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 "Accurate Eigenvalues of a Symmetric Tridiagonal


 Matrix", Report CS41, Computer Science Dept., Stanford


 University, July 21, 1966.

RANGE

          RANGE is CHARACTER*1


          = 'A': ("All")   all eigenvalues will be found.


          = 'V': ("Value") all eigenvalues in the half-open interval


                           (VL, VU] will be found.


          = 'I': ("Index") the IL-th through IU-th eigenvalues (of the


                           entire matrix) will be found.

ORDER

          ORDER is CHARACTER*1


          = 'B': ("By Block") the eigenvalues will be grouped by


                              split-off block (see IBLOCK, ISPLIT) and


                              ordered from smallest to largest within


                              the block.


          = 'E': ("Entire matrix")


                              the eigenvalues for the entire matrix


                              will be ordered from smallest to


                              largest.

N

          N is INTEGER


          The order of the tridiagonal matrix T.  N >= 0.

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'.

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'.

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'.

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'.

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.

D

          D is DOUBLE PRECISION array, dimension (N)


          The n diagonal elements of the tridiagonal matrix T.

E

          E is DOUBLE PRECISION array, dimension (N-1)


          The (n-1) off-diagonal elements of the tridiagonal matrix T.

M

          M is INTEGER


          The actual number of eigenvalues found. 0 <= M <= N.


          (See also the description of INFO=2,3.)

NSPLIT

          NSPLIT is INTEGER


          The number of diagonal blocks in the matrix T.


          1 <= NSPLIT <= N.

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.)

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.)

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.)

WORK

          WORK is DOUBLE PRECISION array, dimension (4*N)

IWORK

          IWORK is INTEGER array, dimension (3*N)

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 "FUDGE" 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 "FUDGE",


                              recompile, and try again.

  RELFAC  DOUBLE PRECISION, default = 2.0e0


          The relative tolerance.  An interval (a,b] lies within


          "relative tolerance" if  b-a < RELFAC*ulp*max(|a|,|b|),


          where "ulp" is the machine precision (distance from 1 to


          the next larger floating point number.)


  FUDGE   DOUBLE PRECISION, default = 2


          A "fudge factor" 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.

Univ. of Tennessee

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NAG Ltd.

June 2016