUBROUTINE ZHEGVX( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB,
002:      $                   VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK,
003:      $                   LWORK, RWORK, 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, ITYPE, IU, LDA, LDB, LDZ, LWORK, M, N
013:       DOUBLE PRECISION   ABSTOL, VL, VU
014: *     ..
015: *     .. Array Arguments ..
016:       INTEGER            IFAIL( * ), IWORK( * )
017:       DOUBLE PRECISION   RWORK( * ), W( * )
018:       COMPLEX*16         A( LDA, * ), B( LDB, * ), WORK( * ),
019:      $                   Z( LDZ, * )
020: *     ..
021: *
022: *  Purpose
023: *  =======
024: *
025: *  ZHEGVX computes selected eigenvalues, and optionally, eigenvectors
026: *  of a complex generalized Hermitian-definite eigenproblem, of the form
027: *  A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A and
028: *  B are assumed to be Hermitian and B is also positive definite.
029: *  Eigenvalues and eigenvectors can be selected by specifying either a
030: *  range of values or a range of indices for the desired eigenvalues.
031: *
032: *  Arguments
033: *  =========
034: *
035: *  ITYPE   (input) INTEGER
036: *          Specifies the problem type to be solved:
037: *          = 1:  A*x = (lambda)*B*x
038: *          = 2:  A*B*x = (lambda)*x
039: *          = 3:  B*A*x = (lambda)*x
040: *
041: *  JOBZ    (input) CHARACTER*1
042: *          = 'N':  Compute eigenvalues only;
043: *          = 'V':  Compute eigenvalues and eigenvectors.
044: *
045: *  RANGE   (input) CHARACTER*1
046: *          = 'A': all eigenvalues will be found.
047: *          = 'V': all eigenvalues in the half-open interval (VL,VU]
048: *                 will be found.
049: *          = 'I': the IL-th through IU-th eigenvalues will be found.
050: **
051: *  UPLO    (input) CHARACTER*1
052: *          = 'U':  Upper triangles of A and B are stored;
053: *          = 'L':  Lower triangles of A and B are stored.
054: *
055: *  N       (input) INTEGER
056: *          The order of the matrices A and B.  N >= 0.
057: *
058: *  A       (input/output) COMPLEX*16 array, dimension (LDA, N)
059: *          On entry, the Hermitian matrix A.  If UPLO = 'U', the
060: *          leading N-by-N upper triangular part of A contains the
061: *          upper triangular part of the matrix A.  If UPLO = 'L',
062: *          the leading N-by-N lower triangular part of A contains
063: *          the lower triangular part of the matrix A.
064: *
065: *          On exit,  the lower triangle (if UPLO='L') or the upper
066: *          triangle (if UPLO='U') of A, including the diagonal, is
067: *          destroyed.
068: *
069: *  LDA     (input) INTEGER
070: *          The leading dimension of the array A.  LDA >= max(1,N).
071: *
072: *  B       (input/output) COMPLEX*16 array, dimension (LDB, N)
073: *          On entry, the Hermitian matrix B.  If UPLO = 'U', the
074: *          leading N-by-N upper triangular part of B contains the
075: *          upper triangular part of the matrix B.  If UPLO = 'L',
076: *          the leading N-by-N lower triangular part of B contains
077: *          the lower triangular part of the matrix B.
078: *
079: *          On exit, if INFO <= N, the part of B containing the matrix is
080: *          overwritten by the triangular factor U or L from the Cholesky
081: *          factorization B = U**H*U or B = L*L**H.
082: *
083: *  LDB     (input) INTEGER
084: *          The leading dimension of the array B.  LDB >= max(1,N).
085: *
086: *  VL      (input) DOUBLE PRECISION
087: *  VU      (input) DOUBLE PRECISION
088: *          If RANGE='V', the lower and upper bounds of the interval to
089: *          be searched for eigenvalues. VL < VU.
090: *          Not referenced if RANGE = 'A' or 'I'.
091: *
092: *  IL      (input) INTEGER
093: *  IU      (input) INTEGER
094: *          If RANGE='I', the indices (in ascending order) of the
095: *          smallest and largest eigenvalues to be returned.
096: *          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
097: *          Not referenced if RANGE = 'A' or 'V'.
098: *
099: *  ABSTOL  (input) DOUBLE PRECISION
100: *          The absolute error tolerance for the eigenvalues.
