SUBROUTINE SSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, \$ M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK, \$ IWORK, LIWORK, INFO ) IMPLICIT NONE * * -- LAPACK computational routine (version 3.2.1) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * -- April 2009 -- * * .. Scalar Arguments .. CHARACTER JOBZ, RANGE LOGICAL TRYRAC INTEGER IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N REAL VL, VU * .. * .. Array Arguments .. INTEGER ISUPPZ( * ), IWORK( * ) REAL D( * ), E( * ), W( * ), WORK( * ) REAL Z( LDZ, * ) * .. * * Purpose * ======= * * SSTEMR computes selected eigenvalues and, optionally, eigenvectors * of a real symmetric tridiagonal matrix T. Any such unreduced matrix has * a well defined set of pairwise different real eigenvalues, the corresponding * real eigenvectors are pairwise orthogonal. * * The spectrum may be computed either completely or partially by specifying * either an interval (VL,VU] or a range of indices IL:IU for the desired * eigenvalues. * * Depending on the number of desired eigenvalues, these are computed either * by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are * computed by the use of various suitable L D L^T factorizations near clusters * of close eigenvalues (referred to as RRRs, Relatively Robust * Representations). An informal sketch of the algorithm follows. * * For each unreduced block (submatrix) of T, * (a) Compute T - sigma I = L D L^T, so that L and D * define all the wanted eigenvalues to high relative accuracy. * This means that small relative changes in the entries of D and L * cause only small relative changes in the eigenvalues and * eigenvectors. The standard (unfactored) representation of the * tridiagonal matrix T does not have this property in general. * (b) Compute the eigenvalues to suitable accuracy. * If the eigenvectors are desired, the algorithm attains full * accuracy of the computed eigenvalues only right before * the corresponding vectors have to be computed, see steps c) and d). * (c) For each cluster of close eigenvalues, select a new * shift close to the cluster, find a new factorization, and refine * the shifted eigenvalues to suitable accuracy. * (d) For each eigenvalue with a large enough relative separation compute * the corresponding eigenvector by forming a rank revealing twisted * factorization. Go back to (c) for any clusters that remain. * * For more details, see: * - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations * to compute orthogonal eigenvectors of symmetric tridiagonal matrices," * Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004. * - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and * Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25, * 2004. Also LAPACK Working Note 154. * - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric * tridiagonal eigenvalue/eigenvector problem", * Computer Science Division Technical Report No. UCB/CSD-97-971, * UC Berkeley, May 1997. * * Further Details * 1.SSTEMR works only on machines which follow IEEE-754 * floating-point standard in their handling of infinities and NaNs. * This permits the use of efficient inner loops avoiding a check for * zero divisors. * * Arguments * ========= * * JOBZ (input) CHARACTER*1 * = 'N': Compute eigenvalues only; * = 'V': Compute eigenvalues and eigenvectors. * * RANGE (input) CHARACTER*1 * = 'A': all eigenvalues will be found. * = 'V': all eigenvalues in the half-open interval (VL,VU] * will be found. * = 'I': the IL-th through IU-th eigenvalues will be found. * * N (input) INTEGER * The order of the matrix. N >= 0. * * D (input/output) REAL array, dimension (N) * On entry, the N diagonal elements of the tridiagonal matrix * T. On exit, D is overwritten. * * E (input/output) REAL array, dimension (N) * On entry, the (N-1) subdiagonal elements of the tridiagonal * matrix T in elements 1 to N-1 of E. E(N) need not be set on * input, but is used internally as workspace. * On exit, E is overwritten. * * VL (input) REAL * VU (input) REAL * If RANGE='V', the lower and upper bounds of the interval to * be searched for eigenvalues. VL < VU. * Not referenced if RANGE = 'A' or 'I'. * * IL (input) INTEGER * IU (input) INTEGER * If RANGE='I', the indices (in ascending order) of the * smallest and largest eigenvalues to be returned. * 1 <= IL <= IU <= N, if N > 0. * Not referenced if RANGE = 'A' or 'V'. * * M (output) INTEGER * The total number of eigenvalues found. 0 <= M <= N. * If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. * * W (output) REAL array, dimension (N) * The first M elements contain the selected eigenvalues in * ascending order. * * Z (output) REAL array, dimension (LDZ, max(1,M) ) * If JOBZ = 'V', and if INFO = 0, then the first M columns of Z * contain the orthonormal eigenvectors of the matrix T * corresponding to the selected eigenvalues, with the i-th * column of Z holding the eigenvector associated with W(i). * If JOBZ = 'N', then Z is not referenced. * Note: the user must ensure that at least max(1,M) columns are * supplied in the array Z; if RANGE = 'V', the exact value of M * is not known in advance and can be computed with a workspace * query by setting NZC = -1, see below. * * LDZ (input) INTEGER * The leading dimension of the array Z. LDZ >= 1, and if * JOBZ = 'V', then LDZ >= max(1,N). * * NZC (input) INTEGER * The number of eigenvectors to be held in the array Z. * If RANGE = 'A', then NZC >= max(1,N). * If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU]. * If RANGE = 'I', then NZC >= IU-IL+1. * If NZC = -1, then a workspace query is assumed; the * routine calculates the number of columns of the array Z that * are needed to hold the eigenvectors. * This value is returned as the first entry of the Z array, and * no error message related to NZC is issued by XERBLA. * * ISUPPZ (output) INTEGER ARRAY, dimension ( 2*max(1,M) ) * The support of the eigenvectors in Z, i.e., the indices * indicating the nonzero elements in Z. The i-th computed eigenvector * is nonzero only in elements ISUPPZ( 2*i-1 ) through * ISUPPZ( 2*i ). This is relevant in the case when the matrix * is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0. * * TRYRAC (input/output) LOGICAL * If TRYRAC.EQ..TRUE., indicates that the code should check whether * the tridiagonal matrix defines its eigenvalues to high relative * accuracy. If so, the code uses relative-accuracy preserving * algorithms that might be (a bit) slower depending on the matrix. * If the matrix does not define its eigenvalues to high relative * accuracy, the code can uses possibly faster algorithms. * If TRYRAC.EQ..FALSE., the code is not required to guarantee * relatively accurate eigenvalues and can use the fastest possible * techniques. * On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix * does not define its eigenvalues to high relative accuracy. * * WORK (workspace/output) REAL array, dimension (LWORK) * On exit, if INFO = 0, WORK(1) returns the optimal * (and minimal) LWORK. * * LWORK (input) INTEGER * The dimension of the array WORK. LWORK >= max(1,18*N) * if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'. * If LWORK = -1, then a workspace query is assumed; the routine * only calculates the optimal size of the WORK array, returns * this value as the first entry of the WORK array, and no error * message related to LWORK is issued by XERBLA. * * IWORK (workspace/output) INTEGER array, dimension (LIWORK) * On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. * * LIWORK (input) INTEGER * The dimension of the array IWORK. LIWORK >= max(1,10*N) * if the eigenvectors are desired, and LIWORK >= max(1,8*N) * if only the eigenvalues are to be computed. * If LIWORK = -1, then a workspace query is assumed; the * routine only calculates the optimal size of the IWORK array, * returns this value as the first entry of the IWORK array, and * no error message related to LIWORK is issued by XERBLA. * * INFO (output) INTEGER * On exit, INFO * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * > 0: if INFO = 1X, internal error in SLARRE, * if INFO = 2X, internal error in SLARRV. * Here, the digit X = ABS( IINFO ) < 10, where IINFO is * the nonzero error code returned by SLARRE or * SLARRV, respectively. * * * Further Details * =============== * * Based on contributions by * Beresford Parlett, University of California, Berkeley, USA * Jim Demmel, University of California, Berkeley, USA * Inderjit Dhillon, University of Texas, Austin, USA * Osni Marques, LBNL/NERSC, USA * Christof Voemel, University of California, Berkeley, USA * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE, FOUR, MINRGP PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0, \$ FOUR = 4.0E0, \$ MINRGP = 3.0E-3 ) * .. * .. Local Scalars .. LOGICAL ALLEIG, INDEIG, LQUERY, VALEIG, WANTZ, ZQUERY INTEGER I, IBEGIN, IEND, IFIRST, IIL, IINDBL, IINDW, \$ IINDWK, IINFO, IINSPL, IIU, ILAST, IN, INDD, \$ INDE2, INDERR, INDGP, INDGRS, INDWRK, ITMP, \$ ITMP2, J, JBLK, JJ, LIWMIN, LWMIN, NSPLIT, \$ NZCMIN, OFFSET, WBEGIN, WEND REAL BIGNUM, CS, EPS, PIVMIN, R1, R2, RMAX, RMIN, \$ RTOL1, RTOL2, SAFMIN, SCALE, SMLNUM, SN, \$ THRESH, TMP, TNRM, WL, WU * .. * .. * .. External Functions .. LOGICAL LSAME REAL SLAMCH, SLANST EXTERNAL LSAME, SLAMCH, SLANST * .. * .. External Subroutines .. EXTERNAL SCOPY, SLAE2, SLAEV2, SLARRC, SLARRE, SLARRJ, \$ SLARRR, SLARRV, SLASRT, SSCAL, SSWAP, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN, SQRT * .. * .. Executable Statements .. * * Test the input parameters. * WANTZ = LSAME( JOBZ, 'V' ) ALLEIG = LSAME( RANGE, 'A' ) VALEIG = LSAME( RANGE, 'V' ) INDEIG = LSAME( RANGE, 'I' ) * LQUERY = ( ( LWORK.EQ.-1 ).OR.( LIWORK.EQ.-1 ) ) ZQUERY = ( NZC.EQ.-1 ) * SSTEMR needs WORK of size 6*N, IWORK of size 3*N. * In addition, SLARRE needs WORK of size 6*N, IWORK of size 5*N. * Furthermore, SLARRV needs WORK of size 12*N, IWORK of size 7*N. IF( WANTZ ) THEN LWMIN = 18*N LIWMIN = 10*N ELSE * need less workspace if only the eigenvalues are wanted LWMIN = 12*N LIWMIN = 8*N ENDIF WL = ZERO WU = ZERO IIL = 0 IIU = 0 IF( VALEIG ) THEN * We do not reference VL, VU in the cases RANGE = 'I','A' * The interval (WL, WU] contains all the wanted eigenvalues. * It is either given by the user or computed in SLARRE. WL = VL WU = VU ELSEIF( INDEIG ) THEN * We do not reference IL, IU in the cases RANGE = 'V','A' IIL = IL IIU = IU ENDIF * INFO = 0 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN INFO = -1 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( VALEIG .AND. N.GT.0 .AND. WU.LE.WL ) THEN INFO = -7 ELSE IF( INDEIG .AND. ( IIL.LT.1 .OR. IIL.GT.N ) ) THEN INFO = -8 ELSE IF( INDEIG .AND. ( IIU.LT.IIL .OR. IIU.GT.N ) ) THEN INFO = -9 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN INFO = -13 ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN INFO = -17 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN INFO = -19 END IF * * Get machine constants. * SAFMIN = SLAMCH( 'Safe minimum' ) EPS = SLAMCH( 'Precision' ) SMLNUM = SAFMIN / EPS BIGNUM = ONE / SMLNUM RMIN = SQRT( SMLNUM ) RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) ) * IF( INFO.EQ.0 ) THEN WORK( 1 ) = LWMIN IWORK( 1 ) = LIWMIN * IF( WANTZ .AND. ALLEIG ) THEN NZCMIN = N ELSE IF( WANTZ .AND. VALEIG ) THEN CALL SLARRC( 'T', N, VL, VU, D, E, SAFMIN, \$ NZCMIN, ITMP, ITMP2, INFO ) ELSE IF( WANTZ .AND. INDEIG ) THEN NZCMIN = IIU-IIL+1 ELSE * WANTZ .EQ. FALSE. NZCMIN = 0 ENDIF IF( ZQUERY .AND. INFO.EQ.0 ) THEN Z( 1,1 ) = NZCMIN ELSE IF( NZC.LT.NZCMIN .AND. .NOT.ZQUERY ) THEN INFO = -14 END IF END IF IF( INFO.NE.0 ) THEN * CALL XERBLA( 'SSTEMR', -INFO ) * RETURN ELSE IF( LQUERY .OR. ZQUERY ) THEN RETURN END IF * * Handle N = 0, 1, and 2 cases immediately * M = 0 IF( N.EQ.0 ) \$ RETURN * IF( N.EQ.1 ) THEN IF( ALLEIG .OR. INDEIG ) THEN M = 1 W( 1 ) = D( 1 ) ELSE IF( WL.LT.D( 1 ) .AND. WU.GE.D( 1 ) ) THEN M = 1 W( 1 ) = D( 1 ) END IF END IF IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN Z( 1, 1 ) = ONE ISUPPZ(1) = 1 ISUPPZ(2) = 1 END IF RETURN END IF * IF( N.EQ.2 ) THEN IF( .NOT.WANTZ ) THEN CALL SLAE2( D(1), E(1), D(2), R1, R2 ) ELSE IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN CALL SLAEV2( D(1), E(1), D(2), R1, R2, CS, SN ) END IF IF( ALLEIG.OR. \$ (VALEIG.AND.(R2.GT.WL).AND. \$ (R2.LE.WU)).OR. \$ (INDEIG.AND.(IIL.EQ.1)) ) THEN M = M+1 W( M ) = R2 IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN Z( 1, M ) = -SN Z( 2, M ) = CS * Note: At most one of SN and CS can be zero. IF (SN.NE.ZERO) THEN IF (CS.NE.ZERO) THEN ISUPPZ(2*M-1) = 1 ISUPPZ(2*M-1) = 2 ELSE ISUPPZ(2*M-1) = 1 ISUPPZ(2*M-1) = 1 END IF ELSE ISUPPZ(2*M-1) = 2 ISUPPZ(2*M) = 2 END IF ENDIF ENDIF IF( ALLEIG.OR. \$ (VALEIG.AND.(R1.GT.WL).AND. \$ (R1.LE.WU)).OR. \$ (INDEIG.AND.(IIU.EQ.2)) ) THEN M = M+1 W( M ) = R1 IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN Z( 1, M ) = CS Z( 2, M ) = SN * Note: At most one of SN and CS can be zero. IF (SN.NE.ZERO) THEN IF (CS.NE.ZERO) THEN ISUPPZ(2*M-1) = 1 ISUPPZ(2*M-1) = 2 ELSE ISUPPZ(2*M-1) = 1 ISUPPZ(2*M-1) = 1 END IF ELSE ISUPPZ(2*M-1) = 2 ISUPPZ(2*M) = 2 END IF ENDIF ENDIF RETURN END IF * Continue with general N INDGRS = 1 INDERR = 2*N + 1 INDGP = 3*N + 1 INDD = 4*N + 1 INDE2 = 5*N + 1 INDWRK = 6*N + 1 * IINSPL = 1 IINDBL = N + 1 IINDW = 2*N + 1 IINDWK = 3*N + 1 * * Scale matrix to allowable range, if necessary. * The allowable range is related to the PIVMIN parameter; see the * comments in SLARRD. The preference for scaling small values * up is heuristic; we expect users' matrices not to be close to the * RMAX threshold. * SCALE = ONE TNRM = SLANST( 'M', N, D, E ) IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN SCALE = RMIN / TNRM ELSE IF( TNRM.GT.RMAX ) THEN SCALE = RMAX / TNRM END IF IF( SCALE.NE.ONE ) THEN CALL SSCAL( N, SCALE, D, 1 ) CALL SSCAL( N-1, SCALE, E, 1 ) TNRM = TNRM*SCALE IF( VALEIG ) THEN * If eigenvalues in interval have to be found, * scale (WL, WU] accordingly WL = WL*SCALE WU = WU*SCALE ENDIF END IF * * Compute the desired eigenvalues of the tridiagonal after splitting * into smaller subblocks if the corresponding off-diagonal elements * are small * THRESH is the splitting parameter for SLARRE * A negative THRESH forces the old splitting criterion based on the * size of the off-diagonal. A positive THRESH switches to splitting * which preserves relative accuracy. * IF( TRYRAC ) THEN * Test whether the matrix warrants the more expensive relative approach. CALL SLARRR( N, D, E, IINFO ) ELSE * The user does not care about relative accurately eigenvalues IINFO = -1 ENDIF * Set the splitting criterion IF (IINFO.EQ.0) THEN THRESH = EPS ELSE THRESH = -EPS * relative accuracy is desired but T does not guarantee it TRYRAC = .FALSE. ENDIF * IF( TRYRAC ) THEN * Copy original diagonal, needed to guarantee relative accuracy CALL SCOPY(N,D,1,WORK(INDD),1) ENDIF * Store the squares of the offdiagonal values of T DO 5 J = 1, N-1 WORK( INDE2+J-1 ) = E(J)**2 5 CONTINUE * Set the tolerance parameters for bisection IF( .NOT.WANTZ ) THEN * SLARRE computes the eigenvalues to full precision. RTOL1 = FOUR * EPS RTOL2 = FOUR * EPS ELSE * SLARRE computes the eigenvalues to less than full precision. * SLARRV will refine the eigenvalue approximations, and we can * need less accurate initial bisection in SLARRE. * Note: these settings do only affect the subset case and SLARRE RTOL1 = MAX( SQRT(EPS)*5.0E-2, FOUR * EPS ) RTOL2 = MAX( SQRT(EPS)*5.0E-3, FOUR * EPS ) ENDIF CALL SLARRE( RANGE, N, WL, WU, IIL, IIU, D, E, \$ WORK(INDE2), RTOL1, RTOL2, THRESH, NSPLIT, \$ IWORK( IINSPL ), M, W, WORK( INDERR ), \$ WORK( INDGP ), IWORK( IINDBL ), \$ IWORK( IINDW ), WORK( INDGRS ), PIVMIN, \$ WORK( INDWRK ), IWORK( IINDWK ), IINFO ) IF( IINFO.NE.0 ) THEN INFO = 10 + ABS( IINFO ) RETURN END IF * Note that if RANGE .NE. 'V', SLARRE computes bounds on the desired * part of the spectrum. All desired eigenvalues are contained in * (WL,WU] IF( WANTZ ) THEN * * Compute the desired eigenvectors corresponding to the computed * eigenvalues * CALL SLARRV( N, WL, WU, D, E, \$ PIVMIN, IWORK( IINSPL ), M, \$ 1, M, MINRGP, RTOL1, RTOL2, \$ W, WORK( INDERR ), WORK( INDGP ), IWORK( IINDBL ), \$ IWORK( IINDW ), WORK( INDGRS ), Z, LDZ, \$ ISUPPZ, WORK( INDWRK ), IWORK( IINDWK ), IINFO ) IF( IINFO.NE.0 ) THEN INFO = 20 + ABS( IINFO ) RETURN END IF ELSE * SLARRE computes eigenvalues of the (shifted) root representation * SLARRV returns the eigenvalues of the unshifted matrix. * However, if the eigenvectors are not desired by the user, we need * to apply the corresponding shifts from SLARRE to obtain the * eigenvalues of the original matrix. DO 20 J = 1, M ITMP = IWORK( IINDBL+J-1 ) W( J ) = W( J ) + E( IWORK( IINSPL+ITMP-1 ) ) 20 CONTINUE END IF * IF ( TRYRAC ) THEN * Refine computed eigenvalues so that they are relatively accurate * with respect to the original matrix T. IBEGIN = 1 WBEGIN = 1 DO 39 JBLK = 1, IWORK( IINDBL+M-1 ) IEND = IWORK( IINSPL+JBLK-1 ) IN = IEND - IBEGIN + 1 WEND = WBEGIN - 1 * check if any eigenvalues have to be refined in this block 36 CONTINUE IF( WEND.LT.M ) THEN IF( IWORK( IINDBL+WEND ).EQ.JBLK ) THEN WEND = WEND + 1 GO TO 36 END IF END IF IF( WEND.LT.WBEGIN ) THEN IBEGIN = IEND + 1 GO TO 39 END IF OFFSET = IWORK(IINDW+WBEGIN-1)-1 IFIRST = IWORK(IINDW+WBEGIN-1) ILAST = IWORK(IINDW+WEND-1) RTOL2 = FOUR * EPS CALL SLARRJ( IN, \$ WORK(INDD+IBEGIN-1), WORK(INDE2+IBEGIN-1), \$ IFIRST, ILAST, RTOL2, OFFSET, W(WBEGIN), \$ WORK( INDERR+WBEGIN-1 ), \$ WORK( INDWRK ), IWORK( IINDWK ), PIVMIN, \$ TNRM, IINFO ) IBEGIN = IEND + 1 WBEGIN = WEND + 1 39 CONTINUE ENDIF * * If matrix was scaled, then rescale eigenvalues appropriately. * IF( SCALE.NE.ONE ) THEN CALL SSCAL( M, ONE / SCALE, W, 1 ) END IF * * If eigenvalues are not in increasing order, then sort them, * possibly along with eigenvectors. * IF( NSPLIT.GT.1 ) THEN IF( .NOT. WANTZ ) THEN CALL SLASRT( 'I', M, W, IINFO ) IF( IINFO.NE.0 ) THEN INFO = 3 RETURN END IF ELSE DO 60 J = 1, M - 1 I = 0 TMP = W( J ) DO 50 JJ = J + 1, M IF( W( JJ ).LT.TMP ) THEN I = JJ TMP = W( JJ ) END IF 50 CONTINUE IF( I.NE.0 ) THEN W( I ) = W( J ) W( J ) = TMP IF( WANTZ ) THEN CALL SSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 ) ITMP = ISUPPZ( 2*I-1 ) ISUPPZ( 2*I-1 ) = ISUPPZ( 2*J-1 ) ISUPPZ( 2*J-1 ) = ITMP ITMP = ISUPPZ( 2*I ) ISUPPZ( 2*I ) = ISUPPZ( 2*J ) ISUPPZ( 2*J ) = ITMP END IF END IF 60 CONTINUE END IF ENDIF * * WORK( 1 ) = LWMIN IWORK( 1 ) = LIWMIN RETURN * * End of SSTEMR * END