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 3.2) --
6: * -- LAPACK is a software package provided by Univ. of Tennessee, --
7: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8: * November 2006
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)'*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'*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
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