1 SUBROUTINE SSTEVX( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, 2 $ M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO ) 3* 4* -- LAPACK driver routine (version 3.0) -- 5* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., 6* Courant Institute, Argonne National Lab, and Rice University 7* June 30, 1999 8* 9* .. Scalar Arguments .. 10 CHARACTER JOBZ, RANGE 11 INTEGER IL, INFO, IU, LDZ, M, N 12 REAL ABSTOL, VL, VU 13* .. 14* .. Array Arguments .. 15 INTEGER IFAIL( * ), IWORK( * ) 16 REAL D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * ) 17* .. 18* 19* Purpose 20* ======= 21* 22* SSTEVX computes selected eigenvalues and, optionally, eigenvectors 23* of a real symmetric tridiagonal matrix A. Eigenvalues and 24* eigenvectors can be selected by specifying either a range of values 25* or a range of indices for the desired eigenvalues. 26* 27* Arguments 28* ========= 29* 30* JOBZ (input) CHARACTER*1 31* = 'N': Compute eigenvalues only; 32* = 'V': Compute eigenvalues and eigenvectors. 33* 34* RANGE (input) CHARACTER*1 35* = 'A': all eigenvalues will be found. 36* = 'V': all eigenvalues in the half-open interval (VL,VU] 37* will be found. 38* = 'I': the IL-th through IU-th eigenvalues will be found. 39* 40* N (input) INTEGER 41* The order of the matrix. N >= 0. 42* 43* D (input/output) REAL array, dimension (N) 44* On entry, the n diagonal elements of the tridiagonal matrix 45* A. 46* On exit, D may be multiplied by a constant factor chosen 47* to avoid over/underflow in computing the eigenvalues. 48* 49* E (input/output) REAL array, dimension (N) 50* On entry, the (n-1) subdiagonal elements of the tridiagonal 51* matrix A in elements 1 to N-1 of E; E(N) need not be set. 52* On exit, E may be multiplied by a constant factor chosen 53* to avoid over/underflow in computing the eigenvalues. 54* 55* VL (input) REAL 56* VU (input) REAL 57* If RANGE='V', the lower and upper bounds of the interval to 58* be searched for eigenvalues. VL < VU. 59* Not referenced if RANGE = 'A' or 'I'. 60* 61* IL (input) INTEGER 62* IU (input) INTEGER 63* If RANGE='I', the indices (in ascending order) of the 64* smallest and largest eigenvalues to be returned. 65* 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. 66* Not referenced if RANGE = 'A' or 'V'. 67* 68* ABSTOL (input) REAL 69* The absolute error tolerance for the eigenvalues. 70* An approximate eigenvalue is accepted as converged 71* when it is determined to lie in an interval [a,b] 72* of width less than or equal to 73* 74* ABSTOL + EPS * max( |a|,|b| ) , 75* 76* where EPS is the machine precision. If ABSTOL is less 77* than or equal to zero, then EPS*|T| will be used in 78* its place, where |T| is the 1-norm of the tridiagonal 79* matrix. 80* 81* Eigenvalues will be computed most accurately when ABSTOL is 82* set to twice the underflow threshold 2*SLAMCH('S'), not zero. 83* If this routine returns with INFO>0, indicating that some 84* eigenvectors did not converge, try setting ABSTOL to 85* 2*SLAMCH('S'). 86* 87* See "Computing Small Singular Values of Bidiagonal Matrices 88* with Guaranteed High Relative Accuracy," by Demmel and 89* Kahan, LAPACK Working Note #3. 90* 91* M (output) INTEGER 92* The total number of eigenvalues found. 0 <= M <= N. 93* If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. 