1*> \brief \b DSBGVD 2* 3* =========== DOCUMENTATION =========== 4* 5* Online html documentation available at 6* http://www.netlib.org/lapack/explore-html/ 7* 8*> \htmlonly 9*> Download DSBGVD + dependencies 10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsbgvd.f"> 11*> [TGZ]</a> 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsbgvd.f"> 13*> [ZIP]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsbgvd.f"> 15*> [TXT]</a> 16*> \endhtmlonly 17* 18* Definition: 19* =========== 20* 21* SUBROUTINE DSBGVD( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, 22* Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO ) 23* 24* .. Scalar Arguments .. 25* CHARACTER JOBZ, UPLO 26* INTEGER INFO, KA, KB, LDAB, LDBB, LDZ, LIWORK, LWORK, N 27* .. 28* .. Array Arguments .. 29* INTEGER IWORK( * ) 30* DOUBLE PRECISION AB( LDAB, * ), BB( LDBB, * ), W( * ), 31* $ WORK( * ), Z( LDZ, * ) 32* .. 33* 34* 35*> \par Purpose: 36* ============= 37*> 38*> \verbatim 39*> 40*> DSBGVD computes all the eigenvalues, and optionally, the eigenvectors 41*> of a real generalized symmetric-definite banded eigenproblem, of the 42*> form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric and 43*> banded, and B is also positive definite. If eigenvectors are 44*> desired, it uses a divide and conquer algorithm. 45*> 46*> The divide and conquer algorithm makes very mild assumptions about 47*> floating point arithmetic. It will work on machines with a guard 48*> digit in add/subtract, or on those binary machines without guard 49*> digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or 50*> Cray-2. It could conceivably fail on hexadecimal or decimal machines 51*> without guard digits, but we know of none. 52*> \endverbatim 53* 54* Arguments: 55* ========== 56* 57*> \param[in] JOBZ 58*> \verbatim 59*> JOBZ is CHARACTER*1 60*> = 'N': Compute eigenvalues only; 61*> = 'V': Compute eigenvalues and eigenvectors. 62*> \endverbatim 63*> 64*> \param[in] UPLO 65*> \verbatim 66*> UPLO is CHARACTER*1 67*> = 'U': Upper triangles of A and B are stored; 68*> = 'L': Lower triangles of A and B are stored. 69*> \endverbatim 70*> 71*> \param[in] N 72*> \verbatim 73*> N is INTEGER 74*> The order of the matrices A and B. N >= 0. 75*> \endverbatim 76*> 77*> \param[in] KA 78*> \verbatim 79*> KA is INTEGER 80*> The number of superdiagonals of the matrix A if UPLO = 'U', 81*> or the number of subdiagonals if UPLO = 'L'. KA >= 0. 82*> \endverbatim 83*> 84*> \param[in] KB 85*> \verbatim 86*> KB is INTEGER 87*> The number of superdiagonals of the matrix B if UPLO = 'U', 88*> or the number of subdiagonals if UPLO = 'L'. KB >= 0. 89*> \endverbatim 90*> 91*> \param[in,out] AB 92*> \verbatim 93*> AB is DOUBLE PRECISION array, dimension (LDAB, N) 94*> On entry, the upper or lower triangle of the symmetric band 95*> matrix A, stored in the first ka+1 rows of the array. The 96*> j-th column of A is stored in the j-th column of the array AB 97*> as follows: 98*> if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j; 99*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka). 100*> 101*> On exit, the contents of AB are destroyed. 102*> \endverbatim 103*> 104*> \param[in] LDAB 105*> \verbatim 106*> LDAB is INTEGER 107*> The leading dimension of the array AB. LDAB >= KA+1. 108*> \endverbatim 109*> 110*> \param[in,out] BB 111*> \verbatim 112*> BB is DOUBLE PRECISION array, dimension (LDBB, N) 113*> On entry, the upper or lower triangle of the symmetric band 114*> matrix B, stored in the first kb+1 rows of the array. The 115*> j-th column of B is stored in the j-th column of the array BB 116*> as follows: 117*> if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j; 118*> if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb). 