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LAPACK 3.3.0
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LAPACK 3.3.0
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00001 SUBROUTINE SSYEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, 00002 $ ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK, 00003 $ IFAIL, INFO ) 00004 * 00005 * -- LAPACK driver routine (version 3.2) -- 00006 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00007 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00008 * November 2006 00009 * 00010 * .. Scalar Arguments .. 00011 CHARACTER JOBZ, RANGE, UPLO 00012 INTEGER IL, INFO, IU, LDA, LDZ, LWORK, M, N 00013 REAL ABSTOL, VL, VU 00014 * .. 00015 * .. Array Arguments .. 00016 INTEGER IFAIL( * ), IWORK( * ) 00017 REAL A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * ) 00018 * .. 00019 * 00020 * Purpose 00021 * ======= 00022 * 00023 * SSYEVX computes selected eigenvalues and, optionally, eigenvectors 00024 * of a real symmetric matrix A. Eigenvalues and eigenvectors can be 00025 * selected by specifying either a range of values or a range of indices 00026 * for the desired eigenvalues. 00027 * 00028 * Arguments 00029 * ========= 00030 * 00031 * JOBZ (input) CHARACTER*1 00032 * = 'N': Compute eigenvalues only; 00033 * = 'V': Compute eigenvalues and eigenvectors. 00034 * 00035 * RANGE (input) CHARACTER*1 00036 * = 'A': all eigenvalues will be found. 00037 * = 'V': all eigenvalues in the half-open interval (VL,VU] 00038 * will be found. 00039 * = 'I': the IL-th through IU-th eigenvalues will be found. 00040 * 00041 * UPLO (input) CHARACTER*1 00042 * = 'U': Upper triangle of A is stored; 00043 * = 'L': Lower triangle of A is stored. 00044 * 00045 * N (input) INTEGER 00046 * The order of the matrix A. N >= 0. 00047 * 00048 * A (input/output) REAL array, dimension (LDA, N) 00049 * On entry, the symmetric matrix A. If UPLO = 'U', the 00050 * leading N-by-N upper triangular part of A contains the 00051 * upper triangular part of the matrix A. If UPLO = 'L', 00052 * the leading N-by-N lower triangular part of A contains 00053 * the lower triangular part of the matrix A. 00054 * On exit, the lower triangle (if UPLO='L') or the upper 00055 * triangle (if UPLO='U') of A, including the diagonal, is 00056 * destroyed. 00057 * 00058 * LDA (input) INTEGER 00059 * The leading dimension of the array A. LDA >= max(1,N). 00060 * 00061 * VL (input) REAL 00062 * VU (input) REAL 00063 * If RANGE='V', the lower and upper bounds of the interval to 00064 * be searched for eigenvalues. VL < VU. 00065 * Not referenced if RANGE = 'A' or 'I'. 00066 * 00067 * IL (input) INTEGER 00068 * IU (input) INTEGER 00069 * If RANGE='I', the indices (in ascending order) of the 00070 * smallest and largest eigenvalues to be returned. 00071 * 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. 00072 * Not referenced if RANGE = 'A' or 'V'. 00073 * 00074 * ABSTOL (input) REAL 00075 * The absolute error tolerance for the eigenvalues. 00076 * An approximate eigenvalue is accepted as converged 00077 * when it is determined to lie in an interval [a,b] 00078 * of width less than or equal to 00079 * 00080 * ABSTOL + EPS * max( |a|,|b| ) , 00081 * 00082 * where EPS is the machine precision. If ABSTOL is less than 00083 * or equal to zero, then EPS*|T| will be used in its place, 00084 * where |T| is the 1-norm of the tridiagonal matrix obtained 00085 * by reducing A to tridiagonal form. 00086 * 00087 * Eigenvalues will be computed most accurately when ABSTOL is 00088 * set to twice the underflow threshold 2*SLAMCH('S'), not zero. 00089 * If this routine returns with INFO>0, indicating that some 00090 * eigenvectors did not converge, try setting ABSTOL to 00091 * 2*SLAMCH('S'). 