1: SUBROUTINE DSBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL,
2: $ VU, 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, IU, KD, LDAB, LDQ, LDZ, M, N
13: DOUBLE PRECISION ABSTOL, VL, VU
14: * ..
15: * .. Array Arguments ..
16: INTEGER IFAIL( * ), IWORK( * )
17: DOUBLE PRECISION AB( LDAB, * ), Q( LDQ, * ), W( * ), WORK( * ),
18: $ Z( LDZ, * )
19: * ..
20: *
21: * Purpose
22: * =======
23: *
24: * DSBEVX computes selected eigenvalues and, optionally, eigenvectors
25: * of a real symmetric band matrix A. Eigenvalues and eigenvectors can
26: * be selected by specifying either a range of values or a range of
27: * 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: * KD (input) INTEGER
50: * The number of superdiagonals of the matrix A if UPLO = 'U',
51: * or the number of subdiagonals if UPLO = 'L'. KD >= 0.
52: *
53: * AB (input/output) DOUBLE PRECISION array, dimension (LDAB, N)
54: * On entry, the upper or lower triangle of the symmetric band
55: * matrix A, stored in the first KD+1 rows of the array. The
56: * j-th column of A is stored in the j-th column of the array AB
57: * as follows:
58: * if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
59: * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
60: *
61: * On exit, AB is overwritten by values generated during the
62: * reduction to tridiagonal form. If UPLO = 'U', the first
63: * superdiagonal and the diagonal of the tridiagonal matrix T
64: * are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
65: * the diagonal and first subdiagonal of T are returned in the
66: * first two rows of AB.
67: *
68: * LDAB (input) INTEGER
69: * The leading dimension of the array AB. LDAB >= KD + 1.
70: *
71: * Q (output) DOUBLE PRECISION array, dimension (LDQ, N)
72: * If JOBZ = 'V', the N-by-N orthogonal matrix used in the
73: * reduction to tridiagonal form.
74: * If JOBZ = 'N', the array Q is not referenced.
75: *
76: * LDQ (input) INTEGER
77: * The leading dimension of the array Q. If JOBZ = 'V', then
78: * LDQ >= max(1,N).
79: *
80: * VL (input) DOUBLE PRECISION
81: * VU (input) DOUBLE PRECISION
82: * If RANGE='V', the lower and upper bounds of the interval to
83: * be searched for eigenvalues. VL < VU.
84: * Not referenced if RANGE = 'A' or 'I'.
85: *
86: * IL (input) INTEGER
87: * IU (input) INTEGER
88: * If RANGE='I', the indices (in ascending order) of the
89: * smallest and largest eigenvalues to be returned.
90: * 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
91: * Not referenced if RANGE = 'A' or 'V'.
92: *
93: * ABSTOL (input) DOUBLE PRECISION
94: * The absolute error tolerance for the eigenvalues.
95: * An approximate eigenvalue is accepted as converged
96: * when it is determined to lie in an interval [a,b]
97: * of width less than or equal to
98: *
99: * ABSTOL + EPS * max( |a|,|b| ) ,
100: *
101: * where EPS is the machine precision. If ABSTOL is less than
102: * or equal to zero, then EPS*|T| will be used in its place,
103: * where |T| is the 1-norm of the tridiagonal matrix obtained
104: * by reducing AB to tridiagonal form.
105: *
106: * Eigenvalues will be computed most accurately when ABSTOL is
107: * set to twice the underflow threshold 2*DLAMCH('S'), not zero.
108: * If this routine returns with INFO>0, indicating that some
109: * eigenvectors did not converge, try setting ABSTOL to
110: * 2*DLAMCH('S').
111: *
112: * See "Computing Small Singular Values of Bidiagonal Matrices
113: * with Guaranteed High Relative Accuracy," by Demmel and
114: * Kahan, LAPACK Working Note #3.
115: *
116: * M (output) INTEGER
117: * The total number of eigenvalues found. 0 <= M <= N.
118: * If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
119: *
120: * W (output) DOUBLE PRECISION array, dimension (N)
121: * The first M elements contain the selected eigenvalues in
122: * ascending order.
123: *
124: * Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M))
125: * If JOBZ = 'V', then if INFO = 0, the first M columns of Z
126: * contain the orthonormal eigenvectors of the matrix A
127: * corresponding to the selected eigenvalues, with the i-th
128: * column of Z holding the eigenvector associated with W(i).
129: * If an eigenvector fails to converge, then that column of Z
130: * contains the latest approximation to the eigenvector, and the
131: * index of the eigenvector is returned in IFAIL.
132: * If JOBZ = 'N', then Z is not referenced.
133: * Note: the user must ensure that at least max(1,M) columns are
134: * supplied in the array Z; if RANGE = 'V', the exact value of M
135: * is not known in advance and an upper bound must be used.
