SUBROUTINE ZHPEVD( JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, LWORK, $ RWORK, LRWORK, IWORK, LIWORK, INFO ) * * -- LAPACK driver routine (version 2.0) -- * Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., * Courant Institute, Argonne National Lab, and Rice University * September 30, 1994 * * .. Scalar Arguments .. CHARACTER JOBZ, UPLO INTEGER INFO, LDZ, LIWORK, LRWORK, LWORK, N * .. * .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION RWORK( * ), W( * ) COMPLEX*16 AP( * ), WORK( * ), Z( LDZ, * ) * .. * * Purpose * ======= * * ZHPEVD computes all the eigenvalues and, optionally, eigenvectors of * a complex Hermitian matrix A in packed storage. If eigenvectors are * desired, it uses a divide and conquer algorithm. * * The divide and conquer algorithm makes very mild assumptions about * floating point arithmetic. It will work on machines with a guard * digit in add/subtract, or on those binary machines without guard * digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or * Cray-2. It could conceivably fail on hexadecimal or decimal machines * without guard digits, but we know of none. * * Arguments * ========= * * JOBZ (input) CHARACTER*1 * = 'N': Compute eigenvalues only; * = 'V': Compute eigenvalues and eigenvectors. * * 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. * * AP (input/output) COMPLEX*16 array, dimension (N*(N+1)/2) * On entry, the upper or lower triangle of the Hermitian matrix * A, packed columnwise in a linear array. The j-th column of A * is stored in the array AP as follows: * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; * if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. * * On exit, AP is overwritten by values generated during the * reduction to tridiagonal form. If UPLO = 'U', the diagonal * and first superdiagonal of the tridiagonal matrix T overwrite * the corresponding elements of A, and if UPLO = 'L', the * diagonal and first subdiagonal of T overwrite the * corresponding elements of A. * * W (output) DOUBLE PRECISION array, dimension (N) * If INFO = 0, the eigenvalues in ascending order. * * Z (output) COMPLEX*16 array, dimension (LDZ, N) * If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal * eigenvectors of the matrix A, with the i-th column of Z * holding the eigenvector associated with W(i). * If JOBZ = 'N', then Z is not referenced. * * LDZ (input) INTEGER * The leading dimension of the array Z. LDZ >= 1, and if * JOBZ = 'V', LDZ >= max(1,N). * * WORK (workspace/output) COMPLEX*16 array, dimension (LWORK) * On exit, if LWORK > 0, WORK(1) returns the optimal LWORK. * * LWORK (input) INTEGER * The dimension of array WORK. * If N <= 1, LWORK must be at least 1. * If JOBZ = 'N' and N > 1, LWORK must be at least N. * If JOBZ = 'V' and N > 1, LWORK must be at least 2*N. * * RWORK (workspace/output) DOUBLE PRECISION array, * dimension (LRWORK) * On exit, if LRWORK > 0, RWORK(1) returns the optimal LRWORK. * * LRWORK (input) INTEGER * The dimension of array RWORK. * If N <= 1, LRWORK must be at least 1. * If JOBZ = 'N' and N > 1, LRWORK must be at least N. * If JOBZ = 'V' and N > 1, LRWORK must be at least * 1 + 4*N + 2*N*lg N + 3*N**2 , * where lg( N ) = smallest integer k such * that 2**k >= N. * * IWORK (workspace/output) INTEGER array, dimension (LIWORK) * On exit, if LIWORK > 0, IWORK(1) returns the optimal LIWORK. * * LIWORK (input) INTEGER * The dimension of array IWORK. * If JOBZ = 'N' or N <= 1, LIWORK must be at least 1. * If JOBZ = 'V' and N > 1, LIWORK must be at least 2 + 5*N. * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value. * > 0: if INFO = i, the algorithm failed to converge; i * off-diagonal elements of an intermediate tridiagonal * form did not converge to zero. