Annotation of rpl/lapack/lapack/zhbevd.f, revision 1.8
1.8 ! bertrand 1: *> \brief <b> ZHBEVD 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 ZHBEVD + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zhbevd.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zhbevd.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zhbevd.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE ZHBEVD( JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK,
! 22: * LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO )
! 23: *
! 24: * .. Scalar Arguments ..
! 25: * CHARACTER JOBZ, UPLO
! 26: * INTEGER INFO, KD, LDAB, LDZ, LIWORK, LRWORK, LWORK, N
! 27: * ..
! 28: * .. Array Arguments ..
! 29: * INTEGER IWORK( * )
! 30: * DOUBLE PRECISION RWORK( * ), W( * )
! 31: * COMPLEX*16 AB( LDAB, * ), WORK( * ), Z( LDZ, * )
! 32: * ..
! 33: *
! 34: *
! 35: *> \par Purpose:
! 36: * =============
! 37: *>
! 38: *> \verbatim
! 39: *>
! 40: *> ZHBEVD computes all the eigenvalues and, optionally, eigenvectors of
! 41: *> a complex Hermitian band matrix A. If eigenvectors are desired, it
! 42: *> uses a divide and conquer algorithm.
! 43: *>
! 44: *> The divide and conquer algorithm makes very mild assumptions about
! 45: *> floating point arithmetic. It will work on machines with a guard
! 46: *> digit in add/subtract, or on those binary machines without guard
! 47: *> digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
! 48: *> Cray-2. It could conceivably fail on hexadecimal or decimal machines
! 49: *> without guard digits, but we know of none.
! 50: *> \endverbatim
! 51: *
! 52: * Arguments:
! 53: * ==========
! 54: *
! 55: *> \param[in] JOBZ
! 56: *> \verbatim
! 57: *> JOBZ is CHARACTER*1
! 58: *> = 'N': Compute eigenvalues only;
! 59: *> = 'V': Compute eigenvalues and eigenvectors.
! 60: *> \endverbatim
! 61: *>
! 62: *> \param[in] UPLO
! 63: *> \verbatim
! 64: *> UPLO is CHARACTER*1
! 65: *> = 'U': Upper triangle of A is stored;
! 66: *> = 'L': Lower triangle of A is stored.
! 67: *> \endverbatim
! 68: *>
! 69: *> \param[in] N
! 70: *> \verbatim
! 71: *> N is INTEGER
! 72: *> The order of the matrix A. N >= 0.
! 73: *> \endverbatim
! 74: *>
! 75: *> \param[in] KD
! 76: *> \verbatim
! 77: *> KD is INTEGER
! 78: *> The number of superdiagonals of the matrix A if UPLO = 'U',
! 79: *> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
! 80: *> \endverbatim
! 81: *>
! 82: *> \param[in,out] AB
! 83: *> \verbatim
! 84: *> AB is COMPLEX*16 array, dimension (LDAB, N)
! 85: *> On entry, the upper or lower triangle of the Hermitian band
! 86: *> matrix A, stored in the first KD+1 rows of the array. The
! 87: *> j-th column of A is stored in the j-th column of the array AB
! 88: *> as follows:
! 89: *> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
! 90: *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
! 91: *>
! 92: *> On exit, AB is overwritten by values generated during the
! 93: *> reduction to tridiagonal form. If UPLO = 'U', the first
! 94: *> superdiagonal and the diagonal of the tridiagonal matrix T
! 95: *> are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
! 96: *> the diagonal and first subdiagonal of T are returned in the
! 97: *> first two rows of AB.
! 98: *> \endverbatim
! 99: *>
! 100: *> \param[in] LDAB
! 101: *> \verbatim
! 102: *> LDAB is INTEGER
! 103: *> The leading dimension of the array AB. LDAB >= KD + 1.
! 104: *> \endverbatim
! 105: *>
! 106: *> \param[out] W
! 107: *> \verbatim
! 108: *> W is DOUBLE PRECISION array, dimension (N)
! 109: *> If INFO = 0, the eigenvalues in ascending order.
