1: *> \brief <b> DSBEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DSBEVD + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsbevd.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsbevd.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsbevd.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DSBEVD( JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK,
22: * LWORK, IWORK, LIWORK, INFO )
23: *
24: * .. Scalar Arguments ..
25: * CHARACTER JOBZ, UPLO
26: * INTEGER INFO, KD, LDAB, LDZ, LIWORK, LWORK, N
27: * ..
28: * .. Array Arguments ..
29: * INTEGER IWORK( * )
30: * DOUBLE PRECISION AB( LDAB, * ), W( * ), WORK( * ), Z( LDZ, * )
31: * ..
32: *
33: *
34: *> \par Purpose:
35: * =============
36: *>
37: *> \verbatim
38: *>
39: *> DSBEVD computes all the eigenvalues and, optionally, eigenvectors of
40: *> a real symmetric band matrix A. If eigenvectors are desired, it uses
41: *> a divide and conquer algorithm.
42: *>
43: *> The divide and conquer algorithm makes very mild assumptions about
44: *> floating point arithmetic. It will work on machines with a guard
45: *> digit in add/subtract, or on those binary machines without guard
46: *> digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
47: *> Cray-2. It could conceivably fail on hexadecimal or decimal machines
48: *> without guard digits, but we know of none.
49: *> \endverbatim
50: *
51: * Arguments:
52: * ==========
53: *
54: *> \param[in] JOBZ
55: *> \verbatim
56: *> JOBZ is CHARACTER*1
57: *> = 'N': Compute eigenvalues only;
58: *> = 'V': Compute eigenvalues and eigenvectors.
59: *> \endverbatim
60: *>
61: *> \param[in] UPLO
62: *> \verbatim
63: *> UPLO is CHARACTER*1
64: *> = 'U': Upper triangle of A is stored;
65: *> = 'L': Lower triangle of A is stored.
66: *> \endverbatim
67: *>
68: *> \param[in] N
69: *> \verbatim
70: *> N is INTEGER
71: *> The order of the matrix A. N >= 0.
72: *> \endverbatim
73: *>
74: *> \param[in] KD
75: *> \verbatim
76: *> KD is INTEGER
77: *> The number of superdiagonals of the matrix A if UPLO = 'U',
78: *> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
79: *> \endverbatim
80: *>
81: *> \param[in,out] AB
82: *> \verbatim
83: *> AB is DOUBLE PRECISION array, dimension (LDAB, N)
84: *> On entry, the upper or lower triangle of the symmetric band
85: *> matrix A, stored in the first KD+1 rows of the array. The
86: *> j-th column of A is stored in the j-th column of the array AB
87: *> as follows:
88: *> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
89: *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
90: *>
91: *> On exit, AB is overwritten by values generated during the
92: *> reduction to tridiagonal form. If UPLO = 'U', the first
93: *> superdiagonal and the diagonal of the tridiagonal matrix T
94: *> are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
95: *> the diagonal and first subdiagonal of T are returned in the
96: *> first two rows of AB.
97: *> \endverbatim
98: *>
99: *> \param[in] LDAB
100: *> \verbatim
101: *> LDAB is INTEGER
102: *> The leading dimension of the array AB. LDAB >= KD + 1.
103: *> \endverbatim
104: *>
105: *> \param[out] W
106: *> \verbatim
107: *> W is DOUBLE PRECISION array, dimension (N)
108: *> If INFO = 0, the eigenvalues in ascending order.
109: *> \endverbatim
110: *>
111: *> \param[out] Z
112: *> \verbatim
113: *> Z is DOUBLE PRECISION array, dimension (LDZ, N)
114: *> If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
115: *> eigenvectors of the matrix A, with the i-th column of Z
116: *> holding the eigenvector associated with W(i).
117: *> If JOBZ = 'N', then Z is not referenced.
118: *> \endverbatim
119: *>
120: *> \param[in] LDZ
121: *> \verbatim
122: *> LDZ is INTEGER
123: *> The leading dimension of the array Z. LDZ >= 1, and if
124: *> JOBZ = 'V', LDZ >= max(1,N).
125: *> \endverbatim
126: *>
127: *> \param[out] WORK
128: *> \verbatim
129: *> WORK is DOUBLE PRECISION array,
130: *> dimension (LWORK)
131: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
132: *> \endverbatim
133: *>
134: *> \param[in] LWORK
135: *> \verbatim
136: *> LWORK is INTEGER
137: *> The dimension of the array WORK.