101: *          An approximate eigenvalue is accepted as converged
102: *          when it is determined to lie in an interval [a,b]
103: *          of width less than or equal to
104: *
105: *                  ABSTOL + EPS *   max( |a|,|b| ) ,
106: *
107: *          where EPS is the machine precision.  If ABSTOL is less than
108: *          or equal to zero, then  EPS*|T|  will be used in its place,
109: *          where |T| is the 1-norm of the tridiagonal matrix obtained
110: *          by reducing A to tridiagonal form.
111: *
112: *          Eigenvalues will be computed most accurately when ABSTOL is
113: *          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
114: *          If this routine returns with INFO>0, indicating that some
115: *          eigenvectors did not converge, try setting ABSTOL to
116: *          2*DLAMCH('S').
117: *
118: *  M       (output) INTEGER
119: *          The total number of eigenvalues found.  0 <= M <= N.
120: *          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
121: *
122: *  W       (output) DOUBLE PRECISION array, dimension (N)
123: *          The first M elements contain the selected
124: *          eigenvalues in ascending order.
125: *
126: *  Z       (output) COMPLEX*16 array, dimension (LDZ, max(1,M))
127: *          If JOBZ = 'N', then Z is not referenced.
128: *          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
129: *          contain the orthonormal eigenvectors of the matrix A
130: *          corresponding to the selected eigenvalues, with the i-th
131: *          column of Z holding the eigenvector associated with W(i).
132: *          The eigenvectors are normalized as follows:
133: *          if ITYPE = 1 or 2, Z**T*B*Z = I;
134: *          if ITYPE = 3, Z**T*inv(B)*Z = I.
135: *
136: *          If an eigenvector fails to converge, then that column of Z
137: *          contains the latest approximation to the eigenvector, and the
138: *          index of the eigenvector is returned in IFAIL.
139: *          Note: the user must ensure that at least max(1,M) columns are
140: *          supplied in the array Z; if RANGE = 'V', the exact value of M
141: *          is not known in advance and an upper bound must be used.
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) COMPLEX*16 array, dimension (MAX(1,LWORK))
148: *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
149: *
150: *  LWORK   (input) INTEGER
151: *          The length of the array WORK.  LWORK >= max(1,2*N).
152: *          For optimal efficiency, LWORK >= (NB+1)*N,
153: *          where NB is the blocksize for ZHETRD returned by ILAENV.
154: *
155: *          If LWORK = -1, then a workspace query is assumed; the routine
156: *          only calculates the optimal size of the WORK array, returns
157: *          this value as the first entry of the WORK array, and no error
158: *          message related to LWORK is issued by XERBLA.
159: *
160: *  RWORK   (workspace) DOUBLE PRECISION array, dimension (7*N)
161: *
162: *  IWORK   (workspace) INTEGER array, dimension (5*N)
163: *
164: *  IFAIL   (output) INTEGER array, dimension (N)
165: *          If JOBZ = 'V', then if INFO = 0, the first M elements of
166: *          IFAIL are zero.  If INFO > 0, then IFAIL contains the
167: *          indices of the eigenvectors that failed to converge.
168: *          If JOBZ = 'N', then IFAIL is not referenced.
169: *
170: *  INFO    (output) INTEGER
171: *          = 0:  successful exit
172: *          < 0:  if INFO = -i, the i-th argument had an illegal value
173: *          > 0:  ZPOTRF or ZHEEVX returned an error code:
174: *             <= N:  if INFO = i, ZHEEVX failed to converge;
175: *                    i eigenvectors failed to converge.  Their indices
176: *                    are stored in array IFAIL.
177: *             > N:   if INFO = N + i, for 1 <= i <= N, then the leading
178: *                    minor of order i of B is not positive definite.
179: *                    The factorization of B could not be completed and
180: *                    no eigenvalues or eigenvectors were computed.
181: *
182: *  Further Details
183: *  ===============
184: *
185: *  Based on contributions by
186: *     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
187: *
188: *  =====================================================================
189: *
190: *     .. Parameters ..
191:       COMPLEX*16         CONE
192:       PARAMETER          ( CONE = ( 1.0D+0, 0.0D+0 ) )
193: *     ..
194: *     .. Local Scalars ..
195:       LOGICAL            ALLEIG, INDEIG, LQUERY, UPPER, VALEIG, WANTZ
196:       CHARACTER          TRANS
197:       INTEGER            LWKOPT, NB
198: *     ..
199: *     .. External Functions ..
200:       LOGICAL            LSAME
201:       INTEGER            ILAENV
202:       EXTERNAL           LSAME, ILAENV
203: *     ..
204: *     .. External Subroutines ..