94* 95* W (output) REAL array, dimension (N) 96* The first M elements contain the selected eigenvalues in 97* ascending order. 98* 99* Z (output) REAL array, dimension (LDZ, max(1,M) ) 100* If JOBZ = 'V', then if INFO = 0, the first M columns of Z 101* contain the orthonormal eigenvectors of the matrix A 102* corresponding to the selected eigenvalues, with the i-th 103* column of Z holding the eigenvector associated with W(i). 104* If an eigenvector fails to converge (INFO > 0), then that 105* column of Z contains the latest approximation to the 106* eigenvector, and the index of the eigenvector is returned 107* in IFAIL. If JOBZ = 'N', then Z is not referenced. 108* Note: the user must ensure that at least max(1,M) columns are 109* supplied in the array Z; if RANGE = 'V', the exact value of M 110* is not known in advance and an upper bound must be used. 111* 112* LDZ (input) INTEGER 113* The leading dimension of the array Z. LDZ >= 1, and if 114* JOBZ = 'V', LDZ >= max(1,N). 115* 116* WORK (workspace) REAL array, dimension (5*N) 117* 118* IWORK (workspace) INTEGER array, dimension (5*N) 119* 120* IFAIL (output) INTEGER array, dimension (N) 121* If JOBZ = 'V', then if INFO = 0, the first M elements of 122* IFAIL are zero. If INFO > 0, then IFAIL contains the 123* indices of the eigenvectors that failed to converge. 124* If JOBZ = 'N', then IFAIL is not referenced. 125* 126* INFO (output) INTEGER 127* = 0: successful exit 128* < 0: if INFO = -i, the i-th argument had an illegal value 129* > 0: if INFO = i, then i eigenvectors failed to converge. 130* Their indices are stored in array IFAIL. 131* 132* ===================================================================== 133* 134* .. Parameters .. 135 REAL ZERO, ONE 136 PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 ) 137* .. 138* .. Local Scalars .. 139 LOGICAL ALLEIG, INDEIG, VALEIG, WANTZ 140 CHARACTER ORDER 141 INTEGER I, IMAX, INDIBL, INDISP, INDIWO, INDWRK, 142 $ ISCALE, ITMP1, J, JJ, NSPLIT 143 REAL BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM, 144 $ TMP1, TNRM, VLL, VUU 145* .. 146* .. External Functions .. 147 LOGICAL LSAME 148 REAL SLAMCH, SLANST 149 EXTERNAL LSAME, SLAMCH, SLANST 150* .. 151* .. External Subroutines .. 152 EXTERNAL SCOPY, SSCAL, SSTEBZ, SSTEIN, SSTEQR, SSTERF, 153 $ SSWAP, XERBLA 154* .. 155* .. Intrinsic Functions .. 156 INTRINSIC MAX, MIN, SQRT 157* .. 158* .. Executable Statements .. 159* 160* Test the input parameters. 161* 162 WANTZ = LSAME( JOBZ, 'V' ) 163 ALLEIG = LSAME( RANGE, 'A' ) 164 VALEIG = LSAME( RANGE, 'V' ) 165 INDEIG = LSAME( RANGE, 'I' ) 166* 167 INFO = 0 168 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN 169 INFO = -1 170 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN 171 INFO = -2 172 ELSE IF( N.LT.0 ) THEN 173 INFO = -3 174 ELSE 175 IF( VALEIG ) THEN 176 IF( N.GT.0 .AND. VU.LE.VL ) 177 $ INFO = -7 178 ELSE IF( INDEIG ) THEN 179 IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN 180 INFO = -8 181 ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN 182 INFO = -9 183 END IF 184 END IF 185 END IF 186 IF( INFO.EQ.0 ) THEN 187 IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) 188 $ INFO = -14 189 END IF 190* 191 IF( INFO.NE.0 ) THEN 192 CALL XERBLA( 'SSTEVX', -INFO ) 193 RETURN 194 END IF 195* 196* Quick return if possible 197* 198 M = 0 199 IF( N.EQ.0 ) 200 $ RETURN 201* 202 IF( N.EQ.1 ) THEN 203 IF( ALLEIG .OR. INDEIG ) THEN 204 M = 1 205 W( 1 ) = D( 1 ) 206 ELSE 207 IF( VL.LT.D( 1 ) .AND. VU.GE.D( 1 ) ) THEN 208 M = 1 209 W( 1 ) = D( 1 ) 210 END IF 211 END IF 212 IF( WANTZ ) 213 $ Z( 1, 1 ) = ONE 214 RETURN 215 END IF 216* 217* Get machine constants. 