138: *> IF N <= 1, LWORK must be at least 1.
139: *> If JOBZ = 'N' and N > 2, LWORK must be at least 2*N.
140: *> If JOBZ = 'V' and N > 2, LWORK must be at least
141: *> ( 1 + 5*N + 2*N**2 ).
142: *>
143: *> If LWORK = -1, then a workspace query is assumed; the routine
144: *> only calculates the optimal sizes of the WORK and IWORK
145: *> arrays, returns these values as the first entries of the WORK
146: *> and IWORK arrays, and no error message related to LWORK or
147: *> LIWORK is issued by XERBLA.
148: *> \endverbatim
149: *>
150: *> \param[out] IWORK
151: *> \verbatim
152: *> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
153: *> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
154: *> \endverbatim
155: *>
156: *> \param[in] LIWORK
157: *> \verbatim
158: *> LIWORK is INTEGER
159: *> The dimension of the array IWORK.
160: *> If JOBZ = 'N' or N <= 1, LIWORK must be at least 1.
161: *> If JOBZ = 'V' and N > 2, LIWORK must be at least 3 + 5*N.
162: *>
163: *> If LIWORK = -1, then a workspace query is assumed; the
164: *> routine only calculates the optimal sizes of the WORK and
165: *> IWORK arrays, returns these values as the first entries of
166: *> the WORK and IWORK arrays, and no error message related to
167: *> LWORK or LIWORK is issued by XERBLA.
168: *> \endverbatim
169: *>
170: *> \param[out] INFO
171: *> \verbatim
172: *> INFO is INTEGER
173: *> = 0: successful exit
174: *> < 0: if INFO = -i, the i-th argument had an illegal value
175: *> > 0: if INFO = i, the algorithm failed to converge; i
176: *> off-diagonal elements of an intermediate tridiagonal
177: *> form did not converge to zero.
178: *> \endverbatim
179: *
180: * Authors:
181: * ========
182: *
183: *> \author Univ. of Tennessee
184: *> \author Univ. of California Berkeley
185: *> \author Univ. of Colorado Denver
186: *> \author NAG Ltd.
187: *
188: *> \ingroup doubleOTHEReigen
189: *
190: * =====================================================================
191: SUBROUTINE DSBEVD( JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK,
192: $ LWORK, IWORK, LIWORK, INFO )
193: *
194: * -- LAPACK driver routine --
195: * -- LAPACK is a software package provided by Univ. of Tennessee, --
196: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
197: *
198: * .. Scalar Arguments ..
199: CHARACTER JOBZ, UPLO
200: INTEGER INFO, KD, LDAB, LDZ, LIWORK, LWORK, N
201: * ..
202: * .. Array Arguments ..
203: INTEGER IWORK( * )
204: DOUBLE PRECISION AB( LDAB, * ), W( * ), WORK( * ), Z( LDZ, * )
205: * ..
206: *
207: * =====================================================================
208: *
209: * .. Parameters ..
210: DOUBLE PRECISION ZERO, ONE
211: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
212: * ..
213: * .. Local Scalars ..
214: LOGICAL LOWER, LQUERY, WANTZ
215: INTEGER IINFO, INDE, INDWK2, INDWRK, ISCALE, LIWMIN,
216: $ LLWRK2, LWMIN
217: DOUBLE PRECISION ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
218: $ SMLNUM
219: * ..
220: * .. External Functions ..
221: LOGICAL LSAME
222: DOUBLE PRECISION DLAMCH, DLANSB
223: EXTERNAL LSAME, DLAMCH, DLANSB
224: * ..
225: * .. External Subroutines ..
226: EXTERNAL DGEMM, DLACPY, DLASCL, DSBTRD, DSCAL, DSTEDC,
227: $ DSTERF, XERBLA
228: * ..
229: * .. Intrinsic Functions ..
230: INTRINSIC SQRT
231: * ..
232: * .. Executable Statements ..
233: *
234: * Test the input parameters.
