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LAPACK 3.3.1
Linear Algebra PACKage
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LAPACK 3.3.1
Linear Algebra PACKage
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00001 SUBROUTINE ZHBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, 00002 $ VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK, 00003 $ IWORK, 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, KD, LDAB, LDQ, LDZ, M, N 00013 DOUBLE PRECISION ABSTOL, VL, VU 00014 * .. 00015 * .. Array Arguments .. 00016 INTEGER IFAIL( * ), IWORK( * ) 00017 DOUBLE PRECISION RWORK( * ), W( * ) 00018 COMPLEX*16 AB( LDAB, * ), Q( LDQ, * ), WORK( * ), 00019 $ Z( LDZ, * ) 00020 * .. 00021 * 00022 * Purpose 00023 * ======= 00024 * 00025 * ZHBEVX computes selected eigenvalues and, optionally, eigenvectors 00026 * of a complex Hermitian band matrix A. Eigenvalues and eigenvectors 00027 * can be selected by specifying either a range of values or a range of 00028 * indices for the desired eigenvalues. 00029 * 00030 * Arguments 00031 * ========= 00032 * 00033 * JOBZ (input) CHARACTER*1 00034 * = 'N': Compute eigenvalues only; 00035 * = 'V': Compute eigenvalues and eigenvectors. 00036 * 00037 * RANGE (input) CHARACTER*1 00038 * = 'A': all eigenvalues will be found; 00039 * = 'V': all eigenvalues in the half-open interval (VL,VU] 00040 * will be found; 00041 * = 'I': the IL-th through IU-th eigenvalues will be found. 00042 * 00043 * UPLO (input) CHARACTER*1 00044 * = 'U': Upper triangle of A is stored; 00045 * = 'L': Lower triangle of A is stored. 00046 * 00047 * N (input) INTEGER 00048 * The order of the matrix A. N >= 0. 00049 * 00050 * KD (input) INTEGER 00051 * The number of superdiagonals of the matrix A if UPLO = 'U', 00052 * or the number of subdiagonals if UPLO = 'L'. KD >= 0. 00053 * 00054 * AB (input/output) COMPLEX*16 array, dimension (LDAB, N) 00055 * On entry, the upper or lower triangle of the Hermitian band 00056 * matrix A, stored in the first KD+1 rows of the array. The 00057 * j-th column of A is stored in the j-th column of the array AB 00058 * as follows: 00059 * if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; 00060 * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). 00061 * 00062 * On exit, AB is overwritten by values generated during the 00063 * reduction to tridiagonal form. 00064 * 00065 * LDAB (input) INTEGER 00066 * The leading dimension of the array AB. LDAB >= KD + 1. 00067 * 00068 * Q (output) COMPLEX*16 array, dimension (LDQ, N) 00069 * If JOBZ = 'V', the N-by-N unitary matrix used in the 00070 * reduction to tridiagonal form. 00071 * If JOBZ = 'N', the array Q is not referenced. 00072 * 00073 * LDQ (input) INTEGER 00074 * The leading dimension of the array Q. If JOBZ = 'V', then 00075 * LDQ >= max(1,N). 00076 * 00077 * VL (input) DOUBLE PRECISION 00078 * VU (input) DOUBLE PRECISION 00079 * If RANGE='V', the lower and upper bounds of the interval to 00080 * be searched for eigenvalues. VL < VU. 00081 * Not referenced if RANGE = 'A' or 'I'. 00082 * 00083 * IL (input) INTEGER 00084 * IU (input) INTEGER 00085 * If RANGE='I', the indices (in ascending order) of the 00086 * smallest and largest eigenvalues to be returned. 00087 * 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. 00088 * Not referenced if RANGE = 'A' or 'V'. 00089 * 00090 * ABSTOL (input) DOUBLE PRECISION 00091 * The absolute error tolerance for the eigenvalues. 