136: *
137: * LDZ (input) INTEGER
138: * The leading dimension of the array Z. LDZ >= 1, and if
139: * JOBZ = 'V', LDZ >= max(1,N).
140: *
141: * WORK (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: if INFO = i, then i eigenvectors failed to converge.
155: * Their indices are stored in array IFAIL.
156: *
157: * =====================================================================
158: *
159: * .. Parameters ..
160: DOUBLE PRECISION ZERO, ONE
161: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
162: * ..
163: * .. Local Scalars ..
164: LOGICAL ALLEIG, INDEIG, LOWER, TEST, VALEIG, WANTZ
165: CHARACTER ORDER
166: INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
167: $ INDISP, INDIWO, INDWRK, ISCALE, ITMP1, J, JJ,
168: $ NSPLIT
169: DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
170: $ SIGMA, SMLNUM, TMP1, VLL, VUU
171: * ..
172: * .. External Functions ..
173: LOGICAL LSAME
174: DOUBLE PRECISION DLAMCH, DLANSB
175: EXTERNAL LSAME, DLAMCH, DLANSB
176: * ..
177: * .. External Subroutines ..
178: EXTERNAL DCOPY, DGEMV, DLACPY, DLASCL, DSBTRD, DSCAL,
179: $ DSTEBZ, DSTEIN, DSTEQR, DSTERF, DSWAP, XERBLA
180: * ..
181: * .. Intrinsic Functions ..
182: INTRINSIC MAX, MIN, SQRT
183: * ..
184: * .. Executable Statements ..
185: *
186: * Test the input parameters.
187: *
188: WANTZ = LSAME( JOBZ, 'V' )
189: ALLEIG = LSAME( RANGE, 'A' )
190: VALEIG = LSAME( RANGE, 'V' )
191: INDEIG = LSAME( RANGE, 'I' )
192: LOWER = LSAME( UPLO, 'L' )
193: *
194: INFO = 0
195: IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
196: INFO = -1
197: ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
198: INFO = -2
199: ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
200: INFO = -3
201: ELSE IF( N.LT.0 ) THEN
202: INFO = -4
203: ELSE IF( KD.LT.0 ) THEN
204: INFO = -5
205: ELSE IF( LDAB.LT.KD+1 ) THEN
206: INFO = -7
207: ELSE IF( WANTZ .AND. LDQ.LT.MAX( 1, N ) ) THEN
208: INFO = -9
209: ELSE
210: IF( VALEIG ) THEN
211: IF( N.GT.0 .AND. VU.LE.VL )
212: $ INFO = -11
213: ELSE IF( INDEIG ) THEN
214: IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
215: INFO = -12
216: ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
217: INFO = -13
218: END IF
219: END IF
220: END IF
221: IF( INFO.EQ.0 ) THEN
222: IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) )
223: $ INFO = -18
224: END IF
225: *
226: IF( INFO.NE.0 ) THEN
227: CALL XERBLA( 'DSBEVX', -INFO )
228: RETURN
229: END IF
230: *
231: * Quick return if possible
232: *
233: M = 0
234: IF( N.EQ.0 )
235: $ RETURN
236: *
237: IF( N.EQ.1 ) THEN
238: M = 1
239: IF( LOWER ) THEN
240: TMP1 = AB( 1, 1 )
241: ELSE
242: TMP1 = AB( KD+1, 1 )
243: END IF
244: IF( VALEIG ) THEN
245: IF( .NOT.( VL.LT.TMP1 .AND. VU.GE.TMP1 ) )
246: $ M = 0
247: END IF
248: IF( M.EQ.1 ) THEN
249: W( 1 ) = TMP1
250: IF( WANTZ )
251: $ Z( 1, 1 ) = ONE
252: END IF
253: RETURN
254: END IF
255: *
256: * Get machine constants.
257: *
258: SAFMIN = DLAMCH( 'Safe minimum' )
259: EPS = DLAMCH( 'Precision' )
260: SMLNUM = SAFMIN / EPS
261: BIGNUM = ONE / SMLNUM
262: RMIN = SQRT( SMLNUM )
263: RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
264: *
265: * Scale matrix to allowable range, if necessary.
266: *
267: ISCALE = 0
268: ABSTLL = ABSTOL
269: IF( VALEIG ) THEN
270: VLL = VL
271: VUU = VU
272: ELSE
273: VLL = ZERO
274: VUU = ZERO
275: END IF
276: ANRM = DLANSB( 'M', UPLO, N, KD, AB, LDAB, WORK )
277: IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
278: ISCALE = 1
279: SIGMA = RMIN / ANRM
280: ELSE IF( ANRM.GT.RMAX ) THEN
281: ISCALE = 1
282: SIGMA = RMAX / ANRM
283: END IF
284: IF( ISCALE.EQ.1 ) THEN
285: IF( LOWER ) THEN
286: CALL DLASCL( 'B', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
287: ELSE
288: CALL DLASCL( 'Q', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
289: END IF
290: IF( ABSTOL.GT.0 )
291: $ ABSTLL = ABSTOL*SIGMA
292: IF( VALEIG ) THEN
293: VLL = VL*SIGMA
294: VUU = VU*SIGMA
295: END IF
296: END IF
297: *
298: * Call DSBTRD to reduce symmetric band matrix to tridiagonal form.