235: *
236: WANTZ = LSAME( JOBZ, 'V' )
237: LOWER = LSAME( UPLO, 'L' )
238: LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
239: *
240: INFO = 0
241: IF( N.LE.1 ) THEN
242: LIWMIN = 1
243: LWMIN = 1
244: ELSE
245: IF( WANTZ ) THEN
246: LIWMIN = 3 + 5*N
247: LWMIN = 1 + 5*N + 2*N**2
248: ELSE
249: LIWMIN = 1
250: LWMIN = 2*N
251: END IF
252: END IF
253: IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
254: INFO = -1
255: ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
256: INFO = -2
257: ELSE IF( N.LT.0 ) THEN
258: INFO = -3
259: ELSE IF( KD.LT.0 ) THEN
260: INFO = -4
261: ELSE IF( LDAB.LT.KD+1 ) THEN
262: INFO = -6
263: ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
264: INFO = -9
265: END IF
266: *
267: IF( INFO.EQ.0 ) THEN
268: WORK( 1 ) = LWMIN
269: IWORK( 1 ) = LIWMIN
270: *
271: IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
272: INFO = -11
273: ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
274: INFO = -13
275: END IF
276: END IF
277: *
278: IF( INFO.NE.0 ) THEN
279: CALL XERBLA( 'DSBEVD', -INFO )
280: RETURN
281: ELSE IF( LQUERY ) THEN
282: RETURN
283: END IF
284: *
285: * Quick return if possible
286: *
287: IF( N.EQ.0 )
288: $ RETURN
289: *
290: IF( N.EQ.1 ) THEN
291: W( 1 ) = AB( 1, 1 )
292: IF( WANTZ )
293: $ Z( 1, 1 ) = ONE
294: RETURN
295: END IF
296: *
297: * Get machine constants.
298: *
299: SAFMIN = DLAMCH( 'Safe minimum' )
300: EPS = DLAMCH( 'Precision' )
301: SMLNUM = SAFMIN / EPS
302: BIGNUM = ONE / SMLNUM
303: RMIN = SQRT( SMLNUM )
304: RMAX = SQRT( BIGNUM )
305: *
306: * Scale matrix to allowable range, if necessary.
307: *
308: ANRM = DLANSB( 'M', UPLO, N, KD, AB, LDAB, WORK )
309: ISCALE = 0
310: IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
311: ISCALE = 1
312: SIGMA = RMIN / ANRM
313: ELSE IF( ANRM.GT.RMAX ) THEN
314: ISCALE = 1
315: SIGMA = RMAX / ANRM
316: END IF
317: IF( ISCALE.EQ.1 ) THEN
318: IF( LOWER ) THEN
319: CALL DLASCL( 'B', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
320: ELSE
321: CALL DLASCL( 'Q', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
322: END IF
323: END IF
324: *
325: * Call DSBTRD to reduce symmetric band matrix to tridiagonal form.
326: *
327: INDE = 1
328: INDWRK = INDE + N
329: INDWK2 = INDWRK + N*N
330: LLWRK2 = LWORK - INDWK2 + 1
331: CALL DSBTRD( JOBZ, UPLO, N, KD, AB, LDAB, W, WORK( INDE ), Z, LDZ,
332: $ WORK( INDWRK ), IINFO )
333: *
334: * For eigenvalues only, call DSTERF. For eigenvectors, call SSTEDC.
335: *
336: IF( .NOT.WANTZ ) THEN
337: CALL DSTERF( N, W, WORK( INDE ), INFO )
338: ELSE
339: CALL DSTEDC( 'I', N, W, WORK( INDE ), WORK( INDWRK ), N,
340: $ WORK( INDWK2 ), LLWRK2, IWORK, LIWORK, INFO )
341: CALL DGEMM( 'N', 'N', N, N, N, ONE, Z, LDZ, WORK( INDWRK ), N,
342: $ ZERO, WORK( INDWK2 ), N )
343: CALL DLACPY( 'A', N, N, WORK( INDWK2 ), N, Z, LDZ )
344: END IF
345: *
346: * If matrix was scaled, then rescale eigenvalues appropriately.
347: *
348: IF( ISCALE.EQ.1 )
349: $ CALL DSCAL( N, ONE / SIGMA, W, 1 )
350: *
351: WORK( 1 ) = LWMIN
352: IWORK( 1 ) = LIWMIN
353: RETURN
354: *
355: * End of DSBEVD
356: *
357: END
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