SUBROUTINE ZHBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, \$ VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK, \$ IWORK, IFAIL, INFO ) * * -- LAPACK driver routine (version 3.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. Scalar Arguments .. CHARACTER JOBZ, RANGE, UPLO INTEGER IL, INFO, IU, KD, LDAB, LDQ, LDZ, M, N DOUBLE PRECISION ABSTOL, VL, VU * .. * .. Array Arguments .. INTEGER IFAIL( * ), IWORK( * ) DOUBLE PRECISION RWORK( * ), W( * ) COMPLEX*16 AB( LDAB, * ), Q( LDQ, * ), WORK( * ), \$ Z( LDZ, * ) * .. * * Purpose * ======= * * ZHBEVX computes selected eigenvalues and, optionally, eigenvectors * of a complex Hermitian band 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. * * UPLO (input) CHARACTER*1 * = 'U': Upper triangle of A is stored; * = 'L': Lower triangle of A is stored. * * N (input) INTEGER * The order of the matrix A. N >= 0. * * KD (input) INTEGER * The number of superdiagonals of the matrix A if UPLO = 'U', * or the number of subdiagonals if UPLO = 'L'. KD >= 0. * * AB (input/output) COMPLEX*16 array, dimension (LDAB, N) * On entry, the upper or lower triangle of the Hermitian band * matrix A, stored in the first KD+1 rows of the array. The * j-th column of A is stored in the j-th column of the array AB * as follows: * if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). * * On exit, AB is overwritten by values generated during the * reduction to tridiagonal form. * * LDAB (input) INTEGER * The leading dimension of the array AB. LDAB >= KD + 1. * * Q (output) COMPLEX*16 array, dimension (LDQ, N) * If JOBZ = 'V', the N-by-N unitary matrix used in the * reduction to tridiagonal form. * If JOBZ = 'N', the array Q is not referenced. * * LDQ (input) INTEGER * The leading dimension of the array Q. If JOBZ = 'V', then * LDQ >= max(1,N). * * VL (input) DOUBLE PRECISION * VU (input) DOUBLE PRECISION * 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) DOUBLE PRECISION * 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 obtained * by reducing AB to tridiagonal form. * * Eigenvalues will be computed most accurately when ABSTOL is * set to twice the underflow threshold 2*DLAMCH('S'), not zero. * If this routine returns with INFO>0, indicating that some * eigenvectors did not converge, try setting ABSTOL to * 2*DLAMCH('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) DOUBLE PRECISION array, dimension (N) * The first M elements contain the selected eigenvalues in * ascending order. * * Z (output) COMPLEX*16 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, 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) COMPLEX*16 array, dimension (N) * * RWORK (workspace) DOUBLE PRECISION array, dimension (7*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 .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) COMPLEX*16 CZERO, CONE PARAMETER ( CZERO = ( 0.0D0, 0.0D0 ), \$ CONE = ( 1.0D0, 0.0D0 ) ) * .. * .. Local Scalars .. LOGICAL ALLEIG, INDEIG, LOWER, TEST, VALEIG, WANTZ CHARACTER ORDER INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL, \$ INDISP, INDIWK, INDRWK, INDWRK, ISCALE, ITMP1, \$ J, JJ, NSPLIT DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, \$ SIGMA, SMLNUM, TMP1, VLL, VUU COMPLEX*16 CTMP1 * .. * .