00092 * An approximate eigenvalue is accepted as converged 00093 * when it is determined to lie in an interval [a,b] 00094 * of width less than or equal to 00095 * 00096 * ABSTOL + EPS * max( |a|,|b| ) , 00097 * 00098 * where EPS is the machine precision. If ABSTOL is less than 00099 * or equal to zero, then EPS*|T| will be used in its place, 00100 * where |T| is the 1-norm of the tridiagonal matrix obtained 00101 * by reducing AB to tridiagonal form. 00102 * 00103 * Eigenvalues will be computed most accurately when ABSTOL is 00104 * set to twice the underflow threshold 2*DLAMCH('S'), not zero. 00105 * If this routine returns with INFO>0, indicating that some 00106 * eigenvectors did not converge, try setting ABSTOL to 00107 * 2*DLAMCH('S'). 00108 * 00109 * See "Computing Small Singular Values of Bidiagonal Matrices 00110 * with Guaranteed High Relative Accuracy," by Demmel and 00111 * Kahan, LAPACK Working Note #3. 00112 * 00113 * M (output) INTEGER 00114 * The total number of eigenvalues found. 0 <= M <= N. 00115 * If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. 00116 * 00117 * W (output) DOUBLE PRECISION array, dimension (N) 00118 * The first M elements contain the selected eigenvalues in 00119 * ascending order. 00120 * 00121 * Z (output) COMPLEX*16 array, dimension (LDZ, max(1,M)) 00122 * If JOBZ = 'V', then if INFO = 0, the first M columns of Z 00123 * contain the orthonormal eigenvectors of the matrix A 00124 * corresponding to the selected eigenvalues, with the i-th 00125 * column of Z holding the eigenvector associated with W(i). 00126 * If an eigenvector fails to converge, then that column of Z 00127 * contains the latest approximation to the eigenvector, and the 00128 * index of the eigenvector is returned in IFAIL. 00129 * If JOBZ = 'N', then Z is not referenced. 00130 * Note: the user must ensure that at least max(1,M) columns are 00131 * supplied in the array Z; if RANGE = 'V', the exact value of M 00132 * is not known in advance and an upper bound must be used. 00133 * 00134 * LDZ (input) INTEGER 00135 * The leading dimension of the array Z. LDZ >= 1, and if 00136 * JOBZ = 'V', LDZ >= max(1,N). 00137 * 00138 * WORK (workspace) COMPLEX*16 array, dimension (N) 00139 * 00140 * RWORK (workspace) DOUBLE PRECISION array, dimension (7*N) 00141 * 00142 * IWORK (workspace) INTEGER array, dimension (5*N) 00143 * 00144 * IFAIL (output) INTEGER array, dimension (N) 00145 * If JOBZ = 'V', then if INFO = 0, the first M elements of 00146 * IFAIL are zero. If INFO > 0, then IFAIL contains the 00147 * indices of the eigenvectors that failed to converge. 00148 * If JOBZ = 'N', then IFAIL is not referenced. 00149 * 00150 * INFO (output) INTEGER 00151 * = 0: successful exit 00152 * < 0: if INFO = -i, the i-th argument had an illegal value 00153 * > 0: if INFO = i, then i eigenvectors failed to converge. 00154 * Their indices are stored in array IFAIL. 00155 * 00156 * ===================================================================== 00157 * 00158 * .. Parameters .. 00159 DOUBLE PRECISION ZERO, ONE 00160 PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) 00161 COMPLEX*16 CZERO, CONE 00162 PARAMETER ( CZERO = ( 0.0D0, 0.0D0 ), 00163 $ CONE = ( 1.0D0, 0.0D0 ) ) 00164 * .. 00165 * .. Local Scalars .. 