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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief <b> DSBEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b> 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 *> \htmlonly 00009 *> Download DSBEVD + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsbevd.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsbevd.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsbevd.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE DSBEVD( JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, 00022 * LWORK, IWORK, LIWORK, INFO ) 00023 * 00024 * .. Scalar Arguments .. 00025 * CHARACTER JOBZ, UPLO 00026 * INTEGER INFO, KD, LDAB, LDZ, LIWORK, LWORK, N 00027 * .. 00028 * .. Array Arguments .. 00029 * INTEGER IWORK( * ) 00030 * DOUBLE PRECISION AB( LDAB, * ), W( * ), WORK( * ), Z( LDZ, * ) 00031 * .. 00032 * 00033 * 00034 *> \par Purpose: 00035 * ============= 00036 *> 00037 *> \verbatim 00038 *> 00039 *> DSBEVD computes all the eigenvalues and, optionally, eigenvectors of 00040 *> a real symmetric band matrix A. If eigenvectors are desired, it uses 00041 *> a divide and conquer algorithm. 00042 *> 00043 *> The divide and conquer algorithm makes very mild assumptions about 00044 *> floating point arithmetic. It will work on machines with a guard 00045 *> digit in add/subtract, or on those binary machines without guard 00046 *> digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or 00047 *> Cray-2. It could conceivably fail on hexadecimal or decimal machines 00048 *> without guard digits, but we know of none. 00049 *> \endverbatim 00050 * 00051 * Arguments: 00052 * ========== 00053 * 00054 *> \param[in] JOBZ 00055 *> \verbatim 00056 *> JOBZ is CHARACTER*1 00057 *> = 'N': Compute eigenvalues only; 00058 *> = 'V': Compute eigenvalues and eigenvectors. 00059 *> \endverbatim 00060 *> 00061 *> \param[in] UPLO 00062 *> \verbatim 00063 *> UPLO is CHARACTER*1 00064 *> = 'U': Upper triangle of A is stored; 00065 *> = 'L': Lower triangle of A is stored. 00066 *> \endverbatim 00067 *> 00068 *> \param[in] N 00069 *> \verbatim 00070 *> N is INTEGER 00071 *> The order of the matrix A. N >= 0. 00072 *> \endverbatim 00073 *> 00074 *> \param[in] KD 00075 *> \verbatim 00076 *> KD is INTEGER 00077 *> The number of superdiagonals of the matrix A if UPLO = 'U', 00078 *> or the number of subdiagonals if UPLO = 'L'. KD >= 0. 00079 *> \endverbatim 00080 *> 00081 *> \param[in,out] AB 00082 *> \verbatim 00083 *> AB is DOUBLE PRECISION array, dimension (LDAB, N) 00084 *> On entry, the upper or lower triangle of the symmetric band 00085 *> matrix A, stored in the first KD+1 rows of the array. The 00086 *> j-th column of A is stored in the j-th column of the array AB 00087 *> as follows: 00088 *> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; 00089 *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). 00090 *> 00091 *> On exit, AB is overwritten by values generated during the 00092 *> reduction to tridiagonal form. If UPLO = 'U', the first 00093 *> superdiagonal and the diagonal of the tridiagonal matrix T 00094 *> are returned in rows KD and KD+1 of AB, and if UPLO = 'L', 00095 *> the diagonal and first subdiagonal of T are returned in the 00096 *> first two rows of AB. 00097 *> \endverbatim 00098 *> 00099 *> \param[in] LDAB 00100 *> \verbatim 00101 *> LDAB is INTEGER 00102 *> The leading dimension of the array AB. LDAB >= KD + 1. 00103 *> \endverbatim 00104 *> 00105 *> \param[out] W 00106 *> \verbatim 00107 *> W is DOUBLE PRECISION array, dimension (N) 00108 *> If INFO = 0, the eigenvalues in ascending order. 