1: SUBROUTINE ZHPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU,
2: $ ABSTOL, M, W, Z, LDZ, WORK, RWORK, 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, 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( * ), WORK( * ), Z( LDZ, * )
19: * ..
20: *
21: * Purpose
22: * =======
23: *
24: * ZHPEVX computes selected eigenvalues and, optionally, eigenvectors
25: * of a complex Hermitian matrix A in packed storage.
26: * Eigenvalues/vectors can be selected by specifying either a range of
27: * values or a range of indices for the desired eigenvalues.
28: *
29: * Arguments
30: * =========
31: *
32: * JOBZ (input) CHARACTER*1
33: * = 'N': Compute eigenvalues only;
34: * = 'V': Compute eigenvalues and eigenvectors.
35: *
36: * RANGE (input) CHARACTER*1
37: * = 'A': all eigenvalues will be found;
38: * = 'V': all eigenvalues in the half-open interval (VL,VU]
39: * will be found;
40: * = 'I': the IL-th through IU-th eigenvalues will be found.
41: *
42: * UPLO (input) CHARACTER*1
43: * = 'U': Upper triangle of A is stored;
44: * = 'L': Lower triangle of A is stored.
45: *
46: * N (input) INTEGER
47: * The order of the matrix A. N >= 0.
48: *
49: * AP (input/output) COMPLEX*16 array, dimension (N*(N+1)/2)
50: * On entry, the upper or lower triangle of the Hermitian matrix
51: * A, packed columnwise in a linear array. The j-th column of A
52: * is stored in the array AP as follows:
53: * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
54: * if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
55: *
56: * On exit, AP is overwritten by values generated during the
57: * reduction to tridiagonal form. If UPLO = 'U', the diagonal
58: * and first superdiagonal of the tridiagonal matrix T overwrite
59: * the corresponding elements of A, and if UPLO = 'L', the
60: * diagonal and first subdiagonal of T overwrite the
61: * corresponding elements of A.
62: *
63: * VL (input) DOUBLE PRECISION
64: * VU (input) DOUBLE PRECISION
65: * If RANGE='V', the lower and upper bounds of the interval to
66: * be searched for eigenvalues. VL < VU.
67: * Not referenced if RANGE = 'A' or 'I'.
68: *
69: * IL (input) INTEGER
70: * IU (input) INTEGER
71: * If RANGE='I', the indices (in ascending order) of the
72: * smallest and largest eigenvalues to be returned.
73: * 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
74: * Not referenced if RANGE = 'A' or 'V'.
75: *
76: * ABSTOL (input) DOUBLE PRECISION
77: * The absolute error tolerance for the eigenvalues.
78: * An approximate eigenvalue is accepted as converged
79: * when it is determined to lie in an interval [a,b]
80: * of width less than or equal to
81: *
82: * ABSTOL + EPS * max( |a|,|b| ) ,
83: *
84: * where EPS is the machine precision. If ABSTOL is less than
85: * or equal to zero, then EPS*|T| will be used in its place,
86: * where |T| is the 1-norm of the tridiagonal matrix obtained
87: * by reducing AP to tridiagonal form.
88: *
89: * Eigenvalues will be computed most accurately when ABSTOL is
90: * set to twice the underflow threshold 2*DLAMCH('S'), not zero.
91: * If this routine returns with INFO>0, indicating that some
92: * eigenvectors did not converge, try setting ABSTOL to
93: * 2*DLAMCH('S').
94: *
95: * See "Computing Small Singular Values of Bidiagonal Matrices
96: * with Guaranteed High Relative Accuracy," by Demmel and
97: * Kahan, LAPACK Working Note #3.
98: *
99: * M (output) INTEGER
100: * The total number of eigenvalues found. 0 <= M <= N.
101: * If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
102: *
103: * W (output) DOUBLE PRECISION array, dimension (N)
104: * If INFO = 0, the selected eigenvalues in ascending order.
105: *
106: * Z (output) COMPLEX*16 array, dimension (LDZ, max(1,M))
107: * If JOBZ = 'V', then if INFO = 0, the first M columns of Z
108: * contain the orthonormal eigenvectors of the matrix A
109: * corresponding to the selected eigenvalues, with the i-th
110: * column of Z holding the eigenvector associated with W(i).
111: * If an eigenvector fails to converge, then that column of Z
112: * contains the latest approximation to the eigenvector, and
113: * the index of the eigenvector is returned in IFAIL.
114: * If JOBZ = 'N', then Z is not referenced.
115: * Note: the user must ensure that at least max(1,M) columns are
116: * supplied in the array Z; if RANGE = 'V', the exact value of M
117: * is not known in advance and an upper bound must be used.
118: *
119: * LDZ (input) INTEGER
120: * The leading dimension of the array Z. LDZ >= 1, and if
121: * JOBZ = 'V', LDZ >= max(1,N).