00092 * 00093 * See "Computing Small Singular Values of Bidiagonal Matrices 00094 * with Guaranteed High Relative Accuracy," by Demmel and 00095 * Kahan, LAPACK Working Note #3. 00096 * 00097 * M (output) INTEGER 00098 * The total number of eigenvalues found. 0 <= M <= N. 00099 * If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. 00100 * 00101 * W (output) REAL array, dimension (N) 00102 * On normal exit, the first M elements contain the selected 00103 * eigenvalues in ascending order. 00104 * 00105 * Z (output) REAL array, dimension (LDZ, max(1,M)) 00106 * If JOBZ = 'V', then if INFO = 0, the first M columns of Z 00107 * contain the orthonormal eigenvectors of the matrix A 00108 * corresponding to the selected eigenvalues, with the i-th 00109 * column of Z holding the eigenvector associated with W(i). 00110 * If an eigenvector fails to converge, then that column of Z 00111 * contains the latest approximation to the eigenvector, and the 00112 * index of the eigenvector is returned in IFAIL. 00113 * If JOBZ = 'N', then Z is not referenced. 00114 * Note: the user must ensure that at least max(1,M) columns are 00115 * supplied in the array Z; if RANGE = 'V', the exact value of M 00116 * is not known in advance and an upper bound must be used. 00117 * 00118 * LDZ (input) INTEGER 00119 * The leading dimension of the array Z. LDZ >= 1, and if 00120 * JOBZ = 'V', LDZ >= max(1,N). 00121 * 00122 * WORK (workspace/output) REAL array, dimension (MAX(1,LWORK)) 00123 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 00124 * 00125 * LWORK (input) INTEGER 00126 * The length of the array WORK. LWORK >= 1, when N <= 1; 00127 * otherwise 8*N. 00128 * For optimal efficiency, LWORK >= (NB+3)*N, 00129 * where NB is the max of the blocksize for SSYTRD and SORMTR 00130 * returned by ILAENV. 00131 * 00132 * If LWORK = -1, then a workspace query is assumed; the routine 00133 * only calculates the optimal size of the WORK array, returns 00134 * this value as the first entry of the WORK array, and no error 00135 * message related to LWORK is issued by XERBLA. 00136 * 00137 * IWORK (workspace) INTEGER array, dimension (5*N) 00138 * 00139 * IFAIL (output) INTEGER array, dimension (N) 00140 * If JOBZ = 'V', then if INFO = 0, the first M elements of 00141 * IFAIL are zero. If INFO > 0, then IFAIL contains the 00142 * indices of the eigenvectors that failed to converge. 00143 * If JOBZ = 'N', then IFAIL is not referenced. 00144 * 00145 * INFO (output) INTEGER 00146 * = 0: successful exit 00147 * < 0: if INFO = -i, the i-th argument had an illegal value 00148 * > 0: if INFO = i, then i eigenvectors failed to converge. 00149 * Their indices are stored in array IFAIL. 00150 * 00151 * ===================================================================== 00152 * 00153 * .. Parameters .. 00154 REAL ZERO, ONE 00155 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 00156 * .. 00157 * .. Local Scalars .. 00158 LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, TEST, VALEIG, 00159 $ WANTZ 00160 CHARACTER ORDER 00161 INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL, 00162 $ INDISP, INDIWO, INDTAU, INDWKN, INDWRK, ISCALE, 00163 $ ITMP1, J, JJ, LLWORK, LLWRKN, LWKMIN, 00164 $ LWKOPT, NB, NSPLIT 00165 REAL ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, 00166 $ SIGMA, SMLNUM, TMP1, VLL, VUU 00167 * .. 00168 * .. External Functions .. 