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE, TWO PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 ) COMPLEX*16 CONE PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) ) * .. * .. Local Scalars .. LOGICAL WANTZ INTEGER IINFO, IMAX, INDE, INDRWK, INDTAU, INDWRK, $ ISCALE, LGN, LIWMIN, LLRWK, LLWRK, LRWMIN, $ LWMIN DOUBLE PRECISION ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, $ SMLNUM * .. * .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, ZLANHP EXTERNAL LSAME, DLAMCH, ZLANHP * .. * .. External Subroutines .. EXTERNAL DSCAL, DSTERF, XERBLA, ZDSCAL, ZHPTRD, ZSTEDC, $ ZUPMTR * .. * .. Intrinsic Functions .. INTRINSIC DBLE, INT, LOG, SQRT * .. * .. Executable Statements .. * * Test the input parameters. * WANTZ = LSAME( JOBZ, 'V' ) * INFO = 0 IF( N.LE.1 ) THEN LWMIN = 1 LIWMIN = 1 LRWMIN = 1 ELSE LGN = INT( LOG( DBLE( N ) ) / LOG( TWO ) ) IF( 2**LGN.LT.N ) $ LGN = LGN + 1 IF( 2**LGN.LT.N ) $ LGN = LGN + 1 IF( WANTZ ) THEN LWMIN = 2*N LRWMIN = 1 + 4*N + 2*N*LGN + 3*N**2 LIWMIN = 2 + 5*N ELSE LWMIN = N LRWMIN = N LIWMIN = 1 END IF END IF IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN INFO = -1 ELSE IF( .NOT.( LSAME( UPLO, 'L' ) .OR. LSAME( UPLO, 'U' ) ) ) $ THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN INFO = -7 ELSE IF( LWORK.LT.LWMIN ) THEN INFO = -9 ELSE IF( LRWORK.LT.LRWMIN ) THEN INFO = -11 ELSE IF( LIWORK.LT.LIWMIN ) THEN INFO = -13 END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'ZHPEVD ', -INFO ) GO TO 10 END IF * * Quick return if possible * IF( N.EQ.0 ) $ GO TO 10 * IF( N.EQ.1 ) THEN W( 1 ) = AP( 1 ) IF( WANTZ ) $ Z( 1, 1 ) = CONE GO TO 10 END IF * * Get machine constants. * SAFMIN = DLAMCH( 'Safe minimum' ) EPS = DLAMCH( 'Precision' ) SMLNUM = SAFMIN / EPS BIGNUM = ONE / SMLNUM RMIN = SQRT( SMLNUM ) RMAX = SQRT( BIGNUM ) * * Scale matrix to allowable range, if necessary. * ANRM = ZLANHP( 'M', UPLO, N, AP, RWORK ) ISCALE = 0 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 CALL ZDSCAL( ( N*( N+1 ) ) / 2, SIGMA, AP, 1 ) END IF * * Call ZHPTRD to reduce Hermitian packed matrix to tridiagonal form. * INDE = 1 INDTAU = 1 INDRWK = INDE + N INDWRK = INDTAU + N LLWRK = LWORK - INDWRK + 1 LLRWK = LRWORK - INDRWK + 1 CALL ZHPTRD( UPLO, N, AP, W, RWORK( INDE ), WORK( INDTAU ), $ IINFO ) * * For eigenvalues only, call DSTERF. For eigenvectors, first call * ZUPGTR to generate the orthogonal matrix, then call ZSTEDC. * IF( .NOT.WANTZ ) THEN CALL DSTERF( N, W, RWORK( INDE ), INFO ) ELSE CALL ZSTEDC( 'I', N, W, RWORK( INDE ), Z, LDZ, WORK( INDWRK ), $ LLWRK, RWORK( INDRWK ), LLRWK, IWORK, LIWORK, $ INFO ) CALL ZUPMTR( 'L', UPLO, 'N', N, N, AP, WORK( INDTAU ), Z, LDZ, $ WORK( INDWRK ), IINFO ) END IF * * If matrix was scaled, then rescale eigenvalues appropriately. * IF( ISCALE.EQ.1 ) THEN IF( INFO.EQ.0 ) THEN IMAX = N ELSE IMAX = INFO - 1 END IF CALL DSCAL( IMAX, ONE / SIGMA, W, 1 ) END IF * 10 CONTINUE IF( LWORK.GT.0 ) $ WORK( 1 ) = LWMIN IF( LRWORK.GT.0 ) $ RWORK( 1 ) = LRWMIN IF( LIWORK.GT.0 ) $ IWORK( 1 ) = LIWMIN RETURN * * End of ZHPEVD * END