001:       SUBROUTINE CHPGVX( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU,
002:      $                   IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK,
003:      $                   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, LDZ, M, N
013:       REAL               ABSTOL, VL, VU
014: *     ..
015: *     .. Array Arguments ..
016:       INTEGER            IFAIL( * ), IWORK( * )
017:       REAL               RWORK( * ), W( * )
018:       COMPLEX            AP( * ), BP( * ), WORK( * ), Z( LDZ, * )
019: *     ..
020: *
021: *  Purpose
022: *  =======
023: *
024: *  CHPGVX computes selected eigenvalues and, optionally, eigenvectors
025: *  of a complex generalized Hermitian-definite eigenproblem, of the form
026: *  A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A and
027: *  B are assumed to be Hermitian, stored in packed format, and B is also
028: *  positive definite.  Eigenvalues and eigenvectors can be selected by
029: *  specifying either a range of values or a range of indices for the
030: *  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: *  AP      (input/output) COMPLEX array, dimension (N*(N+1)/2)
059: *          On entry, the upper or lower triangle of the Hermitian matrix
060: *          A, packed columnwise in a linear array.  The j-th column of A
061: *          is stored in the array AP as follows:
062: *          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
063: *          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
064: *
065: *          On exit, the contents of AP are destroyed.
066: *
067: *  BP      (input/output) COMPLEX array, dimension (N*(N+1)/2)
068: *          On entry, the upper or lower triangle of the Hermitian matrix
069: *          B, packed columnwise in a linear array.  The j-th column of B
070: *          is stored in the array BP as follows:
071: *          if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
072: *          if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
073: *
074: *          On exit, the triangular factor U or L from the Cholesky
075: *          factorization B = U**H*U or B = L*L**H, in the same storage
076: *          format as B.
077: *
078: *  VL      (input) REAL
079: *  VU      (input) REAL
080: *          If RANGE='V', the lower and upper bounds of the interval to
081: *          be searched for eigenvalues. VL < VU.
082: *          Not referenced if RANGE = 'A' or 'I'.
083: *
084: *  IL      (input) INTEGER
085: *  IU      (input) INTEGER
086: *          If RANGE='I', the indices (in ascending order) of the
087: *          smallest and largest eigenvalues to be returned.
088: *          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
089: *          Not referenced if RANGE = 'A' or 'V'.
090: *
091: *  ABSTOL  (input) REAL
092: *          The absolute error tolerance for the eigenvalues.
093: *          An approximate eigenvalue is accepted as converged
094: *          when it is determined to lie in an interval [a,b]
095: *          of width less than or equal to
096: *
097: *                  ABSTOL + EPS *   max( |a|,|b| ) ,
098: *
099: *          where EPS is the machine precision.  If ABSTOL is less than
100: *          or equal to zero, then  EPS*|T|  will be used in its place,
101: *          where |T| is the 1-norm of the tridiagonal matrix obtained
102: *          by reducing AP to tridiagonal form.
103: *
104: *          Eigenvalues will be computed most accurately when ABSTOL is
105: *          set to twice the underflow threshold 2*SLAMCH('S'), not zero.
106: *          If this routine returns with INFO>0, indicating that some
107: *          eigenvectors did not converge, try setting ABSTOL to
108: *          2*SLAMCH('S').
109: *
110: *  M       (output) INTEGER
111: *          The total number of eigenvalues found.  0 <= M <= N.
112: *          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
113: *
114: *  W       (output) REAL array, dimension (N)
115: *          On normal exit, the first M elements contain the selected
116: *          eigenvalues in ascending order.
117: *
118: *  Z       (output) COMPLEX array, dimension (LDZ, N)
119: *          If JOBZ = 'N', then Z is not referenced.
120: *          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
121: *          contain the orthonormal eigenvectors of the matrix A
122: *          corresponding to the selected eigenvalues, with the i-th
123: *          column of Z holding the eigenvector associated with W(i).
124: *          The eigenvectors are normalized as follows:
125: *          if ITYPE = 1 or 2, Z**H*B*Z = I;
126: *          if ITYPE = 3, Z**H*inv(B)*Z = I.
127: *
128: *          If an eigenvector fails to converge, then that column of Z
129: *          contains the latest approximation to the eigenvector, and the
130: *          index of the eigenvector is returned in IFAIL.
131: *          Note: the user must ensure that at least max(1,M) columns are
132: *          supplied in the array Z; if RANGE = 'V', the exact value of M
133: *          is not known in advance and an upper bound must be used.
134: *
135: *  LDZ     (input) INTEGER
136: *          The leading dimension of the array Z.  LDZ >= 1, and if
137: *          JOBZ = 'V', LDZ >= max(1,N).
138: *
139: *  WORK    (workspace) COMPLEX array, dimension (2*N)
140: *
141: *  RWORK   (workspace) REAL array, dimension (7*N)
142: *
143: *  IWORK   (workspace) INTEGER array, dimension (5*N)
144: *
145: *  IFAIL   (output) INTEGER array, dimension (N)
146: *          If JOBZ = 'V', then if INFO = 0, the first M elements of
147: *          IFAIL are zero.  If INFO > 0, then IFAIL contains the
148: *          indices of the eigenvectors that failed to converge.
149: *          If JOBZ = 'N', then IFAIL is not referenced.
150: *
151: *  INFO    (output) INTEGER
152: *          = 0:  successful exit
153: *          < 0:  if INFO = -i, the i-th argument had an illegal value
154: *          > 0:  CPPTRF or CHPEVX returned an error code:
155: *             <= N:  if INFO = i, CHPEVX failed to converge;
156: *                    i eigenvectors failed to converge.  Their indices
157: *                    are stored in array IFAIL.
158: *             > N:   if INFO = N + i, for 1 <= i <= n, then the leading
159: *                    minor of order i of B is not positive definite.
160: *                    The factorization of B could not be completed and
161: *                    no eigenvalues or eigenvectors were computed.
162: *
163: *  Further Details
164: *  ===============
165: *
166: *  Based on contributions by
167: *     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
168: *
169: *  =====================================================================
170: *
171: *     .. Local Scalars ..
172:       LOGICAL            ALLEIG, INDEIG, UPPER, VALEIG, WANTZ
173:       CHARACTER          TRANS
174:       INTEGER            J
175: *     ..
176: *     .. External Functions ..
177:       LOGICAL            LSAME
178:       EXTERNAL           LSAME
179: *     ..
180: *     .. External Subroutines ..
181:       EXTERNAL           CHPEVX, CHPGST, CPPTRF, CTPMV, CTPSV, XERBLA
182: *     ..
183: *     .. Intrinsic Functions ..
184:       INTRINSIC          MIN
185: *     ..
186: *     .. Executable Statements ..
187: *
188: *     Test the input parameters.
189: *
190:       WANTZ = LSAME( JOBZ, 'V' )
191:       UPPER = LSAME( UPLO, 'U' )
192:       ALLEIG = LSAME( RANGE, 'A' )
193:       VALEIG = LSAME( RANGE, 'V' )
194:       INDEIG = LSAME( RANGE, 'I' )
195: *
196:       INFO = 0
197:       IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
198:          INFO = -1
199:       ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
200:          INFO = -2
201:       ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
202:          INFO = -3
203:       ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
204:          INFO = -4
205:       ELSE IF( N.LT.0 ) THEN
206:          INFO = -5
207:       ELSE 
208:          IF( VALEIG ) THEN
209:             IF( N.GT.0 .AND. VU.LE.VL ) THEN
210:                INFO = -9
211:             END IF
212:          ELSE IF( INDEIG ) THEN
213:             IF( IL.LT.1 ) THEN
214:                INFO = -10
215:             ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
216:                INFO = -11
217:             END IF
218:          END IF
219:       END IF
220:       IF( INFO.EQ.0 ) THEN
221:          IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
222:             INFO = -16
223:          END IF
224:       END IF
225: *
226:       IF( INFO.NE.0 ) THEN
227:          CALL XERBLA( 'CHPGVX', -INFO )
228:          RETURN
229:       END IF
230: *
231: *     Quick return if possible
232: *
233:       IF( N.EQ.0 )
234:      $   RETURN
235: *
236: *     Form a Cholesky factorization of B.
237: *
238:       CALL CPPTRF( UPLO, N, BP, INFO )
239:       IF( INFO.NE.0 ) THEN
240:          INFO = N + INFO
241:          RETURN
242:       END IF
243: *
244: *     Transform problem to standard eigenvalue problem and solve.
245: *
246:       CALL CHPGST( ITYPE, UPLO, N, AP, BP, INFO )
247:       CALL CHPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, ABSTOL, M,
248:      $             W, Z, LDZ, WORK, RWORK, IWORK, IFAIL, INFO )
249: *
250:       IF( WANTZ ) THEN
251: *
252: *        Backtransform eigenvectors to the original problem.
253: *
254:          IF( INFO.GT.0 )
255:      $      M = INFO - 1
256:          IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN
257: *
258: *           For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
259: *           backtransform eigenvectors: x = inv(L)'*y or inv(U)*y
260: *
261:             IF( UPPER ) THEN
262:                TRANS = 'N'
263:             ELSE
264:                TRANS = 'C'
265:             END IF
266: *
267:             DO 10 J = 1, M
268:                CALL CTPSV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ),
269:      $                     1 )
270:    10       CONTINUE
271: *
272:          ELSE IF( ITYPE.EQ.3 ) THEN
273: *
274: *           For B*A*x=(lambda)*x;
275: *           backtransform eigenvectors: x = L*y or U'*y
276: *
277:             IF( UPPER ) THEN
278:                TRANS = 'C'
279:             ELSE
280:                TRANS = 'N'
281:             END IF
282: *
283:             DO 20 J = 1, M
284:                CALL CTPMV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ),
285:      $                     1 )
286:    20       CONTINUE
287:          END IF
288:       END IF
289: *
290:       RETURN
291: *
292: *     End of CHPGVX
293: *
294:       END
295: