LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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◆ dstebz()

subroutine dstebz ( character range,
character order,
integer n,
double precision vl,
double precision vu,
integer il,
integer iu,
double precision abstol,
double precision, dimension( * ) d,
double precision, dimension( * ) e,
integer m,
integer nsplit,
double precision, dimension( * ) w,
integer, dimension( * ) iblock,
integer, dimension( * ) isplit,
double precision, dimension( * ) work,
integer, dimension( * ) iwork,
integer info )

DSTEBZ

Download DSTEBZ + dependencies [TGZ] [ZIP] [TXT]

Purpose:
!>
!> DSTEBZ computes the eigenvalues of a symmetric tridiagonal
!> matrix T.  The user may ask for all eigenvalues, all eigenvalues
!> in the half-open interval (VL, VU], or the IL-th through IU-th
!> eigenvalues.
!>
!> To avoid overflow, the matrix must be scaled so that its
!> largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest
!> accuracy, it should not be much smaller than that.
!>
!> See W. Kahan , Report CS41, Computer Science Dept., Stanford
!> University, July 21, 1966.
!>
Parameters
[in]RANGE
!>          RANGE is 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 (of the
!>                           entire matrix) will be found.
!>
[in]ORDER
!>          ORDER is CHARACTER*1
!>          = 'B': () the eigenvalues will be grouped by
!>                              split-off block (see IBLOCK, ISPLIT) and
!>                              ordered from smallest to largest within
!>                              the block.
!>          = 'E': ()
!>                              the eigenvalues for the entire matrix
!>                              will be ordered from smallest to
!>                              largest.
!>
[in]N
!>          N is INTEGER
!>          The order of the tridiagonal matrix T.  N >= 0.
!>
[in]VL
!>          VL is DOUBLE PRECISION
!>
!>          If RANGE='V', the lower bound of the interval to
!>          be searched for eigenvalues.  Eigenvalues less than or equal
!>          to VL, or greater than VU, will not be returned.  VL < VU.
!>          Not referenced if RANGE = 'A' or 'I'.
!>
[in]VU
!>          VU is DOUBLE PRECISION
!>
!>          If RANGE='V', the upper bound of the interval to
!>          be searched for eigenvalues.  Eigenvalues less than or equal
!>          to VL, or greater than VU, will not be returned.  VL < VU.
!>          Not referenced if RANGE = 'A' or 'I'.
!>
[in]IL
!>          IL is INTEGER
!>
!>          If RANGE='I', the index of the
!>          smallest eigenvalue 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'.
!>
[in]IU
!>          IU is INTEGER
!>
!>          If RANGE='I', the index of the
!>          largest eigenvalue 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'.
!>
[in]ABSTOL
!>          ABSTOL is DOUBLE PRECISION
!>          The absolute tolerance for the eigenvalues.  An eigenvalue
!>          (or cluster) is considered to be located if it has been
!>          determined to lie in an interval whose width is ABSTOL or
!>          less.  If ABSTOL is less than or equal to zero, then ULP*|T|
!>          will be used, where |T| means the 1-norm of T.
!>
!>          Eigenvalues will be computed most accurately when ABSTOL is
!>          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
!>
[in]D
!>          D is DOUBLE PRECISION array, dimension (N)
!>          The n diagonal elements of the tridiagonal matrix T.
!>
[in]E
!>          E is DOUBLE PRECISION array, dimension (N-1)
!>          The (n-1) off-diagonal elements of the tridiagonal matrix T.
!>
[out]M
!>          M is INTEGER
!>          The actual number of eigenvalues found. 0 <= M <= N.
!>          (See also the description of INFO=2,3.)
!>
[out]NSPLIT
!>          NSPLIT is INTEGER
!>          The number of diagonal blocks in the matrix T.
!>          1 <= NSPLIT <= N.
!>
[out]W
!>          W is DOUBLE PRECISION array, dimension (N)
!>          On exit, the first M elements of W will contain the
!>          eigenvalues.  (DSTEBZ may use the remaining N-M elements as
!>          workspace.)
!>
[out]IBLOCK
!>          IBLOCK is INTEGER array, dimension (N)
!>          At each row/column j where E(j) is zero or small, the
!>          matrix T is considered to split into a block diagonal
!>          matrix.  On exit, if INFO = 0, IBLOCK(i) specifies to which
!>          block (from 1 to the number of blocks) the eigenvalue W(i)
!>          belongs.  (DSTEBZ may use the remaining N-M elements as
!>          workspace.)
!>
[out]ISPLIT
!>          ISPLIT is INTEGER array, dimension (N)
!>          The splitting points, at which T breaks up into submatrices.
!>          The first submatrix consists of rows/columns 1 to ISPLIT(1),
!>          the second of rows/columns ISPLIT(1)+1 through ISPLIT(2),
!>          etc., and the NSPLIT-th consists of rows/columns
!>          ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.
!>          (Only the first NSPLIT elements will actually be used, but
!>          since the user cannot know a priori what value NSPLIT will
!>          have, N words must be reserved for ISPLIT.)
!>
[out]WORK
!>          WORK is DOUBLE PRECISION array, dimension (4*N)
!>
[out]IWORK
!>          IWORK is INTEGER array, dimension (3*N)
!>
[out]INFO
!>          INFO is INTEGER
!>          = 0:  successful exit
!>          < 0:  if INFO = -i, the i-th argument had an illegal value
!>          > 0:  some or all of the eigenvalues failed to converge or
!>                were not computed:
!>                =1 or 3: Bisection failed to converge for some
!>                        eigenvalues; these eigenvalues are flagged by a
!>                        negative block number.  The effect is that the
!>                        eigenvalues may not be as accurate as the
!>                        absolute and relative tolerances.  This is
!>                        generally caused by unexpectedly inaccurate
!>                        arithmetic.
!>                =2 or 3: RANGE='I' only: Not all of the eigenvalues
!>                        IL:IU were found.
!>                        Effect: M < IU+1-IL
!>                        Cause:  non-monotonic arithmetic, causing the
!>                                Sturm sequence to be non-monotonic.
!>                        Cure:   recalculate, using RANGE='A', and pick
!>                                out eigenvalues IL:IU.  In some cases,
!>                                increasing the PARAMETER  may
!>                                make things work.
!>                = 4:    RANGE='I', and the Gershgorin interval
!>                        initially used was too small.  No eigenvalues
!>                        were computed.
!>                        Probable cause: your machine has sloppy
!>                                        floating-point arithmetic.
!>                        Cure: Increase the PARAMETER ,
!>                              recompile, and try again.
!>
Internal Parameters:
!>  RELFAC  DOUBLE PRECISION, default = 2.0e0
!>          The relative tolerance.  An interval (a,b] lies within
!>           if  b-a < RELFAC*ulp*max(|a|,|b|),
!>          where  is the machine precision (distance from 1 to
!>          the next larger floating point number.)
!>
!>  FUDGE   DOUBLE PRECISION, default = 2
!>          A  to widen the Gershgorin intervals.  Ideally,
!>          a value of 1 should work, but on machines with sloppy
!>          arithmetic, this needs to be larger.  The default for
!>          publicly released versions should be large enough to handle
!>          the worst machine around.  Note that this has no effect
!>          on accuracy of the solution.
!>
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.