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, ZLANHB EXTERNAL LSAME, DLAMCH, ZLANHB * .. * .. External Subroutines .. EXTERNAL DCOPY, DSCAL, DSTEBZ, DSTERF, XERBLA, ZCOPY, \$ ZGEMV, ZHBTRD, ZLACPY, ZLASCL, ZSTEIN, ZSTEQR, \$ ZSWAP * .. * .. Intrinsic Functions .. INTRINSIC DBLE, 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' ) LOWER = LSAME( UPLO, 'L' ) * 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( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( KD.LT.0 ) THEN INFO = -5 ELSE IF( LDAB.LT.KD+1 ) THEN INFO = -7 ELSE IF( WANTZ .AND. LDQ.LT.MAX( 1, N ) ) THEN INFO = -9 ELSE IF( VALEIG ) THEN IF( N.GT.0 .AND. VU.LE.VL ) \$ INFO = -11 ELSE IF( INDEIG ) THEN IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN INFO = -12 ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN INFO = -13 END IF END IF END IF IF( INFO.EQ.0 ) THEN IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) \$ INFO = -18 END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'ZHBEVX', -INFO ) RETURN END IF * * Quick return if possible * M = 0 IF( N.EQ.0 ) \$ RETURN * IF( N.EQ.1 ) THEN M = 1 IF( LOWER ) THEN CTMP1 = AB( 1, 1 ) ELSE CTMP1 = AB( KD+1, 1 ) END IF TMP1 = DBLE( CTMP1 ) IF( VALEIG ) THEN IF( .NOT.( VL.LT.TMP1 .AND. VU.GE.TMP1 ) ) \$ M = 0 END IF IF( M.EQ.1 ) THEN W( 1 ) = CTMP1 IF( WANTZ ) \$ Z( 1, 1 ) = CONE END IF RETURN END IF * * Get machine constants. * SAFMIN = DLAMCH( 'Safe minimum' ) EPS = DLAMCH( '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 ABSTLL = ABSTOL IF( VALEIG ) THEN VLL = VL VUU = VU ELSE VLL = ZERO VUU = ZERO END IF ANRM = ZLANHB( 'M', UPLO, N, KD, AB, LDAB, RWORK ) IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN ISCALE = 1 SIGMA = RMIN / ANRM ELSE IF( ANRM.GT.RMAX ) THEN ISCALE = 1 SIGMA = RMAX / ANRM END IF IF( ISCALE.EQ.1 ) THEN IF( LOWER ) THEN CALL ZLASCL( 'B', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO ) ELSE CALL ZLASCL( 'Q', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO ) END IF IF( ABSTOL.GT.0 ) \$ ABSTLL = ABSTOL*SIGMA IF( VALEIG ) THEN VLL = VL*SIGMA VUU = VU*SIGMA END IF END IF * * Call ZHBTRD to reduce Hermitian band matrix to tridiagonal form. * INDD = 1 INDE = INDD + N INDRWK = INDE + N INDWRK = 1 CALL ZHBTRD( JOBZ, UPLO, N, KD, AB, LDAB, RWORK( INDD ), \$ RWORK( INDE ), Q, LDQ, WORK( INDWRK ), IINFO ) * * If all eigenvalues are desired and ABSTOL is less than or equal * to zero, then call DSTERF or ZSTEQR. If this fails for some * eigenvalue, then try DSTEBZ. * 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 DCOPY( N, RWORK( INDD ), 1, W, 1 ) INDEE = INDRWK + 2*N IF( .NOT.WANTZ ) THEN CALL DCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 ) CALL DSTERF( N, W, RWORK( INDEE ), INFO ) ELSE CALL ZLACPY( 'A', N, N, Q, LDQ, Z, LDZ ) CALL DCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 ) CALL ZSTEQR( JOBZ, N, W, RWORK( INDEE ), Z, LDZ, \$ RWORK( INDRWK ), 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 30 END IF INFO = 0 END IF * * Otherwise, call DSTEBZ and, if eigenvectors are desired, ZSTEIN. * IF( WANTZ ) THEN ORDER = 'B' ELSE ORDER = 'E' END IF INDIBL = 1 INDISP = INDIBL + N INDIWK = INDISP + N CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL, \$ RWORK( INDD ), RWORK( INDE ), M, NSPLIT, W, \$ IWORK( INDIBL ), IWORK( INDISP ), RWORK( INDRWK ), \$ IWORK( INDIWK ), INFO ) * IF( WANTZ ) THEN CALL ZSTEIN( N, RWORK( INDD ), RWORK( INDE ), M, W, \$ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ, \$ RWORK( INDRWK ), IWORK( INDIWK ), IFAIL, INFO ) * * Apply unitary matrix used in reduction to tridiagonal * form to eigenvectors returned by ZSTEIN. * DO 20 J = 1, M CALL ZCOPY( N, Z( 1, J ), 1, WORK( 1 ), 1 ) CALL ZGEMV( 'N', N, N, CONE, Q, LDQ, WORK, 1, CZERO, \$ Z( 1, J ), 1 ) 20 CONTINUE END IF * * If matrix was scaled, then rescale eigenvalues appropriately. * 30 CONTINUE IF( ISCALE.EQ.1 ) THEN IF( INFO.EQ.0 ) THEN IMAX = M ELSE IMAX = INFO - 1 END IF CALL DSCAL( IMAX, ONE / SIGMA, W, 1 ) END IF * * If eigenvalues are not in order, then sort them, along with * eigenvectors. * IF( WANTZ ) THEN DO 50 J = 1, M - 1 I = 0 TMP1 = W( J ) DO 40 JJ = J + 1, M IF( W( JJ ).LT.TMP1 ) THEN I = JJ TMP1 = W( JJ ) END IF 40 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 ZSWAP( 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 50 CONTINUE END IF * RETURN * * End of ZHBEVX * END