SUBROUTINE SSTEVX( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, \$ M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO ) * * -- LAPACK driver routine (version 3.2) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * November 2006 * * .. Scalar Arguments .. CHARACTER JOBZ, RANGE INTEGER IL, INFO, IU, LDZ, M, N REAL ABSTOL, VL, VU * .. * .. Array Arguments .. INTEGER IFAIL( * ), IWORK( * ) REAL D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * ) * .. * * Purpose * ======= * * SSTEVX computes selected eigenvalues and, optionally, eigenvectors * of a real symmetric tridiagonal matrix A. Eigenvalues and * eigenvectors can be selected by specifying either a range of values * or a range of indices for the desired eigenvalues. * * 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 * A. * On exit, D may be multiplied by a constant factor chosen * to avoid over/underflow in computing the eigenvalues. * * E (input/output) REAL array, dimension (max(1,N-1)) * On entry, the (n-1) subdiagonal elements of the tridiagonal * matrix A in elements 1 to N-1 of E. * On exit, E may be multiplied by a constant factor chosen * to avoid over/underflow in computing the eigenvalues. * * 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; IL = 1 and IU = 0 if N = 0. * Not referenced if RANGE = 'A' or 'V'. * * ABSTOL (input) REAL * The absolute error tolerance for the eigenvalues. * An approximate eigenvalue is accepted as converged * when it is determined to lie in an interval [a,b] * of width less than or equal to * * ABSTOL + EPS * max( |a|,|b| ) , * * where EPS is the machine precision. If ABSTOL is less * than or equal to zero, then EPS*|T| will be used in * its place, where |T| is the 1-norm of the tridiagonal * matrix. * * Eigenvalues will be computed most accurately when ABSTOL is * set to twice the underflow threshold 2*SLAMCH('S'), not zero. * If this routine returns with INFO>0, indicating that some * eigenvectors did not converge, try setting ABSTOL to * 2*SLAMCH('S'). * * See "Computing Small Singular Values of Bidiagonal Matrices * with Guaranteed High Relative Accuracy," by Demmel and * Kahan, LAPACK Working Note #3. * * 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', then if INFO = 0, the first M columns of Z * contain the orthonormal eigenvectors of the matrix A * corresponding to the selected eigenvalues, with the i-th * column of Z holding the eigenvector associated with W(i). * If an eigenvector fails to converge (INFO > 0), then that * column of Z contains the latest approximation to the * eigenvector, and the index of the eigenvector is returned * in IFAIL. 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 an upper bound must be used. * * LDZ (input) INTEGER * The leading dimension of the array Z. LDZ >= 1, and if * JOBZ = 'V', LDZ >= max(1,N). * * WORK (workspace) REAL array, dimension (5*N) * * IWORK (workspace) INTEGER array, dimension (5*N) * * IFAIL (output) INTEGER array, dimension (N) * If JOBZ = 'V', then if INFO = 0, the first M elements of * IFAIL are zero. If INFO > 0, then IFAIL contains the * indices of the eigenvectors that failed to converge. * If JOBZ = 'N', then IFAIL is not referenced. * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * > 0: if INFO = i, then i eigenvectors failed to converge. * Their indices are stored in array IFAIL. * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 ) * .. * .. Local Scalars .. LOGICAL ALLEIG, INDEIG, TEST, VALEIG, WANTZ CHARACTER ORDER INTEGER