122: *
123: * WORK (workspace) COMPLEX*16 array, dimension (2*N)
124: *
125: * RWORK (workspace) DOUBLE PRECISION array, dimension (7*N)
126: *
127: * IWORK (workspace) INTEGER array, dimension (5*N)
128: *
129: * IFAIL (output) INTEGER array, dimension (N)
130: * If JOBZ = 'V', then if INFO = 0, the first M elements of
131: * IFAIL are zero. If INFO > 0, then IFAIL contains the
132: * indices of the eigenvectors that failed to converge.
133: * If JOBZ = 'N', then IFAIL is not referenced.
134: *
135: * INFO (output) INTEGER
136: * = 0: successful exit
137: * < 0: if INFO = -i, the i-th argument had an illegal value
138: * > 0: if INFO = i, then i eigenvectors failed to converge.
139: * Their indices are stored in array IFAIL.
140: *
141: * =====================================================================
142: *
143: * .. Parameters ..
144: DOUBLE PRECISION ZERO, ONE
145: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
146: COMPLEX*16 CONE
147: PARAMETER ( CONE = ( 1.0D0, 0.0D0 ) )
148: * ..
149: * .. Local Scalars ..
150: LOGICAL ALLEIG, INDEIG, TEST, VALEIG, WANTZ
151: CHARACTER ORDER
152: INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
153: $ INDISP, INDIWK, INDRWK, INDTAU, INDWRK, ISCALE,
154: $ ITMP1, J, JJ, NSPLIT
155: DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
156: $ SIGMA, SMLNUM, TMP1, VLL, VUU
157: * ..
158: * .. External Functions ..
159: LOGICAL LSAME
160: DOUBLE PRECISION DLAMCH, ZLANHP
161: EXTERNAL LSAME, DLAMCH, ZLANHP
162: * ..
163: * .. External Subroutines ..
164: EXTERNAL DCOPY, DSCAL, DSTEBZ, DSTERF, XERBLA, ZDSCAL,
165: $ ZHPTRD, ZSTEIN, ZSTEQR, ZSWAP, ZUPGTR, ZUPMTR
166: * ..
167: * .. Intrinsic Functions ..
168: INTRINSIC DBLE, MAX, MIN, SQRT
169: * ..
170: * .. Executable Statements ..
171: *
172: * Test the input parameters.
173: *
174: WANTZ = LSAME( JOBZ, 'V' )
175: ALLEIG = LSAME( RANGE, 'A' )
176: VALEIG = LSAME( RANGE, 'V' )
177: INDEIG = LSAME( RANGE, 'I' )
178: *
179: INFO = 0
180: IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
181: INFO = -1
182: ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
183: INFO = -2
184: ELSE IF( .NOT.( LSAME( UPLO, 'L' ) .OR. LSAME( UPLO, 'U' ) ) )
185: $ THEN
186: INFO = -3
187: ELSE IF( N.LT.0 ) THEN
188: INFO = -4
189: ELSE
190: IF( VALEIG ) THEN
191: IF( N.GT.0 .AND. VU.LE.VL )
192: $ INFO = -7
193: ELSE IF( INDEIG ) THEN
194: IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
195: INFO = -8
196: ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
197: INFO = -9
198: END IF
199: END IF
200: END IF
201: IF( INFO.EQ.0 ) THEN
202: IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) )
203: $ INFO = -14
204: END IF
205: *
206: IF( INFO.NE.0 ) THEN
207: CALL XERBLA( 'ZHPEVX', -INFO )
208: RETURN
209: END IF
210: *
211: * Quick return if possible
212: *
213: M = 0
214: IF( N.EQ.0 )
215: $ RETURN
216: *
217: IF( N.EQ.1 ) THEN
218: IF( ALLEIG .OR. INDEIG ) THEN
219: M = 1
220: W( 1 ) = AP( 1 )
221: ELSE
222: IF( VL.LT.DBLE( AP( 1 ) ) .AND. VU.GE.DBLE( AP( 1 ) ) ) THEN
223: M = 1
224: W( 1 ) = AP( 1 )
225: END IF
226: END IF
227: IF( WANTZ )
228: $ Z( 1, 1 ) = CONE
229: RETURN
230: END IF
231: *
232: * Get machine constants.
233: *
234: SAFMIN = DLAMCH( 'Safe minimum' )
235: EPS = DLAMCH( 'Precision' )
236: SMLNUM = SAFMIN / EPS
237: BIGNUM = ONE / SMLNUM
238: RMIN = SQRT( SMLNUM )
239: RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
240: *
241: * Scale matrix to allowable range, if necessary.
242: *
243: ISCALE = 0
244: ABSTLL = ABSTOL
245: IF( VALEIG ) THEN
246: VLL = VL
247: VUU = VU
248: ELSE
249: VLL = ZERO
250: VUU = ZERO
251: END IF
252: ANRM = ZLANHP( 'M', UPLO, N, AP, RWORK )
253: IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
254: ISCALE = 1
255: SIGMA = RMIN / ANRM
256: ELSE IF( ANRM.GT.RMAX ) THEN
257: ISCALE = 1
258: SIGMA = RMAX / ANRM
259: END IF
260: IF( ISCALE.EQ.1 ) THEN
261: CALL ZDSCAL( ( N*( N+1 ) ) / 2, SIGMA, AP, 1 )
262: IF( ABSTOL.GT.0 )
263: $ ABSTLL = ABSTOL*SIGMA
264: IF( VALEIG ) THEN
265: VLL = VL*SIGMA
266: VUU = VU*SIGMA
267: END IF
268: END IF
269: *
270: * Call ZHPTRD to reduce Hermitian packed matrix to tridiagonal form.