218* 219 SAFMIN = SLAMCH( 'Safe minimum' ) 220 EPS = SLAMCH( 'Precision' ) 221 SMLNUM = SAFMIN / EPS 222 BIGNUM = ONE / SMLNUM 223 RMIN = SQRT( SMLNUM ) 224 RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) ) 225* 226* Scale matrix to allowable range, if necessary. 227* 228 ISCALE = 0 229 IF ( VALEIG ) THEN 230 VLL = VL 231 VUU = VU 232 ELSE 233 VLL = ZERO 234 VUU = ZERO 235 ENDIF 236 TNRM = SLANST( 'M', N, D, E ) 237 IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN 238 ISCALE = 1 239 SIGMA = RMIN / TNRM 240 ELSE IF( TNRM.GT.RMAX ) THEN 241 ISCALE = 1 242 SIGMA = RMAX / TNRM 243 END IF 244 IF( ISCALE.EQ.1 ) THEN 245 CALL SSCAL( N, SIGMA, D, 1 ) 246 CALL SSCAL( N-1, SIGMA, E( 1 ), 1 ) 247 IF( VALEIG ) THEN 248 VLL = VL*SIGMA 249 VUU = VU*SIGMA 250 END IF 251 END IF 252* 253* If all eigenvalues are desired and ABSTOL is less than zero, then 254* call SSTERF or SSTEQR. If this fails for some eigenvalue, then 255* try SSTEBZ. 256* 257 IF( ( ALLEIG .OR. ( INDEIG .AND. IL.EQ.1 .AND. IU.EQ.N ) ) .AND. 258 $ ( ABSTOL.LE.ZERO ) ) THEN 259 CALL SCOPY( N, D, 1, W, 1 ) 260 CALL SCOPY( N-1, E( 1 ), 1, WORK( 1 ), 1 ) 261 INDWRK = N + 1 262 IF( .NOT.WANTZ ) THEN 263 CALL SSTERF( N, W, WORK, INFO ) 264 ELSE 265 CALL SSTEQR( 'I', N, W, WORK, Z, LDZ, WORK( INDWRK ), INFO ) 266 IF( INFO.EQ.0 ) THEN 267 DO 10 I = 1, N 268 IFAIL( I ) = 0 269 10 CONTINUE 270 END IF 271 END IF 272 IF( INFO.EQ.0 ) THEN 273 M = N 274 GO TO 20 275 END IF 276 INFO = 0 277 END IF 278* 279* Otherwise, call SSTEBZ and, if eigenvectors are desired, SSTEIN. 280* 281 IF( WANTZ ) THEN 282 ORDER = 'B' 283 ELSE 284 ORDER = 'E' 285 END IF 286 INDWRK = 1 287 INDIBL = 1 288 INDISP = INDIBL + N 289 INDIWO = INDISP + N 290 CALL SSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTOL, D, E, M, 291 $ NSPLIT, W, IWORK( INDIBL ), IWORK( INDISP ), 292 $ WORK( INDWRK ), IWORK( INDIWO ), INFO ) 293* 294 IF( WANTZ ) THEN 295 CALL SSTEIN( N, D, E, M, W, IWORK( INDIBL ), IWORK( INDISP ), 296 $ Z, LDZ, WORK( INDWRK ), IWORK( INDIWO ), IFAIL, 297 $ INFO ) 298 END IF 299* 300* If matrix was scaled, then rescale eigenvalues appropriately. 301* 302 20 CONTINUE 303 IF( ISCALE.EQ.1 ) THEN 304 IF( INFO.EQ.0 ) THEN 305 IMAX = M 306 ELSE 307 IMAX = INFO - 1 308 END IF 309 CALL SSCAL( IMAX, ONE / SIGMA, W, 1 ) 310 END IF 311* 312* If eigenvalues are not in order, then sort them, along with 313* eigenvectors. 314* 315 IF( WANTZ ) THEN 316 DO 40 J = 1, M - 1 317 I = 0 318 TMP1 = W( J ) 319 DO 30 JJ = J + 1, M 320 IF( W( JJ ).LT.TMP1 ) THEN 321 I = JJ 322 TMP1 = W( JJ ) 323 END IF 324 30 CONTINUE 325* 326 IF( I.NE.0 ) THEN 327 ITMP1 = IWORK( INDIBL+I-1 ) 328 W( I ) = W( J ) 329 IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 ) 330 W( J ) = TMP1 331 IWORK( INDIBL+J-1 ) = ITMP1 332 CALL SSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 ) 333 IF( INFO.NE.0 ) THEN 334 ITMP1 = IFAIL( I ) 335 IFAIL( I ) = IFAIL( J ) 336 IFAIL( J ) = ITMP1 337 END IF 338 END IF 339 40 CONTINUE 340 END IF 341* 342 RETURN 343* 344* End of SSTEVX 345* 346 END 347