I, IMAX, INDIBL, INDISP, INDIWO, INDWRK, \$ ISCALE, ITMP1, J, JJ, NSPLIT REAL BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM, \$ TMP1, TNRM, VLL, VUU * .. * .. External Functions .. LOGICAL LSAME REAL SLAMCH, SLANST EXTERNAL LSAME, SLAMCH, SLANST * .. * .. External Subroutines .. EXTERNAL SCOPY, SSCAL, SSTEBZ, SSTEIN, SSTEQR, SSTERF, \$ 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' ) * 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 ) THEN IF( N.GT.0 .AND. VU.LE.VL ) \$ INFO = -7 ELSE IF( INDEIG ) THEN IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN INFO = -8 ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN INFO = -9 END IF END IF END IF IF( INFO.EQ.0 ) THEN IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) \$ INFO = -14 END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'SSTEVX', -INFO ) RETURN END IF * * Quick return if possible * 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( VL.LT.D( 1 ) .AND. VU.GE.D( 1 ) ) THEN M = 1 W( 1 ) = D( 1 ) END IF END IF IF( WANTZ ) \$ Z( 1, 1 ) = ONE RETURN 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 ) ) ) * * Scale matrix to allowable range, if necessary. * ISCALE = 0 IF ( VALEIG ) THEN VLL = VL VUU = VU ELSE VLL = ZERO VUU = ZERO ENDIF TNRM = SLANST( 'M', N, D, E ) IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN ISCALE = 1 SIGMA = RMIN / TNRM ELSE IF( TNRM.GT.RMAX ) THEN ISCALE = 1 SIGMA = RMAX / TNRM END IF IF( ISCALE.EQ.1 ) THEN CALL SSCAL( N, SIGMA, D, 1 ) CALL SSCAL( N-1, SIGMA, E( 1 ), 1 ) IF( VALEIG ) THEN VLL = VL*SIGMA VUU = VU*SIGMA END IF END IF * * If all eigenvalues are desired and ABSTOL is less than zero, then * call SSTERF or SSTEQR. If this fails for some eigenvalue, then * try SSTEBZ. * TEST = .FALSE. IF( INDEIG ) THEN IF( IL.EQ.1 .AND. IU.EQ.N ) THEN TEST = .TRUE. END IF END IF IF( ( ALLEIG .OR. TEST ) .AND. ( ABSTOL.LE.ZERO ) ) THEN CALL SCOPY( N, D, 1, W, 1 ) CALL SCOPY( N-1, E( 1 ), 1, WORK( 1 ), 1 ) INDWRK = N + 1 IF( .NOT.WANTZ ) THEN CALL SSTERF( N, W, WORK, INFO ) ELSE CALL SSTEQR( 'I', N, W, WORK, Z, LDZ, WORK( INDWRK ), INFO ) IF( INFO.EQ.0 ) THEN DO 10 I = 1, N IFAIL( I ) = 0 10 CONTINUE END IF END IF IF( INFO.EQ.0 ) THEN M = N GO TO 20 END IF INFO = 0 END IF * * Otherwise, call SSTEBZ and, if eigenvectors are desired, SSTEIN. * IF( WANTZ ) THEN ORDER = 'B' ELSE ORDER = 'E' END IF INDWRK = 1 INDIBL = 1 INDISP = INDIBL + N INDIWO = INDISP + N CALL SSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTOL, D, E, M, \$ NSPLIT, W, IWORK( INDIBL ), IWORK( INDISP ), \$ WORK( INDWRK ), IWORK( INDIWO ), INFO ) * IF( WANTZ ) THEN CALL SSTEIN( N, D, E, M, W, IWORK( INDIBL ), IWORK( INDISP ), \$ Z, LDZ, WORK( INDWRK ), IWORK( INDIWO ), IFAIL, \$ INFO ) END IF * * If matrix was scaled, then rescale eigenvalues appropriately. * 20 CONTINUE IF( ISCALE.EQ.1 ) THEN IF( INFO.EQ.0 ) THEN IMAX = M ELSE IMAX = INFO - 1 END IF CALL SSCAL( IMAX, ONE / SIGMA, W, 1 ) END IF * * If eigenvalues are not in order, then sort them, along with * eigenvectors. * IF( WANTZ ) THEN DO 40 J = 1, M - 1 I = 0 TMP1 = W( J ) DO 30 JJ = J + 1, M IF( W( JJ ).LT.TMP1 ) THEN I = JJ TMP1 = W( JJ ) END IF 30 CONTINUE * IF( I.NE.0 ) THEN ITMP1 = IWORK( INDIBL+I-1 ) W( I ) = W( J ) IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 ) W( J ) = TMP1 IWORK( INDIBL+J-1 ) = ITMP1 CALL SSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 ) IF( INFO.NE.0 ) THEN ITMP1 = IFAIL( I ) IFAIL( I ) = IFAIL( J ) IFAIL( J ) = ITMP1 END IF END IF 40 CONTINUE END IF * RETURN * * End of SSTEVX * END