119*> 120*> On exit, the factor S from the split Cholesky factorization 121*> B = S**T*S, as returned by DPBSTF. 122*> \endverbatim 123*> 124*> \param[in] LDBB 125*> \verbatim 126*> LDBB is INTEGER 127*> The leading dimension of the array BB. LDBB >= KB+1. 128*> \endverbatim 129*> 130*> \param[out] W 131*> \verbatim 132*> W is DOUBLE PRECISION array, dimension (N) 133*> If INFO = 0, the eigenvalues in ascending order. 134*> \endverbatim 135*> 136*> \param[out] Z 137*> \verbatim 138*> Z is DOUBLE PRECISION array, dimension (LDZ, N) 139*> If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of 140*> eigenvectors, with the i-th column of Z holding the 141*> eigenvector associated with W(i). The eigenvectors are 142*> normalized so Z**T*B*Z = I. 143*> If JOBZ = 'N', then Z is not referenced. 144*> \endverbatim 145*> 146*> \param[in] LDZ 147*> \verbatim 148*> LDZ is INTEGER 149*> The leading dimension of the array Z. LDZ >= 1, and if 150*> JOBZ = 'V', LDZ >= max(1,N). 151*> \endverbatim 152*> 153*> \param[out] WORK 154*> \verbatim 155*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) 156*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 157*> \endverbatim 158*> 159*> \param[in] LWORK 160*> \verbatim 161*> LWORK is INTEGER 162*> The dimension of the array WORK. 163*> If N <= 1, LWORK >= 1. 164*> If JOBZ = 'N' and N > 1, LWORK >= 2*N. 165*> If JOBZ = 'V' and N > 1, LWORK >= 1 + 5*N + 2*N**2. 166*> 167*> If LWORK = -1, then a workspace query is assumed; the routine 168*> only calculates the optimal sizes of the WORK and IWORK 169*> arrays, returns these values as the first entries of the WORK 170*> and IWORK arrays, and no error message related to LWORK or 171*> LIWORK is issued by XERBLA. 172*> \endverbatim 173*> 174*> \param[out] IWORK 175*> \verbatim 176*> IWORK is INTEGER array, dimension (MAX(1,LIWORK)) 177*> On exit, if LIWORK > 0, IWORK(1) returns the optimal LIWORK. 178*> \endverbatim 179*> 180*> \param[in] LIWORK 181*> \verbatim 182*> LIWORK is INTEGER 183*> The dimension of the array IWORK. 184*> If JOBZ = 'N' or N <= 1, LIWORK >= 1. 185*> If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N. 186*> 187*> If LIWORK = -1, then a workspace query is assumed; the 188*> routine only calculates the optimal sizes of the WORK and 189*> IWORK arrays, returns these values as the first entries of 190*> the WORK and IWORK arrays, and no error message related to 191*> LWORK or LIWORK is issued by XERBLA. 192*> \endverbatim 193*> 194*> \param[out] INFO 195*> \verbatim 196*> INFO is INTEGER 197*> = 0: successful exit 198*> < 0: if INFO = -i, the i-th argument had an illegal value 199*> > 0: if INFO = i, and i is: 200*> <= N: the algorithm failed to converge: 201*> i off-diagonal elements of an intermediate 202*> tridiagonal form did not converge to zero; 203*> > N: if INFO = N + i, for 1 <= i <= N, then DPBSTF 204*> returned INFO = i: B is not positive definite. 205*> The factorization of B could not be completed and 206*> no eigenvalues or eigenvectors were computed. 207*> \endverbatim 208* 209* Authors: 210* ======== 211* 212*> \author Univ. of Tennessee 213*> \author Univ. of California Berkeley 214*> \author Univ. of Colorado Denver 215*> \author NAG Ltd. 216* 217*> \ingroup doubleOTHEReigen 218* 219*> \par Contributors: 220* ================== 221*> 222*> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA 223* 224* ===================================================================== 225 SUBROUTINE DSBGVD( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, 226 $ Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO ) 227* 228* -- LAPACK driver routine -- 229* -- LAPACK is a software package provided by Univ. of Tennessee, -- 230* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 231* 232* .. Scalar Arguments .. 