00169 LOGICAL LSAME 00170 INTEGER ILAENV 00171 REAL SLAMCH, SLANSY 00172 EXTERNAL LSAME, ILAENV, SLAMCH, SLANSY 00173 * .. 00174 * .. External Subroutines .. 00175 EXTERNAL SCOPY, SLACPY, SORGTR, SORMTR, SSCAL, SSTEBZ, 00176 $ SSTEIN, SSTEQR, SSTERF, SSWAP, SSYTRD, XERBLA 00177 * .. 00178 * .. Intrinsic Functions .. 00179 INTRINSIC MAX, MIN, SQRT 00180 * .. 00181 * .. Executable Statements .. 00182 * 00183 * Test the input parameters. 00184 * 00185 LOWER = LSAME( UPLO, 'L' ) 00186 WANTZ = LSAME( JOBZ, 'V' ) 00187 ALLEIG = LSAME( RANGE, 'A' ) 00188 VALEIG = LSAME( RANGE, 'V' ) 00189 INDEIG = LSAME( RANGE, 'I' ) 00190 LQUERY = ( LWORK.EQ.-1 ) 00191 * 00192 INFO = 0 00193 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN 00194 INFO = -1 00195 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN 00196 INFO = -2 00197 ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN 00198 INFO = -3 00199 ELSE IF( N.LT.0 ) THEN 00200 INFO = -4 00201 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 00202 INFO = -6 00203 ELSE 00204 IF( VALEIG ) THEN 00205 IF( N.GT.0 .AND. VU.LE.VL ) 00206 $ INFO = -8 00207 ELSE IF( INDEIG ) THEN 00208 IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN 00209 INFO = -9 00210 ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN 00211 INFO = -10 00212 END IF 00213 END IF 00214 END IF 00215 IF( INFO.EQ.0 ) THEN 00216 IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN 00217 INFO = -15 00218 END IF 00219 END IF 00220 * 00221 IF( INFO.EQ.0 ) THEN 00222 IF( N.LE.1 ) THEN 00223 LWKMIN = 1 00224 WORK( 1 ) = LWKMIN 00225 ELSE 00226 LWKMIN = 8*N 00227 NB = ILAENV( 1, 'SSYTRD', UPLO, N, -1, -1, -1 ) 00228 NB = MAX( NB, ILAENV( 1, 'SORMTR', UPLO, N, -1, -1, -1 ) ) 00229 LWKOPT = MAX( LWKMIN, ( NB + 3 )*N ) 00230 WORK( 1 ) = LWKOPT 00231 END IF 00232 * 00233 IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) 00234 $ INFO = -17 00235 END IF 00236 * 00237 IF( INFO.NE.0 ) THEN 00238 CALL XERBLA( 'SSYEVX', -INFO ) 00239 RETURN 00240 ELSE IF( LQUERY ) THEN 00241 RETURN 00242 END IF 00243 * 00244 * Quick return if possible 00245 * 00246 M = 0 00247 IF( N.EQ.0 ) THEN 00248 RETURN 00249 END IF 00250 * 00251 IF( N.EQ.1 ) THEN 00252 IF( ALLEIG .OR. INDEIG ) THEN 00253 M = 1 00254 W( 1 ) = A( 1, 1 ) 00255 ELSE 00256 IF( VL.LT.A( 1, 1 ) .AND. VU.GE.A( 1, 1 ) ) THEN 00257 M = 1 00258 W( 1 ) = A( 1, 1 ) 00259 END IF 00260 END IF 00261 IF( WANTZ ) 00262 $ Z( 1, 1 ) = ONE 00263 RETURN 00264 END IF 00265 * 00266 * Get machine constants. 00267 * 00268 SAFMIN = SLAMCH( 'Safe minimum' ) 00269 EPS = SLAMCH( 'Precision' ) 00270 SMLNUM = SAFMIN / EPS 00271 BIGNUM = ONE / SMLNUM 00272 RMIN = SQRT( SMLNUM ) 00273 RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) ) 00274 * 00275 * Scale matrix to allowable range, if necessary. 00276 * 00277 ISCALE = 0 00278 ABSTLL = ABSTOL 00279 IF( VALEIG ) THEN 00280 VLL = VL 00281 VUU = VU 00282 END IF 00283 ANRM = SLANSY( 'M', UPLO, N, A, LDA, WORK ) 00284 IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN 00285 ISCALE = 1 00286 SIGMA = RMIN / ANRM 00287 ELSE IF( ANRM.GT.RMAX ) THEN 00288 ISCALE = 1 00289 SIGMA = RMAX / ANRM 00290 END IF 00291 IF( ISCALE.EQ.1 ) THEN 00292 IF( LOWER ) THEN 00293 DO 10 J = 1, N 00294 CALL SSCAL( N-J+1, SIGMA, A( J, J ), 1 ) 00295 10 CONTINUE 00296 ELSE 00297 DO 20 J = 1, N 00298 CALL SSCAL( J, SIGMA, A( 1, J ), 1 ) 00299 20 CONTINUE 00300 END IF 00301 IF( ABSTOL.GT.0 ) 00302 $ ABSTLL = ABSTOL*SIGMA 00303 IF( VALEIG ) THEN 00304 VLL = VL*SIGMA 00305 VUU = VU*SIGMA 00306 END IF 00307 END IF 00308 * 00309 * Call SSYTRD to reduce symmetric matrix to tridiagonal form. 00310 * 00311 INDTAU = 1 00312 INDE = INDTAU + N 00313 INDD = INDE + N 00314 INDWRK = INDD + N 00315 LLWORK = LWORK - INDWRK + 1 00316 CALL SSYTRD( UPLO, N, A, LDA, WORK( INDD ), WORK( INDE ), 00317 $ WORK( INDTAU ), WORK( INDWRK ), LLWORK, IINFO ) 00318 * 00319 * If all eigenvalues are desired and ABSTOL is less than or equal to 00320 * zero, then call SSTERF or SORGTR and SSTEQR. If this fails for 00321 * some eigenvalue, then try SSTEBZ. 