Annotation of rpl/lapack/lapack/dspgvx.f, revision 1.7
1.1 bertrand 1: SUBROUTINE DSPGVX( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU,
2: $ IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK,
3: $ 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 AP( * ), BP( * ), W( * ), WORK( * ),
18: $ Z( LDZ, * )
19: * ..
20: *
21: * Purpose
22: * =======
23: *
24: * DSPGVX computes selected eigenvalues, and optionally, eigenvectors
25: * of a real generalized symmetric-definite eigenproblem, of the form
26: * A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A
27: * and B are assumed to be symmetric, stored in packed storage, and B
28: * is also positive definite. Eigenvalues and eigenvectors can be
29: * selected by specifying either a range of values or a range of indices
30: * for the 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 triangle of A and B are stored;
53: * = 'L': Lower triangle of A and B are stored.
54: *
55: * N (input) INTEGER
56: * The order of the matrix pencil (A,B). N >= 0.
57: *
58: * AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
59: * On entry, the upper or lower triangle of the symmetric 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) DOUBLE PRECISION array, dimension (N*(N+1)/2)
68: * On entry, the upper or lower triangle of the symmetric 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**T*U or B = L*L**T, 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 A 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) DOUBLE PRECISION array, dimension (LDZ, max(1,M))
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**T*B*Z = I;
126: * if ITYPE = 3, Z**T*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) DOUBLE PRECISION array, dimension (8*N)
140: *
141: * IWORK (workspace) INTEGER array, dimension (5*N)
142: *
143: * IFAIL (output) INTEGER array, dimension (N)
144: * If JOBZ = 'V', then if INFO = 0, the first M elements of
145: * IFAIL are zero. If INFO > 0, then IFAIL contains the
146: * indices of the eigenvectors that failed to converge.
147: * If JOBZ = 'N', then IFAIL is not referenced.
148: *
149: * INFO (output) INTEGER
150: * = 0: successful exit
151: * < 0: if INFO = -i, the i-th argument had an illegal value
152: * > 0: DPPTRF or DSPEVX returned an error code:
153: * <= N: if INFO = i, DSPEVX failed to converge;
154: * i eigenvectors failed to converge. Their indices
155: * are stored in array IFAIL.
156: * > N: if INFO = N + i, for 1 <= i <= N, then the leading
157: * minor of order i of B is not positive definite.
158: * The factorization of B could not be completed and
159: * no eigenvalues or eigenvectors were computed.
160: *
161: * Further Details
162: * ===============
163: *
164: * Based on contributions by
165: * Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
166: *
167: * =====================================================================
168: *
169: * .. Local Scalars ..
170: LOGICAL ALLEIG, INDEIG, UPPER, VALEIG, WANTZ
171: CHARACTER TRANS
172: INTEGER J
173: * ..
174: * .. External Functions ..
175: LOGICAL LSAME
176: EXTERNAL LSAME
177: * ..
178: * .. External Subroutines ..
179: EXTERNAL DPPTRF, DSPEVX, DSPGST, DTPMV, DTPSV, XERBLA
180: * ..
181: * .. Intrinsic Functions ..
182: INTRINSIC MIN
183: * ..
184: * .. Executable Statements ..
185: *
186: * Test the input parameters.
187: *
188: UPPER = LSAME( UPLO, 'U' )
189: WANTZ = LSAME( JOBZ, 'V' )
190: ALLEIG = LSAME( RANGE, 'A' )
191: VALEIG = LSAME( RANGE, 'V' )
192: INDEIG = LSAME( RANGE, 'I' )
193: *
194: INFO = 0
195: IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
196: INFO = -1
197: ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
198: INFO = -2
199: ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
200: INFO = -3
201: ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
202: INFO = -4
203: ELSE IF( N.LT.0 ) THEN
204: INFO = -5
205: ELSE
206: IF( VALEIG ) THEN
207: IF( N.GT.0 .AND. VU.LE.VL ) THEN
208: INFO = -9
209: END IF
210: ELSE IF( INDEIG ) THEN
211: IF( IL.LT.1 ) THEN
212: INFO = -10
213: ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
214: INFO = -11
215: END IF
216: END IF
217: END IF
218: IF( INFO.EQ.0 ) THEN
219: IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
220: INFO = -16
221: END IF
222: END IF
223: *
224: IF( INFO.NE.0 ) THEN
225: CALL XERBLA( 'DSPGVX', -INFO )
226: RETURN
227: END IF
228: *
229: * Quick return if possible
230: *
231: M = 0
232: IF( N.EQ.0 )
233: $ RETURN
234: *
235: * Form a Cholesky factorization of B.
236: *
237: CALL DPPTRF( UPLO, N, BP, INFO )
238: IF( INFO.NE.0 ) THEN
239: INFO = N + INFO
240: RETURN
241: END IF
242: *
243: * Transform problem to standard eigenvalue problem and solve.
244: *
245: CALL DSPGST( ITYPE, UPLO, N, AP, BP, INFO )
246: CALL DSPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, ABSTOL, M,
247: $ W, Z, LDZ, WORK, IWORK, IFAIL, INFO )
248: *
249: IF( WANTZ ) THEN
250: *
251: * Backtransform eigenvectors to the original problem.
252: *
253: IF( INFO.GT.0 )
254: $ M = INFO - 1
255: IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN
256: *
257: * For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
258: * backtransform eigenvectors: x = inv(L)'*y or inv(U)*y
259: *
260: IF( UPPER ) THEN
261: TRANS = 'N'
262: ELSE
263: TRANS = 'T'
264: END IF
265: *
266: DO 10 J = 1, M
267: CALL DTPSV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ),
268: $ 1 )
269: 10 CONTINUE
270: *
271: ELSE IF( ITYPE.EQ.3 ) THEN
272: *
273: * For B*A*x=(lambda)*x;
274: * backtransform eigenvectors: x = L*y or U'*y
275: *
276: IF( UPPER ) THEN
277: TRANS = 'T'
278: ELSE
279: TRANS = 'N'
280: END IF
281: *
282: DO 20 J = 1, M
283: CALL DTPMV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ),
284: $ 1 )
285: 20 CONTINUE
286: END IF
287: END IF
288: *
289: RETURN
290: *
291: * End of DSPGVX
292: *
293: END
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