00109 *> \endverbatim 00110 *> 00111 *> \param[out] Z 00112 *> \verbatim 00113 *> Z is DOUBLE PRECISION array, dimension (LDZ, N) 00114 *> If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal 00115 *> eigenvectors of the matrix A, with the i-th column of Z 00116 *> holding the eigenvector associated with W(i). 00117 *> If JOBZ = 'N', then Z is not referenced. 00118 *> \endverbatim 00119 *> 00120 *> \param[in] LDZ 00121 *> \verbatim 00122 *> LDZ is INTEGER 00123 *> The leading dimension of the array Z. LDZ >= 1, and if 00124 *> JOBZ = 'V', LDZ >= max(1,N). 00125 *> \endverbatim 00126 *> 00127 *> \param[out] WORK 00128 *> \verbatim 00129 *> WORK is DOUBLE PRECISION array, 00130 *> dimension (LWORK) 00131 *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 00132 *> \endverbatim 00133 *> 00134 *> \param[in] LWORK 00135 *> \verbatim 00136 *> LWORK is INTEGER 00137 *> The dimension of the array WORK. 00138 *> IF N <= 1, LWORK must be at least 1. 00139 *> If JOBZ = 'N' and N > 2, LWORK must be at least 2*N. 00140 *> If JOBZ = 'V' and N > 2, LWORK must be at least 00141 *> ( 1 + 5*N + 2*N**2 ). 00142 *> 00143 *> If LWORK = -1, then a workspace query is assumed; the routine 00144 *> only calculates the optimal sizes of the WORK and IWORK 00145 *> arrays, returns these values as the first entries of the WORK 00146 *> and IWORK arrays, and no error message related to LWORK or 00147 *> LIWORK is issued by XERBLA. 00148 *> \endverbatim 00149 *> 00150 *> \param[out] IWORK 00151 *> \verbatim 00152 *> IWORK is INTEGER array, dimension (MAX(1,LIWORK)) 00153 *> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. 00154 *> \endverbatim 00155 *> 00156 *> \param[in] LIWORK 00157 *> \verbatim 00158 *> LIWORK is INTEGER 00159 *> The dimension of the array LIWORK. 00160 *> If JOBZ = 'N' or N <= 1, LIWORK must be at least 1. 00161 *> If JOBZ = 'V' and N > 2, LIWORK must be at least 3 + 5*N. 00162 *> 00163 *> If LIWORK = -1, then a workspace query is assumed; the 00164 *> routine only calculates the optimal sizes of the WORK and 00165 *> IWORK arrays, returns these values as the first entries of 00166 *> the WORK and IWORK arrays, and no error message related to 00167 *> LWORK or LIWORK is issued by XERBLA. 00168 *> \endverbatim 00169 *> 00170 *> \param[out] INFO 00171 *> \verbatim 00172 *> INFO is INTEGER 00173 *> = 0: successful exit 00174 *> < 0: if INFO = -i, the i-th argument had an illegal value 00175 *> > 0: if INFO = i, the algorithm failed to converge; i 00176 *> off-diagonal elements of an intermediate tridiagonal 00177 *> form did not converge to zero. 00178 *> \endverbatim 00179 * 00180 * Authors: 00181 * ======== 00182 * 00183 *> \author Univ. of Tennessee 00184 *> \author Univ. of California Berkeley 00185 *> \author Univ. of Colorado Denver 00186 *> \author NAG Ltd. 00187 * 00188 *> \date November 2011 00189 * 00190 *> \ingroup doubleOTHEReigen 00191 * 00192 * ===================================================================== 00193 SUBROUTINE DSBEVD( JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, 00194 $ LWORK, IWORK, LIWORK, INFO ) 00195 * 00196 * -- LAPACK driver routine (version 3.4.0) -- 00197 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00198 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00199 * November 2011 00200 * 00201 * .. Scalar Arguments .. 00202 CHARACTER JOBZ, UPLO 00203 INTEGER INFO, KD, LDAB, LDZ, LIWORK, LWORK, N 00204 * .. 00205 * .. Array Arguments .. 00206 INTEGER IWORK( * ) 00207 DOUBLE PRECISION AB( LDAB, * ), W( * ), WORK( * ), Z( LDZ, * ) 00208 * .. 00209 * 00210 * ===================================================================== 00211 * 00212 * .. Parameters .. 00213 DOUBLE PRECISION ZERO, ONE 00214 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 00215 * .. 00216 * .. Local Scalars .. 