! 110: *> \endverbatim
! 111: *>
! 112: *> \param[out] Z
! 113: *> \verbatim
! 114: *> Z is COMPLEX*16 array, dimension (LDZ, N)
! 115: *> If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
! 116: *> eigenvectors of the matrix A, with the i-th column of Z
! 117: *> holding the eigenvector associated with W(i).
! 118: *> If JOBZ = 'N', then Z is not referenced.
! 119: *> \endverbatim
! 120: *>
! 121: *> \param[in] LDZ
! 122: *> \verbatim
! 123: *> LDZ is INTEGER
! 124: *> The leading dimension of the array Z. LDZ >= 1, and if
! 125: *> JOBZ = 'V', LDZ >= max(1,N).
! 126: *> \endverbatim
! 127: *>
! 128: *> \param[out] WORK
! 129: *> \verbatim
! 130: *> WORK is COMPLEX*16 array, dimension (MAX(1,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 > 1, LWORK must be at least N.
! 140: *> If JOBZ = 'V' and N > 1, LWORK must be at least 2*N**2.
! 141: *>
! 142: *> If LWORK = -1, then a workspace query is assumed; the routine
! 143: *> only calculates the optimal sizes of the WORK, RWORK and
! 144: *> IWORK arrays, returns these values as the first entries of
! 145: *> the WORK, RWORK and IWORK arrays, and no error message
! 146: *> related to LWORK or LRWORK or LIWORK is issued by XERBLA.
! 147: *> \endverbatim
! 148: *>
! 149: *> \param[out] RWORK
! 150: *> \verbatim
! 151: *> RWORK is DOUBLE PRECISION array,
! 152: *> dimension (LRWORK)
! 153: *> On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.
! 154: *> \endverbatim
! 155: *>
! 156: *> \param[in] LRWORK
! 157: *> \verbatim
! 158: *> LRWORK is INTEGER
! 159: *> The dimension of array RWORK.
! 160: *> If N <= 1, LRWORK must be at least 1.
! 161: *> If JOBZ = 'N' and N > 1, LRWORK must be at least N.
! 162: *> If JOBZ = 'V' and N > 1, LRWORK must be at least
! 163: *> 1 + 5*N + 2*N**2.
! 164: *>
! 165: *> If LRWORK = -1, then a workspace query is assumed; the
! 166: *> routine only calculates the optimal sizes of the WORK, RWORK
! 167: *> and IWORK arrays, returns these values as the first entries
! 168: *> of the WORK, RWORK and IWORK arrays, and no error message
! 169: *> related to LWORK or LRWORK or LIWORK is issued by XERBLA.
! 170: *> \endverbatim
! 171: *>
! 172: *> \param[out] IWORK
! 173: *> \verbatim
! 174: *> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
! 175: *> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
! 176: *> \endverbatim
! 177: *>
! 178: *> \param[in] LIWORK
! 179: *> \verbatim
! 180: *> LIWORK is INTEGER
! 181: *> The dimension of array IWORK.
! 182: *> If JOBZ = 'N' or N <= 1, LIWORK must be at least 1.
! 183: *> If JOBZ = 'V' and N > 1, LIWORK must be at least 3 + 5*N .
! 184: *>
! 185: *> If LIWORK = -1, then a workspace query is assumed; the
! 186: *> routine only calculates the optimal sizes of the WORK, RWORK
! 187: *> and IWORK arrays, returns these values as the first entries
! 188: *> of the WORK, RWORK and IWORK arrays, and no error message
! 189: *> related to LWORK or LRWORK or LIWORK is issued by XERBLA.
! 190: *> \endverbatim
! 191: *>
! 192: *> \param[out] INFO
! 193: *> \verbatim
! 194: *> INFO is INTEGER
! 195: *> = 0: successful exit.
! 196: *> < 0: if INFO = -i, the i-th argument had an illegal value.