00322 * 00323 TEST = .FALSE. 00324 IF( INDEIG ) THEN 00325 IF( IL.EQ.1 .AND. IU.EQ.N ) THEN 00326 TEST = .TRUE. 00327 END IF 00328 END IF 00329 IF( ( ALLEIG .OR. TEST ) .AND. ( ABSTOL.LE.ZERO ) ) THEN 00330 CALL SCOPY( N, WORK( INDD ), 1, W, 1 ) 00331 INDEE = INDWRK + 2*N 00332 IF( .NOT.WANTZ ) THEN 00333 CALL SCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 ) 00334 CALL SSTERF( N, W, WORK( INDEE ), INFO ) 00335 ELSE 00336 CALL SLACPY( 'A', N, N, A, LDA, Z, LDZ ) 00337 CALL SORGTR( UPLO, N, Z, LDZ, WORK( INDTAU ), 00338 $ WORK( INDWRK ), LLWORK, IINFO ) 00339 CALL SCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 ) 00340 CALL SSTEQR( JOBZ, N, W, WORK( INDEE ), Z, LDZ, 00341 $ WORK( INDWRK ), INFO ) 00342 IF( INFO.EQ.0 ) THEN 00343 DO 30 I = 1, N 00344 IFAIL( I ) = 0 00345 30 CONTINUE 00346 END IF 00347 END IF 00348 IF( INFO.EQ.0 ) THEN 00349 M = N 00350 GO TO 40 00351 END IF 00352 INFO = 0 00353 END IF 00354 * 00355 * Otherwise, call SSTEBZ and, if eigenvectors are desired, SSTEIN. 00356 * 00357 IF( WANTZ ) THEN 00358 ORDER = 'B' 00359 ELSE 00360 ORDER = 'E' 00361 END IF 00362 INDIBL = 1 00363 INDISP = INDIBL + N 00364 INDIWO = INDISP + N 00365 CALL SSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL, 00366 $ WORK( INDD ), WORK( INDE ), M, NSPLIT, W, 00367 $ IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWRK ), 00368 $ IWORK( INDIWO ), INFO ) 00369 * 00370 IF( WANTZ ) THEN 00371 CALL SSTEIN( N, WORK( INDD ), WORK( INDE ), M, W, 00372 $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ, 00373 $ WORK( INDWRK ), IWORK( INDIWO ), IFAIL, INFO ) 00374 * 00375 * Apply orthogonal matrix used in reduction to tridiagonal 00376 * form to eigenvectors returned by SSTEIN. 00377 * 00378 INDWKN = INDE 00379 LLWRKN = LWORK - INDWKN + 1 00380 CALL SORMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ), Z, 00381 $ LDZ, WORK( INDWKN ), LLWRKN, IINFO ) 00382 END IF 00383 * 00384 * If matrix was scaled, then rescale eigenvalues appropriately. 00385 * 00386 40 CONTINUE 00387 IF( ISCALE.EQ.1 ) THEN 00388 IF( INFO.EQ.0 ) THEN 00389 IMAX = M 00390 ELSE 00391 IMAX = INFO - 1 00392 END IF 00393 CALL SSCAL( IMAX, ONE / SIGMA, W, 1 ) 00394 END IF 00395 * 00396 * If eigenvalues are not in order, then sort them, along with 00397 * eigenvectors. 00398 * 00399 IF( WANTZ ) THEN 00400 DO 60 J = 1, M - 1 00401 I = 0 00402 TMP1 = W( J ) 00403 DO 50 JJ = J + 1, M 00404 IF( W( JJ ).LT.TMP1 ) THEN 00405 I = JJ 00406 TMP1 = W( JJ ) 00407 END IF 00408 50 CONTINUE 00409 * 00410 IF( I.NE.0 ) THEN 00411 ITMP1 = IWORK( INDIBL+I-1 ) 00412 W( I ) = W( J ) 00413 IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 ) 00414 W( J ) = TMP1 00415 IWORK( INDIBL+J-1 ) = ITMP1 00416 CALL SSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 ) 00417 IF( INFO.NE.0 ) THEN 00418 ITMP1 = IFAIL( I ) 00419 IFAIL( I ) = IFAIL( J ) 00420 IFAIL( J ) = ITMP1 00421 END IF 00422 END IF 00423 60 CONTINUE 00424 END IF 00425 * 00426 * Set WORK(1) to optimal workspace size. 00427 * 00428 WORK( 1 ) = LWKOPT 00429 * 00430 RETURN 00431 * 00432 * End of SSYEVX 00433 * 00434 END
1.7.3 