233 CHARACTER JOBZ, UPLO 234 INTEGER INFO, KA, KB, LDAB, LDBB, LDZ, LIWORK, LWORK, N 235* .. 236* .. Array Arguments .. 237 INTEGER IWORK( * ) 238 DOUBLE PRECISION AB( LDAB, * ), BB( LDBB, * ), W( * ), 239 $ WORK( * ), Z( LDZ, * ) 240* .. 241* 242* ===================================================================== 243* 244* .. Parameters .. 245 DOUBLE PRECISION ONE, ZERO 246 PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) 247* .. 248* .. Local Scalars .. 249 LOGICAL LQUERY, UPPER, WANTZ 250 CHARACTER VECT 251 INTEGER IINFO, INDE, INDWK2, INDWRK, LIWMIN, LLWRK2, 252 $ LWMIN 253* .. 254* .. External Functions .. 255 LOGICAL LSAME 256 EXTERNAL LSAME 257* .. 258* .. External Subroutines .. 259 EXTERNAL DGEMM, DLACPY, DPBSTF, DSBGST, DSBTRD, DSTEDC, 260 $ DSTERF, XERBLA 261* .. 262* .. Executable Statements .. 263* 264* Test the input parameters. 265* 266 WANTZ = LSAME( JOBZ, 'V' ) 267 UPPER = LSAME( UPLO, 'U' ) 268 LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 ) 269* 270 INFO = 0 271 IF( N.LE.1 ) THEN 272 LIWMIN = 1 273 LWMIN = 1 274 ELSE IF( WANTZ ) THEN 275 LIWMIN = 3 + 5*N 276 LWMIN = 1 + 5*N + 2*N**2 277 ELSE 278 LIWMIN = 1 279 LWMIN = 2*N 280 END IF 281* 282 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN 283 INFO = -1 284 ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN 285 INFO = -2 286 ELSE IF( N.LT.0 ) THEN 287 INFO = -3 288 ELSE IF( KA.LT.0 ) THEN 289 INFO = -4 290 ELSE IF( KB.LT.0 .OR. KB.GT.KA ) THEN 291 INFO = -5 292 ELSE IF( LDAB.LT.KA+1 ) THEN 293 INFO = -7 294 ELSE IF( LDBB.LT.KB+1 ) THEN 295 INFO = -9 296 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN 297 INFO = -12 298 END IF 299* 300 IF( INFO.EQ.0 ) THEN 301 WORK( 1 ) = LWMIN 302 IWORK( 1 ) = LIWMIN 303* 304 IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN 305 INFO = -14 306 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN 307 INFO = -16 308 END IF 309 END IF 310* 311 IF( INFO.NE.0 ) THEN 312 CALL XERBLA( 'DSBGVD', -INFO ) 313 RETURN 314 ELSE IF( LQUERY ) THEN 315 RETURN 316 END IF 317* 318* Quick return if possible 319* 320 IF( N.EQ.0 ) 321 $ RETURN 322* 323* Form a split Cholesky factorization of B. 324* 325 CALL DPBSTF( UPLO, N, KB, BB, LDBB, INFO ) 326 IF( INFO.NE.0 ) THEN 327 INFO = N + INFO 328 RETURN 329 END IF 330* 331* Transform problem to standard eigenvalue problem. 332* 333 INDE = 1 334 INDWRK = INDE + N 335 INDWK2 = INDWRK + N*N 336 LLWRK2 = LWORK - INDWK2 + 1 337 CALL DSBGST( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Z, LDZ, 338 $ WORK, IINFO ) 339* 340* Reduce to tridiagonal form. 341* 342 IF( WANTZ ) THEN 343 VECT = 'U' 344 ELSE 345 VECT = 'N' 346 END IF 347 CALL DSBTRD( VECT, UPLO, N, KA, AB, LDAB, W, WORK( INDE ), Z, LDZ, 348 $ WORK( INDWRK ), IINFO ) 349* 350* For eigenvalues only, call DSTERF. For eigenvectors, call SSTEDC. 351* 352 IF( .NOT.WANTZ ) THEN 353 CALL DSTERF( N, W, WORK( INDE ), INFO ) 354 ELSE 355 CALL DSTEDC( 'I', N, W, WORK( INDE ), WORK( INDWRK ), N, 356 $ WORK( INDWK2 ), LLWRK2, IWORK, LIWORK, INFO ) 357 CALL DGEMM( 'N', 'N', N, N, N, ONE, Z, LDZ, WORK( INDWRK ), N, 358 $ ZERO, WORK( INDWK2 ), N ) 359 CALL DLACPY( 'A', N, N, WORK( INDWK2 ), N, Z, LDZ ) 360 END IF 361* 362 WORK( 1 ) = LWMIN 363 IWORK( 1 ) = LIWMIN 364* 365 RETURN 366* 367* End of DSBGVD 368* 369 END 370