! 197: *> > 0: if INFO = i, the algorithm failed to converge; i
! 198: *> off-diagonal elements of an intermediate tridiagonal
! 199: *> form did not converge to zero.
! 200: *> \endverbatim
! 201: *
! 202: * Authors:
! 203: * ========
! 204: *
! 205: *> \author Univ. of Tennessee
! 206: *> \author Univ. of California Berkeley
! 207: *> \author Univ. of Colorado Denver
! 208: *> \author NAG Ltd.
! 209: *
! 210: *> \date November 2011
! 211: *
! 212: *> \ingroup complex16OTHEReigen
! 213: *
! 214: * =====================================================================
1.1 bertrand 215: SUBROUTINE ZHBEVD( JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK,
216: $ LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO )
217: *
1.8 ! bertrand 218: * -- LAPACK driver routine (version 3.4.0) --
1.1 bertrand 219: * -- LAPACK is a software package provided by Univ. of Tennessee, --
220: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.8 ! bertrand 221: * November 2011
1.1 bertrand 222: *
223: * .. Scalar Arguments ..
224: CHARACTER JOBZ, UPLO
225: INTEGER INFO, KD, LDAB, LDZ, LIWORK, LRWORK, LWORK, N
226: * ..
227: * .. Array Arguments ..
228: INTEGER IWORK( * )
229: DOUBLE PRECISION RWORK( * ), W( * )
230: COMPLEX*16 AB( LDAB, * ), WORK( * ), Z( LDZ, * )
231: * ..
232: *
233: * =====================================================================
234: *
235: * .. Parameters ..
236: DOUBLE PRECISION ZERO, ONE
237: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
238: COMPLEX*16 CZERO, CONE
239: PARAMETER ( CZERO = ( 0.0D0, 0.0D0 ),
240: $ CONE = ( 1.0D0, 0.0D0 ) )
241: * ..
242: * .. Local Scalars ..
243: LOGICAL LOWER, LQUERY, WANTZ
244: INTEGER IINFO, IMAX, INDE, INDWK2, INDWRK, ISCALE,
245: $ LIWMIN, LLRWK, LLWK2, LRWMIN, LWMIN
246: DOUBLE PRECISION ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
247: $ SMLNUM
248: * ..
249: * .. External Functions ..
250: LOGICAL LSAME
251: DOUBLE PRECISION DLAMCH, ZLANHB
252: EXTERNAL LSAME, DLAMCH, ZLANHB
253: * ..
254: * .. External Subroutines ..
255: EXTERNAL DSCAL, DSTERF, XERBLA, ZGEMM, ZHBTRD, ZLACPY,
256: $ ZLASCL, ZSTEDC
257: * ..
258: * .. Intrinsic Functions ..
259: INTRINSIC SQRT
260: * ..
261: * .. Executable Statements ..
262: *
263: * Test the input parameters.
264: *
265: WANTZ = LSAME( JOBZ, 'V' )
266: LOWER = LSAME( UPLO, 'L' )
267: LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 .OR. LRWORK.EQ.-1 )
268: *
269: INFO = 0
270: IF( N.LE.1 ) THEN
271: LWMIN = 1
272: LRWMIN = 1
273: LIWMIN = 1
274: ELSE
275: IF( WANTZ ) THEN
276: LWMIN = 2*N**2
277: LRWMIN = 1 + 5*N + 2*N**2
278: LIWMIN = 3 + 5*N
279: ELSE
280: LWMIN = N
281: LRWMIN = N
282: LIWMIN = 1
283: END IF
284: END IF
285: IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
286: INFO = -1
287: ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
288: INFO = -2
289: ELSE IF( N.LT.0 ) THEN
290: INFO = -3
291: ELSE IF( KD.LT.0 ) THEN
292: INFO = -4
293: ELSE IF( LDAB.LT.KD+1 ) THEN
294: INFO = -6
295: ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
296: INFO = -9
297: END IF
298: *
299: IF( INFO.EQ.0 ) THEN
300: WORK( 1 ) = LWMIN
301: RWORK( 1 ) = LRWMIN
302: IWORK( 1 ) = LIWMIN
303: *
304: IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
305: INFO = -11
306: ELSE IF( LRWORK.LT.LRWMIN .AND. .NOT.LQUERY ) THEN
307: INFO = -13
308: ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
309: INFO = -15
310: END IF
311: END IF
312: *
313: IF( INFO.NE.0 ) THEN
314: CALL XERBLA( 'ZHBEVD', -INFO )
315: RETURN
316: ELSE IF( LQUERY ) THEN
317: RETURN
318: END IF
319: *
320: * Quick return if possible
321: *
322: IF( N.EQ.0 )
323: $ RETURN
324: *
325: IF( N.EQ.1 ) THEN
326: W( 1 ) = AB( 1, 1 )
327: IF( WANTZ )
328: $ Z( 1, 1 ) = CONE
329: RETURN
330: END IF
331: *
332: * Get machine constants.