205:       EXTERNAL           XERBLA, ZHEEVX, ZHEGST, ZPOTRF, ZTRMM, ZTRSM
206: *     ..
207: *     .. Intrinsic Functions ..
208:       INTRINSIC          MAX, MIN
209: *     ..
210: *     .. Executable Statements ..
211: *
212: *     Test the input parameters.
213: *
214:       WANTZ = LSAME( JOBZ, 'V' )
215:       UPPER = LSAME( UPLO, 'U' )
216:       ALLEIG = LSAME( RANGE, 'A' )
217:       VALEIG = LSAME( RANGE, 'V' )
218:       INDEIG = LSAME( RANGE, 'I' )
219:       LQUERY = ( LWORK.EQ.-1 )
220: *
221:       INFO = 0
222:       IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
223:          INFO = -1
224:       ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
225:          INFO = -2
226:       ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
227:          INFO = -3
228:       ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
229:          INFO = -4
230:       ELSE IF( N.LT.0 ) THEN
231:          INFO = -5
232:       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
233:          INFO = -7
234:       ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
235:          INFO = -9
236:       ELSE
237:          IF( VALEIG ) THEN
238:             IF( N.GT.0 .AND. VU.LE.VL )
239:      $         INFO = -11
240:          ELSE IF( INDEIG ) THEN
241:             IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
242:                INFO = -12
243:             ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
244:                INFO = -13
245:             END IF
246:          END IF
247:       END IF
248:       IF (INFO.EQ.0) THEN
249:          IF (LDZ.LT.1 .OR. (WANTZ .AND. LDZ.LT.N)) THEN
250:             INFO = -18
251:          END IF
252:       END IF
253: *
254:       IF( INFO.EQ.0 ) THEN
255:          NB = ILAENV( 1, 'ZHETRD', UPLO, N, -1, -1, -1 )
256:          LWKOPT = MAX( 1, ( NB + 1 )*N )
257:          WORK( 1 ) = LWKOPT
258: *
259:          IF( LWORK.LT.MAX( 1, 2*N ) .AND. .NOT.LQUERY ) THEN
260:             INFO = -20
261:          END IF
262:       END IF
263: *
264:       IF( INFO.NE.0 ) THEN
265:          CALL XERBLA( 'ZHEGVX', -INFO )
266:          RETURN
267:       ELSE IF( LQUERY ) THEN
268:          RETURN
269:       END IF
270: *
271: *     Quick return if possible
272: *
273:       M = 0
274:       IF( N.EQ.0 ) THEN
275:          RETURN
276:       END IF
277: *
278: *     Form a Cholesky factorization of B.
279: *
280:       CALL ZPOTRF( UPLO, N, B, LDB, INFO )
281:       IF( INFO.NE.0 ) THEN
282:          INFO = N + INFO
283:          RETURN
284:       END IF
285: *
286: *     Transform problem to standard eigenvalue problem and solve.
287: *
288:       CALL ZHEGST( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
289:       CALL ZHEEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL,
290:      $             M, W, Z, LDZ, WORK, LWORK, RWORK, IWORK, IFAIL,
291:      $             INFO )
292: *
293:       IF( WANTZ ) THEN
294: *
295: *        Backtransform eigenvectors to the original problem.
296: *
297:          IF( INFO.GT.0 )
298:      $      M = INFO - 1
299:          IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN
300: *
301: *           For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
302: *           backtransform eigenvectors: x = inv(L)'*y or inv(U)*y
303: *
304:             IF( UPPER ) THEN
305:                TRANS = 'N'
306:             ELSE
307:                TRANS = 'C'
308:             END IF
309: *
310:             CALL ZTRSM( 'Left', UPLO, TRANS, 'Non-unit', N, M, CONE, B,
311:      $                  LDB, Z, LDZ )
312: *
313:          ELSE IF( ITYPE.EQ.3 ) THEN
314: *
315: *           For B*A*x=(lambda)*x;
316: *           backtransform eigenvectors: x = L*y or U'*y
317: *
318:             IF( UPPER ) THEN
319:                TRANS = 'C'
320:             ELSE
321:                TRANS = 'N'
322:             END IF
323: *
324:             CALL ZTRMM( 'Left', UPLO, TRANS, 'Non-unit', N, M, CONE, B,
325:      $                  LDB, Z, LDZ )
326:          END IF
327:       END IF
328: *
329: *     Set WORK(1) to optimal complex workspace size.
330: *
331:       WORK( 1 ) = LWKOPT
332: *
333:       RETURN
334: *
335: *     End of ZHEGVX
336: *
337:       END
338: