1 SUBROUTINE ZHPGVX( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU,
2 $ IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK,
3 $ IWORK, IFAIL, INFO )
4 *
5 * -- LAPACK driver routine (version 3.3.1) --
6 * -- LAPACK is a software package provided by Univ. of Tennessee, --
7 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8 * -- April 2011 --
9 *
10 * .. Scalar Arguments ..
11 CHARACTER JOBZ, RANGE, UPLO
12 INTEGER IL, INFO, ITYPE, IU, LDZ, M, N
13 DOUBLE PRECISION ABSTOL, VL, VU
14 * ..
15 * .. Array Arguments ..
16 INTEGER IFAIL( * ), IWORK( * )
17 DOUBLE PRECISION RWORK( * ), W( * )
18 COMPLEX*16 AP( * ), BP( * ), WORK( * ), Z( LDZ, * )
19 * ..
20 *
21 * Purpose
22 * =======
23 *
24 * ZHPGVX computes selected eigenvalues and, optionally, eigenvectors
25 * of a complex generalized Hermitian-definite eigenproblem, of the form
26 * A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and
27 * B are assumed to be Hermitian, stored in packed format, and B is also
28 * positive definite. Eigenvalues and eigenvectors can be selected by
29 * specifying either a range of values or a range of indices for the
30 * desired eigenvalues.
31 *
32 * Arguments
33 * =========
34 *
35 * ITYPE (input) INTEGER
36 * Specifies the problem type to be solved:
37 * = 1: A*x = (lambda)*B*x
38 * = 2: A*B*x = (lambda)*x
39 * = 3: B*A*x = (lambda)*x
40 *
41 * JOBZ (input) CHARACTER*1
42 * = 'N': Compute eigenvalues only;
43 * = 'V': Compute eigenvalues and eigenvectors.
44 *
45 * RANGE (input) CHARACTER*1
46 * = 'A': all eigenvalues will be found;
47 * = 'V': all eigenvalues in the half-open interval (VL,VU]
48 * will be found;
49 * = 'I': the IL-th through IU-th eigenvalues will be found.
50 *
51 * UPLO (input) CHARACTER*1
52 * = 'U': Upper triangles of A and B are stored;
53 * = 'L': Lower triangles of A and B are stored.
54 *
55 * N (input) INTEGER
56 * The order of the matrices A and B. N >= 0.
57 *
58 * AP (input/output) COMPLEX*16 array, dimension (N*(N+1)/2)
59 * On entry, the upper or lower triangle of the Hermitian matrix
60 * A, packed columnwise in a linear array. The j-th column of A
61 * is stored in the array AP as follows:
62 * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
63 * if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
64 *
65 * On exit, the contents of AP are destroyed.
66 *
67 * BP (input/output) COMPLEX*16 array, dimension (N*(N+1)/2)
68 * On entry, the upper or lower triangle of the Hermitian matrix
69 * B, packed columnwise in a linear array. The j-th column of B
70 * is stored in the array BP as follows:
71 * if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
72 * if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
73 *
74 * On exit, the triangular factor U or L from the Cholesky
75 * factorization B = U**H*U or B = L*L**H, in the same storage
76 * format as B.
77 *
78 * VL (input) DOUBLE PRECISION
79 * VU (input) DOUBLE PRECISION
80 * If RANGE='V', the lower and upper bounds of the interval to
81 * be searched for eigenvalues. VL < VU.
82 * Not referenced if RANGE = 'A' or 'I'.
83 *
84 * IL (input) INTEGER
85 * IU (input) INTEGER
86 * If RANGE='I', the indices (in ascending order) of the
87 * smallest and largest eigenvalues to be returned.
88 * 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
89 * Not referenced if RANGE = 'A' or 'V'.
90 *
91 * ABSTOL (input) DOUBLE PRECISION
92 * The absolute error tolerance for the eigenvalues.
93 * An approximate eigenvalue is accepted as converged
94 * when it is determined to lie in an interval [a,b]
95 * of width less than or equal to
96 *
97 * ABSTOL + EPS * max( |a|,|b| ) ,
98 *
99 * 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*DLAMCH('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*DLAMCH('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) DOUBLE PRECISION array, dimension (N)
115 * On normal exit, the first M elements contain the selected
116 * eigenvalues in ascending order.
117 *
118 * Z (output) COMPLEX*16 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*16 array, dimension (2*N)
140 *
141 * RWORK (workspace) DOUBLE PRECISION 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: ZPPTRF or ZHPEVX returned an error code:
155 * <= N: if INFO = i, ZHPEVX 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 XERBLA, ZHPEVX, ZHPGST, ZPPTRF, ZTPMV, ZTPSV
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( 'ZHPGVX', -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 ZPPTRF( 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 ZHPGST( ITYPE, UPLO, N, AP, BP, INFO )
247 CALL ZHPEVX( 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)**H *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 ZTPSV( 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**H *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 ZTPMV( 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 ZHPGVX
293 *
294 END
2 $ IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK,
3 $ IWORK, IFAIL, INFO )
4 *
5 * -- LAPACK driver routine (version 3.3.1) --
6 * -- LAPACK is a software package provided by Univ. of Tennessee, --
7 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8 * -- April 2011 --
9 *
10 * .. Scalar Arguments ..
11 CHARACTER JOBZ, RANGE, UPLO
12 INTEGER IL, INFO, ITYPE, IU, LDZ, M, N
13 DOUBLE PRECISION ABSTOL, VL, VU
14 * ..
15 * .. Array Arguments ..
16 INTEGER IFAIL( * ), IWORK( * )
17 DOUBLE PRECISION RWORK( * ), W( * )
18 COMPLEX*16 AP( * ), BP( * ), WORK( * ), Z( LDZ, * )
19 * ..
20 *
21 * Purpose
22 * =======
23 *
24 * ZHPGVX computes selected eigenvalues and, optionally, eigenvectors
25 * of a complex generalized Hermitian-definite eigenproblem, of the form
26 * A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and
27 * B are assumed to be Hermitian, stored in packed format, and B is also
28 * positive definite. Eigenvalues and eigenvectors can be selected by
29 * specifying either a range of values or a range of indices for the
30 * desired eigenvalues.
31 *
32 * Arguments
33 * =========
34 *
35 * ITYPE (input) INTEGER
36 * Specifies the problem type to be solved:
37 * = 1: A*x = (lambda)*B*x
38 * = 2: A*B*x = (lambda)*x
39 * = 3: B*A*x = (lambda)*x
40 *
41 * JOBZ (input) CHARACTER*1
42 * = 'N': Compute eigenvalues only;
43 * = 'V': Compute eigenvalues and eigenvectors.
44 *
45 * RANGE (input) CHARACTER*1
46 * = 'A': all eigenvalues will be found;
47 * = 'V': all eigenvalues in the half-open interval (VL,VU]
48 * will be found;
49 * = 'I': the IL-th through IU-th eigenvalues will be found.
50 *
51 * UPLO (input) CHARACTER*1
52 * = 'U': Upper triangles of A and B are stored;
53 * = 'L': Lower triangles of A and B are stored.
54 *
55 * N (input) INTEGER
56 * The order of the matrices A and B. N >= 0.
57 *
58 * AP (input/output) COMPLEX*16 array, dimension (N*(N+1)/2)
59 * On entry, the upper or lower triangle of the Hermitian matrix
60 * A, packed columnwise in a linear array. The j-th column of A
61 * is stored in the array AP as follows:
62 * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
63 * if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
64 *
65 * On exit, the contents of AP are destroyed.
66 *
67 * BP (input/output) COMPLEX*16 array, dimension (N*(N+1)/2)
68 * On entry, the upper or lower triangle of the Hermitian matrix
69 * B, packed columnwise in a linear array. The j-th column of B
70 * is stored in the array BP as follows:
71 * if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
72 * if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
73 *
74 * On exit, the triangular factor U or L from the Cholesky
75 * factorization B = U**H*U or B = L*L**H, in the same storage
76 * format as B.
77 *
78 * VL (input) DOUBLE PRECISION
79 * VU (input) DOUBLE PRECISION
80 * If RANGE='V', the lower and upper bounds of the interval to
81 * be searched for eigenvalues. VL < VU.
82 * Not referenced if RANGE = 'A' or 'I'.
83 *
84 * IL (input) INTEGER
85 * IU (input) INTEGER
86 * If RANGE='I', the indices (in ascending order) of the
87 * smallest and largest eigenvalues to be returned.
88 * 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
89 * Not referenced if RANGE = 'A' or 'V'.
90 *
91 * ABSTOL (input) DOUBLE PRECISION
92 * The absolute error tolerance for the eigenvalues.
93 * An approximate eigenvalue is accepted as converged
94 * when it is determined to lie in an interval [a,b]
95 * of width less than or equal to
96 *
97 * ABSTOL + EPS * max( |a|,|b| ) ,
98 *
99 * 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*DLAMCH('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*DLAMCH('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) DOUBLE PRECISION array, dimension (N)
115 * On normal exit, the first M elements contain the selected
116 * eigenvalues in ascending order.
117 *
118 * Z (output) COMPLEX*16 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*16 array, dimension (2*N)
140 *
141 * RWORK (workspace) DOUBLE PRECISION 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: ZPPTRF or ZHPEVX returned an error code:
155 * <= N: if INFO = i, ZHPEVX 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 XERBLA, ZHPEVX, ZHPGST, ZPPTRF, ZTPMV, ZTPSV
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( 'ZHPGVX', -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 ZPPTRF( 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 ZHPGST( ITYPE, UPLO, N, AP, BP, INFO )
247 CALL ZHPEVX( 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)**H *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 ZTPSV( 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**H *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 ZTPMV( 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 ZHPGVX
293 *
294 END