271: *
272: INDD = 1
273: INDE = INDD + N
274: INDRWK = INDE + N
275: INDTAU = 1
276: INDWRK = INDTAU + N
277: CALL ZHPTRD( UPLO, N, AP, RWORK( INDD ), RWORK( INDE ),
278: $ WORK( INDTAU ), IINFO )
279: *
280: * If all eigenvalues are desired and ABSTOL is less than or equal
281: * to zero, then call DSTERF or ZUPGTR and ZSTEQR. If this fails
282: * for some eigenvalue, then try DSTEBZ.
283: *
284: TEST = .FALSE.
285: IF (INDEIG) THEN
286: IF (IL.EQ.1 .AND. IU.EQ.N) THEN
287: TEST = .TRUE.
288: END IF
289: END IF
290: IF ((ALLEIG .OR. TEST) .AND. (ABSTOL.LE.ZERO)) THEN
291: CALL DCOPY( N, RWORK( INDD ), 1, W, 1 )
292: INDEE = INDRWK + 2*N
293: IF( .NOT.WANTZ ) THEN
294: CALL DCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
295: CALL DSTERF( N, W, RWORK( INDEE ), INFO )
296: ELSE
297: CALL ZUPGTR( UPLO, N, AP, WORK( INDTAU ), Z, LDZ,
298: $ WORK( INDWRK ), IINFO )
299: CALL DCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
300: CALL ZSTEQR( JOBZ, N, W, RWORK( INDEE ), Z, LDZ,
301: $ RWORK( INDRWK ), INFO )
302: IF( INFO.EQ.0 ) THEN
303: DO 10 I = 1, N
304: IFAIL( I ) = 0
305: 10 CONTINUE
306: END IF
307: END IF
308: IF( INFO.EQ.0 ) THEN
309: M = N
310: GO TO 20
311: END IF
312: INFO = 0
313: END IF
314: *
315: * Otherwise, call DSTEBZ and, if eigenvectors are desired, ZSTEIN.
316: *
317: IF( WANTZ ) THEN
318: ORDER = 'B'
319: ELSE
320: ORDER = 'E'
321: END IF
322: INDIBL = 1
323: INDISP = INDIBL + N
324: INDIWK = INDISP + N
325: CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
326: $ RWORK( INDD ), RWORK( INDE ), M, NSPLIT, W,
327: $ IWORK( INDIBL ), IWORK( INDISP ), RWORK( INDRWK ),
328: $ IWORK( INDIWK ), INFO )
329: *
330: IF( WANTZ ) THEN
331: CALL ZSTEIN( N, RWORK( INDD ), RWORK( INDE ), M, W,
332: $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
333: $ RWORK( INDRWK ), IWORK( INDIWK ), IFAIL, INFO )
334: *
335: * Apply unitary matrix used in reduction to tridiagonal
336: * form to eigenvectors returned by ZSTEIN.
337: *
338: INDWRK = INDTAU + N
339: CALL ZUPMTR( 'L', UPLO, 'N', N, M, AP, WORK( INDTAU ), Z, LDZ,
340: $ WORK( INDWRK ), IINFO )
341: END IF
342: *
343: * If matrix was scaled, then rescale eigenvalues appropriately.
344: *
345: 20 CONTINUE
346: IF( ISCALE.EQ.1 ) THEN
347: IF( INFO.EQ.0 ) THEN
348: IMAX = M
349: ELSE
350: IMAX = INFO - 1
351: END IF
352: CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
353: END IF
354: *
355: * If eigenvalues are not in order, then sort them, along with
356: * eigenvectors.
357: *
358: IF( WANTZ ) THEN
359: DO 40 J = 1, M - 1
360: I = 0
361: TMP1 = W( J )
362: DO 30 JJ = J + 1, M
363: IF( W( JJ ).LT.TMP1 ) THEN
364: I = JJ
365: TMP1 = W( JJ )
366: END IF
367: 30 CONTINUE
368: *
369: IF( I.NE.0 ) THEN
370: ITMP1 = IWORK( INDIBL+I-1 )
371: W( I ) = W( J )
372: IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
373: W( J ) = TMP1
374: IWORK( INDIBL+J-1 ) = ITMP1
375: CALL ZSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
376: IF( INFO.NE.0 ) THEN
377: ITMP1 = IFAIL( I )
378: IFAIL( I ) = IFAIL( J )
379: IFAIL( J ) = ITMP1
380: END IF
381: END IF
382: 40 CONTINUE
383: END IF
384: *
385: RETURN
386: *
387: * End of ZHPEVX
388: *
389: END
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