333: *
334: SAFMIN = DLAMCH( 'Safe minimum' )
335: EPS = DLAMCH( 'Precision' )
336: SMLNUM = SAFMIN / EPS
337: BIGNUM = ONE / SMLNUM
338: RMIN = SQRT( SMLNUM )
339: RMAX = SQRT( BIGNUM )
340: *
341: * Scale matrix to allowable range, if necessary.
342: *
343: ANRM = ZLANHB( 'M', UPLO, N, KD, AB, LDAB, RWORK )
344: ISCALE = 0
345: IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
346: ISCALE = 1
347: SIGMA = RMIN / ANRM
348: ELSE IF( ANRM.GT.RMAX ) THEN
349: ISCALE = 1
350: SIGMA = RMAX / ANRM
351: END IF
352: IF( ISCALE.EQ.1 ) THEN
353: IF( LOWER ) THEN
354: CALL ZLASCL( 'B', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
355: ELSE
356: CALL ZLASCL( 'Q', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
357: END IF
358: END IF
359: *
360: * Call ZHBTRD to reduce Hermitian band matrix to tridiagonal form.
361: *
362: INDE = 1
363: INDWRK = INDE + N
364: INDWK2 = 1 + N*N
365: LLWK2 = LWORK - INDWK2 + 1
366: LLRWK = LRWORK - INDWRK + 1
367: CALL ZHBTRD( JOBZ, UPLO, N, KD, AB, LDAB, W, RWORK( INDE ), Z,
368: $ LDZ, WORK, IINFO )
369: *
370: * For eigenvalues only, call DSTERF. For eigenvectors, call ZSTEDC.
371: *
372: IF( .NOT.WANTZ ) THEN
373: CALL DSTERF( N, W, RWORK( INDE ), INFO )
374: ELSE
375: CALL ZSTEDC( 'I', N, W, RWORK( INDE ), WORK, N, WORK( INDWK2 ),
376: $ LLWK2, RWORK( INDWRK ), LLRWK, IWORK, LIWORK,
377: $ INFO )
378: CALL ZGEMM( 'N', 'N', N, N, N, CONE, Z, LDZ, WORK, N, CZERO,
379: $ WORK( INDWK2 ), N )
380: CALL ZLACPY( 'A', N, N, WORK( INDWK2 ), N, Z, LDZ )
381: END IF
382: *
383: * If matrix was scaled, then rescale eigenvalues appropriately.
384: *
385: IF( ISCALE.EQ.1 ) THEN
386: IF( INFO.EQ.0 ) THEN
387: IMAX = N
388: ELSE
389: IMAX = INFO - 1
390: END IF
391: CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
392: END IF
393: *
394: WORK( 1 ) = LWMIN
395: RWORK( 1 ) = LRWMIN
396: IWORK( 1 ) = LIWMIN
397: RETURN
398: *
399: * End of ZHBEVD
400: *
401: END
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