00166 LOGICAL ALLEIG, INDEIG, LOWER, TEST, VALEIG, WANTZ 00167 CHARACTER ORDER 00168 INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL, 00169 $ INDISP, INDIWK, INDRWK, INDWRK, ISCALE, ITMP1, 00170 $ J, JJ, NSPLIT 00171 DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, 00172 $ SIGMA, SMLNUM, TMP1, VLL, VUU 00173 COMPLEX*16 CTMP1 00174 * .. 00175 * .. External Functions .. 00176 LOGICAL LSAME 00177 DOUBLE PRECISION DLAMCH, ZLANHB 00178 EXTERNAL LSAME, DLAMCH, ZLANHB 00179 * .. 00180 * .. External Subroutines .. 00181 EXTERNAL DCOPY, DSCAL, DSTEBZ, DSTERF, XERBLA, ZCOPY, 00182 $ ZGEMV, ZHBTRD, ZLACPY, ZLASCL, ZSTEIN, ZSTEQR, 00183 $ ZSWAP 00184 * .. 00185 * .. Intrinsic Functions .. 00186 INTRINSIC DBLE, MAX, MIN, SQRT 00187 * .. 00188 * .. Executable Statements .. 00189 * 00190 * Test the input parameters. 00191 * 00192 WANTZ = LSAME( JOBZ, 'V' ) 00193 ALLEIG = LSAME( RANGE, 'A' ) 00194 VALEIG = LSAME( RANGE, 'V' ) 00195 INDEIG = LSAME( RANGE, 'I' ) 00196 LOWER = LSAME( UPLO, 'L' ) 00197 * 00198 INFO = 0 00199 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN 00200 INFO = -1 00201 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN 00202 INFO = -2 00203 ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN 00204 INFO = -3 00205 ELSE IF( N.LT.0 ) THEN 00206 INFO = -4 00207 ELSE IF( KD.LT.0 ) THEN 00208 INFO = -5 00209 ELSE IF( LDAB.LT.KD+1 ) THEN 00210 INFO = -7 00211 ELSE IF( WANTZ .AND. LDQ.LT.MAX( 1, N ) ) THEN 00212 INFO = -9 00213 ELSE 00214 IF( VALEIG ) THEN 00215 IF( N.GT.0 .AND. VU.LE.VL ) 00216 $ INFO = -11 00217 ELSE IF( INDEIG ) THEN 00218 IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN 00219 INFO = -12 00220 ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN 00221 INFO = -13 00222 END IF 00223 END IF 00224 END IF 00225 IF( INFO.EQ.0 ) THEN 00226 IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) 00227 $ INFO = -18 00228 END IF 00229 * 00230 IF( INFO.NE.0 ) THEN 00231 CALL XERBLA( 'ZHBEVX', -INFO ) 00232 RETURN 00233 END IF 00234 * 00235 * Quick return if possible 00236 * 00237 M = 0 00238 IF( N.EQ.0 ) 00239 $ RETURN 00240 * 00241 IF( N.EQ.1 ) THEN 00242 M = 1 00243 IF( LOWER ) THEN 00244 CTMP1 = AB( 1, 1 ) 00245 ELSE 00246 CTMP1 = AB( KD+1, 1 ) 00247 END IF 00248 TMP1 = DBLE( CTMP1 ) 00249 IF( VALEIG ) THEN 00250 IF( .NOT.( VL.LT.TMP1 .AND. VU.GE.TMP1 ) ) 00251 $ M = 0 00252 END IF 00253 IF( M.EQ.1 ) THEN 00254 W( 1 ) = CTMP1 00255 IF( WANTZ ) 00256 $ Z( 1, 1 ) = CONE 00257 END IF 00258 RETURN 00259 END IF 00260 * 00261 * Get machine constants. 00262 * 00263 SAFMIN = DLAMCH( 'Safe minimum' ) 00264 EPS = DLAMCH( 'Precision' ) 00265 SMLNUM = SAFMIN / EPS 00266 BIGNUM = ONE / SMLNUM 00267 RMIN = SQRT( SMLNUM ) 00268 RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) ) 00269 * 00270 * Scale matrix to allowable range, if necessary. 00271 * 00272 ISCALE = 0 00273 ABSTLL = ABSTOL 00274 IF( VALEIG ) THEN 00275 VLL = VL 00276 VUU = VU 00277 ELSE 00278 VLL = ZERO 00279 VUU = ZERO 00280 END IF 00281 ANRM = ZLANHB( 'M', UPLO, N, KD, AB, LDAB, RWORK ) 00282 IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN 00283 ISCALE = 1 00284 SIGMA = RMIN / ANRM 00285 ELSE IF( ANRM.GT.RMAX ) THEN 00286 ISCALE = 1 00287 SIGMA = RMAX / ANRM 00288 END IF 00289 IF( ISCALE.EQ.1 ) THEN 00290 IF( LOWER ) THEN 00291 CALL ZLASCL( 'B', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO ) 00292 ELSE 00293 CALL ZLASCL( 'Q', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO ) 00294 END IF 00295 IF( ABSTOL.GT.0 ) 00296 $ ABSTLL = ABSTOL*SIGMA 00297 IF( VALEIG ) THEN 00298 VLL = VL*SIGMA 00299 VUU = VU*SIGMA 00300 END IF 00301 END IF 00302 * 00303 * Call ZHBTRD to reduce Hermitian band matrix to tridiagonal form. 00304 * 00305 INDD = 1 00306 INDE = INDD + N 00307 INDRWK = INDE + N 00308 INDWRK = 1 00309 CALL ZHBTRD( JOBZ, UPLO, N, KD, AB, LDAB, RWORK( INDD ), 00310 $ RWORK( INDE ), Q, LDQ, WORK( INDWRK ), IINFO ) 00311 * 00312 * If all eigenvalues are desired and ABSTOL is less than or equal 00313 * to zero, then call DSTERF or ZSTEQR. If this fails for some 00314 * eigenvalue, then try DSTEBZ. 00315 * 00316 TEST = .FALSE. 00317 IF (INDEIG) THEN 00318 IF (IL.EQ.1 .AND. IU.EQ.N) THEN 00319 TEST = .TRUE. 00320 END IF 00321 END IF 00322 IF ((ALLEIG .OR. TEST) .AND. (ABSTOL.LE.ZERO)) THEN 00323 CALL DCOPY( N, RWORK( INDD ), 1, W, 1 ) 00324 INDEE = INDRWK + 2*N 00325 IF( .NOT.WANTZ ) THEN 00326 CALL DCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 ) 00327 CALL DSTERF( N, W, RWORK( INDEE ), INFO ) 00328 ELSE 00329 CALL ZLACPY( 'A', N, N, Q, LDQ, Z, LDZ ) 00330 CALL DCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 ) 00331 CALL ZSTEQR( JOBZ, N, W, RWORK( INDEE ), Z, LDZ, 00332 $ RWORK( INDRWK ), INFO ) 00333 IF( INFO.EQ.0 ) THEN 00334 DO 10 I = 1, N 00335 IFAIL( I ) = 0 00336 10 CONTINUE 00337 END IF 00338 END IF 00339 IF( INFO.EQ.0 ) THEN 00340 M = N 00341 GO TO 30 00342 END IF 00343 INFO = 0 00344 END IF 00345 * 00346 * Otherwise, call DSTEBZ and, if eigenvectors are desired, ZSTEIN. 00347 * 00348 IF( WANTZ ) THEN 00349 ORDER = 'B' 00350 ELSE 00351 ORDER = 'E' 00352 END IF 00353 INDIBL = 1 00354 INDISP = INDIBL + N 00355 INDIWK = INDISP + N 00356 CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL, 00357 $ RWORK( INDD ), RWORK( INDE ), M, NSPLIT, W, 00358 $ IWORK( INDIBL ), IWORK( INDISP ), RWORK( INDRWK ), 00359 $ IWORK( INDIWK ), INFO ) 00360 * 00361 IF( WANTZ ) THEN 00362 CALL ZSTEIN( N, RWORK( INDD ), RWORK( INDE ), M, W, 00363 $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ, 00364 $ RWORK( INDRWK ), IWORK( INDIWK ), IFAIL, INFO ) 00365 * 00366 * Apply unitary matrix used in reduction to tridiagonal 00367 * form to eigenvectors returned by ZSTEIN. 00368 * 00369 DO 20 J = 1, M 00370 CALL ZCOPY( N, Z( 1, J ), 1, WORK( 1 ), 1 ) 00371 CALL ZGEMV( 'N', N, N, CONE, Q, LDQ, WORK, 1, CZERO, 00372 $ Z( 1, J ), 1 ) 00373 20 CONTINUE 00374 END IF 00375 * 00376 * If matrix was scaled, then rescale eigenvalues appropriately. 00377 * 00378 30 CONTINUE 00379 IF( ISCALE.EQ.1 ) THEN 00380 IF( INFO.EQ.0 ) THEN 00381 IMAX = M 00382 ELSE 00383 IMAX = INFO - 1 00384 END IF 00385 CALL DSCAL( IMAX, ONE / SIGMA, W, 1 ) 00386 END IF 00387 * 00388 * If eigenvalues are not in order, then sort them, along with 00389 * eigenvectors. 00390 * 00391 IF( WANTZ ) THEN 00392 DO 50 J = 1, M - 1 00393 I = 0 00394 TMP1 = W( J ) 00395 DO 40 JJ = J + 1, M 00396 IF( W( JJ ).LT.TMP1 ) THEN 00397 I = JJ 00398 TMP1 = W( JJ ) 00399 END IF 00400 40 CONTINUE 00401 * 00402 IF( I.NE.0 ) THEN 00403 ITMP1 = IWORK( INDIBL+I-1 ) 00404 W( I ) = W( J ) 00405 IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 ) 00406 W( J ) = TMP1 00407 IWORK( INDIBL+J-1 ) = ITMP1 00408 CALL ZSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 ) 00409 IF( INFO.NE.0 ) THEN 00410 ITMP1 = IFAIL( I ) 00411 IFAIL( I ) = IFAIL( J ) 00412 IFAIL( J ) = ITMP1 00413 END IF 00414 END IF 00415 50 CONTINUE 00416 END IF 00417 * 00418 RETURN 00419 * 00420 * End of ZHBEVX 00421 * 00422 END
1.7.3 