299: *
300: INDD = 1
301: INDE = INDD + N
302: INDWRK = INDE + N
303: CALL DSBTRD( JOBZ, UPLO, N, KD, AB, LDAB, WORK( INDD ),
304: $ WORK( INDE ), Q, LDQ, WORK( INDWRK ), IINFO )
305: *
306: * If all eigenvalues are desired and ABSTOL is less than or equal
307: * to zero, then call DSTERF or SSTEQR. If this fails for some
308: * eigenvalue, then try DSTEBZ.
309: *
310: TEST = .FALSE.
311: IF (INDEIG) THEN
312: IF (IL.EQ.1 .AND. IU.EQ.N) THEN
313: TEST = .TRUE.
314: END IF
315: END IF
316: IF ((ALLEIG .OR. TEST) .AND. (ABSTOL.LE.ZERO)) THEN
317: CALL DCOPY( N, WORK( INDD ), 1, W, 1 )
318: INDEE = INDWRK + 2*N
319: IF( .NOT.WANTZ ) THEN
320: CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
321: CALL DSTERF( N, W, WORK( INDEE ), INFO )
322: ELSE
323: CALL DLACPY( 'A', N, N, Q, LDQ, Z, LDZ )
324: CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
325: CALL DSTEQR( JOBZ, N, W, WORK( INDEE ), Z, LDZ,
326: $ WORK( INDWRK ), INFO )
327: IF( INFO.EQ.0 ) THEN
328: DO 10 I = 1, N
329: IFAIL( I ) = 0
330: 10 CONTINUE
331: END IF
332: END IF
333: IF( INFO.EQ.0 ) THEN
334: M = N
335: GO TO 30
336: END IF
337: INFO = 0
338: END IF
339: *
340: * Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN.
341: *
342: IF( WANTZ ) THEN
343: ORDER = 'B'
344: ELSE
345: ORDER = 'E'
346: END IF
347: INDIBL = 1
348: INDISP = INDIBL + N
349: INDIWO = INDISP + N
350: CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
351: $ WORK( INDD ), WORK( INDE ), M, NSPLIT, W,
352: $ IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWRK ),
353: $ IWORK( INDIWO ), INFO )
354: *
355: IF( WANTZ ) THEN
356: CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W,
357: $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
358: $ WORK( INDWRK ), IWORK( INDIWO ), IFAIL, INFO )
359: *
360: * Apply orthogonal matrix used in reduction to tridiagonal
361: * form to eigenvectors returned by DSTEIN.
362: *
363: DO 20 J = 1, M
364: CALL DCOPY( N, Z( 1, J ), 1, WORK( 1 ), 1 )
365: CALL DGEMV( 'N', N, N, ONE, Q, LDQ, WORK, 1, ZERO,
366: $ Z( 1, J ), 1 )
367: 20 CONTINUE
368: END IF
369: *
370: * If matrix was scaled, then rescale eigenvalues appropriately.
371: *
372: 30 CONTINUE
373: IF( ISCALE.EQ.1 ) THEN
374: IF( INFO.EQ.0 ) THEN
375: IMAX = M
376: ELSE
377: IMAX = INFO - 1
378: END IF
379: CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
380: END IF
381: *
382: * If eigenvalues are not in order, then sort them, along with
383: * eigenvectors.
384: *
385: IF( WANTZ ) THEN
386: DO 50 J = 1, M - 1
387: I = 0
388: TMP1 = W( J )
389: DO 40 JJ = J + 1, M
390: IF( W( JJ ).LT.TMP1 ) THEN
391: I = JJ
392: TMP1 = W( JJ )
393: END IF
394: 40 CONTINUE
395: *
396: IF( I.NE.0 ) THEN
397: ITMP1 = IWORK( INDIBL+I-1 )
398: W( I ) = W( J )
399: IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
400: W( J ) = TMP1
401: IWORK( INDIBL+J-1 ) = ITMP1
402: CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
403: IF( INFO.NE.0 ) THEN
404: ITMP1 = IFAIL( I )
405: IFAIL( I ) = IFAIL( J )
406: IFAIL( J ) = ITMP1
407: END IF
408: END IF
409: 50 CONTINUE
410: END IF
411: *
412: RETURN
413: *
414: * End of DSBEVX
415: *
416: END
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