00217 LOGICAL LOWER, LQUERY, WANTZ 00218 INTEGER IINFO, INDE, INDWK2, INDWRK, ISCALE, LIWMIN, 00219 $ LLWRK2, LWMIN 00220 DOUBLE PRECISION ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, 00221 $ SMLNUM 00222 * .. 00223 * .. External Functions .. 00224 LOGICAL LSAME 00225 DOUBLE PRECISION DLAMCH, DLANSB 00226 EXTERNAL LSAME, DLAMCH, DLANSB 00227 * .. 00228 * .. External Subroutines .. 00229 EXTERNAL DGEMM, DLACPY, DLASCL, DSBTRD, DSCAL, DSTEDC, 00230 $ DSTERF, XERBLA 00231 * .. 00232 * .. Intrinsic Functions .. 00233 INTRINSIC SQRT 00234 * .. 00235 * .. Executable Statements .. 00236 * 00237 * Test the input parameters. 00238 * 00239 WANTZ = LSAME( JOBZ, 'V' ) 00240 LOWER = LSAME( UPLO, 'L' ) 00241 LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 ) 00242 * 00243 INFO = 0 00244 IF( N.LE.1 ) THEN 00245 LIWMIN = 1 00246 LWMIN = 1 00247 ELSE 00248 IF( WANTZ ) THEN 00249 LIWMIN = 3 + 5*N 00250 LWMIN = 1 + 5*N + 2*N**2 00251 ELSE 00252 LIWMIN = 1 00253 LWMIN = 2*N 00254 END IF 00255 END IF 00256 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN 00257 INFO = -1 00258 ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN 00259 INFO = -2 00260 ELSE IF( N.LT.0 ) THEN 00261 INFO = -3 00262 ELSE IF( KD.LT.0 ) THEN 00263 INFO = -4 00264 ELSE IF( LDAB.LT.KD+1 ) THEN 00265 INFO = -6 00266 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN 00267 INFO = -9 00268 END IF 00269 * 00270 IF( INFO.EQ.0 ) THEN 00271 WORK( 1 ) = LWMIN 00272 IWORK( 1 ) = LIWMIN 00273 * 00274 IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN 00275 INFO = -11 00276 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN 00277 INFO = -13 00278 END IF 00279 END IF 00280 * 00281 IF( INFO.NE.0 ) THEN 00282 CALL XERBLA( 'DSBEVD', -INFO ) 00283 RETURN 00284 ELSE IF( LQUERY ) THEN 00285 RETURN 00286 END IF 00287 * 00288 * Quick return if possible 00289 * 00290 IF( N.EQ.0 ) 00291 $ RETURN 00292 * 00293 IF( N.EQ.1 ) THEN 00294 W( 1 ) = AB( 1, 1 ) 00295 IF( WANTZ ) 00296 $ Z( 1, 1 ) = ONE 00297 RETURN 00298 END IF 00299 * 00300 * Get machine constants. 00301 * 00302 SAFMIN = DLAMCH( 'Safe minimum' ) 00303 EPS = DLAMCH( 'Precision' ) 00304 SMLNUM = SAFMIN / EPS 00305 BIGNUM = ONE / SMLNUM 00306 RMIN = SQRT( SMLNUM ) 00307 RMAX = SQRT( BIGNUM ) 00308 * 00309 * Scale matrix to allowable range, if necessary. 00310 * 00311 ANRM = DLANSB( 'M', UPLO, N, KD, AB, LDAB, WORK ) 00312 ISCALE = 0 00313 IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN 00314 ISCALE = 1 00315 SIGMA = RMIN / ANRM 00316 ELSE IF( ANRM.GT.RMAX ) THEN 00317 ISCALE = 1 00318 SIGMA = RMAX / ANRM 00319 END IF 00320 IF( ISCALE.EQ.1 ) THEN 00321 IF( LOWER ) THEN 00322 CALL DLASCL( 'B', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO ) 00323 ELSE 00324 CALL DLASCL( 'Q', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO ) 00325 END IF 00326 END IF 00327 * 00328 * Call DSBTRD to reduce symmetric band matrix to tridiagonal form. 00329 * 00330 INDE = 1 00331 INDWRK = INDE + N 00332 INDWK2 = INDWRK + N*N 00333 LLWRK2 = LWORK - INDWK2 + 1 00334 CALL DSBTRD( JOBZ, UPLO, N, KD, AB, LDAB, W, WORK( INDE ), Z, LDZ, 00335 $ WORK( INDWRK ), IINFO ) 00336 * 00337 * For eigenvalues only, call DSTERF. For eigenvectors, call SSTEDC. 00338 * 00339 IF( .NOT.WANTZ ) THEN 00340 CALL DSTERF( N, W, WORK( INDE ), INFO ) 00341 ELSE 00342 CALL DSTEDC( 'I', N, W, WORK( INDE ), WORK( INDWRK ), N, 00343 $ WORK( INDWK2 ), LLWRK2, IWORK, LIWORK, INFO ) 00344 CALL DGEMM( 'N', 'N', N, N, N, ONE, Z, LDZ, WORK( INDWRK ), N, 00345 $ ZERO, WORK( INDWK2 ), N ) 00346 CALL DLACPY( 'A', N, N, WORK( INDWK2 ), N, Z, LDZ ) 00347 END IF 00348 * 00349 * If matrix was scaled, then rescale eigenvalues appropriately. 00350 * 00351 IF( ISCALE.EQ.1 ) 00352 $ CALL DSCAL( N, ONE / SIGMA, W, 1 ) 00353 * 00354 WORK( 1 ) = LWMIN 00355 IWORK( 1 ) = LIWMIN 00356 RETURN 00357 * 00358 * End of DSBEVD 00359 * 00360 END