DLARRD computes the eigenvalues of a symmetric tridiagonal


 matrix T to suitable accuracy. This is an auxiliary code to be


 called from DSTEMR.


 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'.

GERS

          GERS is DOUBLE PRECISION array, dimension (2*N)


          The N Gerschgorin intervals (the i-th Gerschgorin interval


          is (GERS(2*i-1), GERS(2*i)).

RELTOL

          RELTOL is DOUBLE PRECISION


          The minimum relative width of an interval.  When an interval


          is narrower than RELTOL times the larger (in


          magnitude) endpoint, then it is considered to be


          sufficiently small, i.e., converged.  Note: this should


          always be at least radix*machine epsilon.

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.

E2

          E2 is DOUBLE PRECISION array, dimension (N-1)


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

PIVMIN

          PIVMIN is DOUBLE PRECISION


          The minimum pivot allowed in the Sturm sequence for T.

NSPLIT

          NSPLIT is INTEGER


          The number of diagonal blocks in the matrix T.


          1 <= NSPLIT <= N.

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.)

M

          M is INTEGER


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


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

W

          W is DOUBLE PRECISION array, dimension (N)


          On exit, the first M elements of W will contain the


          eigenvalue approximations. DLARRD computes an interval


          I_j = (a_j, b_j] that includes eigenvalue j. The eigenvalue


          approximation is given as the interval midpoint


          W(j)= ( a_j + b_j)/2. The corresponding error is bounded by


          WERR(j) = abs( a_j - b_j)/2

WERR

          WERR is DOUBLE PRECISION array, dimension (N)


          The error bound on the corresponding eigenvalue approximation


          in W.

WL

          WL is DOUBLE PRECISION

WU

          WU is DOUBLE PRECISION


          The interval (WL, WU] contains all the wanted eigenvalues.


          If RANGE='V', then WL=VL and WU=VU.


          If RANGE='A', then WL and WU are the global Gerschgorin bounds


                        on the spectrum.


          If RANGE='I', then WL and WU are computed by DLAEBZ from the


                        index range specified.

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.  (DLARRD may use the remaining N-M elements as


          workspace.)

INDEXW

          INDEXW is INTEGER array, dimension (N)


          The indices of the eigenvalues within each block (submatrix);


          for example, INDEXW(i)= j and IBLOCK(i)=k imply that the


          i-th eigenvalue W(i) is the j-th eigenvalue in block k.

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.

  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.

W. Kahan, University of California, Berkeley, USA
Beresford Parlett, University of California, Berkeley, USA
Jim Demmel, University of California, Berkeley, USA
Inderjit Dhillon, University of Texas, Austin, USA
Osni Marques, LBNL/NERSC, USA
Christof Voemel, University of California, Berkeley, USA

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 SLARRD computes the eigenvalues of a symmetric tridiagonal


 matrix T to suitable accuracy. This is an auxiliary code to be


 called from SSTEMR.


 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 REAL


          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 REAL


          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'.

GERS

          GERS is REAL array, dimension (2*N)


          The N Gerschgorin intervals (the i-th Gerschgorin interval


          is (GERS(2*i-1), GERS(2*i)).

RELTOL

          RELTOL is REAL


          The minimum relative width of an interval.  When an interval


          is narrower than RELTOL times the larger (in


          magnitude) endpoint, then it is considered to be


          sufficiently small, i.e., converged.  Note: this should


          always be at least radix*machine epsilon.

D

          D is REAL array, dimension (N)


          The n diagonal elements of the tridiagonal matrix T.

E

          E is REAL array, dimension (N-1)


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

E2

          E2 is REAL array, dimension (N-1)


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

PIVMIN

          PIVMIN is REAL


          The minimum pivot allowed in the Sturm sequence for T.

NSPLIT

          NSPLIT is INTEGER


          The number of diagonal blocks in the matrix T.


          1 <= NSPLIT <= N.

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.)

M

          M is INTEGER


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


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

W

          W is REAL array, dimension (N)


          On exit, the first M elements of W will contain the


          eigenvalue approximations. SLARRD computes an interval


          I_j = (a_j, b_j] that includes eigenvalue j. The eigenvalue


          approximation is given as the interval midpoint


          W(j)= ( a_j + b_j)/2. The corresponding error is bounded by


          WERR(j) = abs( a_j - b_j)/2

WERR

          WERR is REAL array, dimension (N)


          The error bound on the corresponding eigenvalue approximation


          in W.

WL

          WL is REAL

WU

          WU is REAL


          The interval (WL, WU] contains all the wanted eigenvalues.


          If RANGE='V', then WL=VL and WU=VU.


          If RANGE='A', then WL and WU are the global Gerschgorin bounds


                        on the spectrum.


          If RANGE='I', then WL and WU are computed by SLAEBZ from the


                        index range specified.

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.  (SLARRD may use the remaining N-M elements as


          workspace.)

INDEXW

          INDEXW is INTEGER array, dimension (N)


          The indices of the eigenvalues within each block (submatrix);


          for example, INDEXW(i)= j and IBLOCK(i)=k imply that the


          i-th eigenvalue W(i) is the j-th eigenvalue in block k.

WORK

          WORK is REAL 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.

  FUDGE   REAL, 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.

W. Kahan, University of California, Berkeley, USA
Beresford Parlett, University of California, Berkeley, USA
Jim Demmel, University of California, Berkeley, USA
Inderjit Dhillon, University of Texas, Austin, USA
Osni Marques, LBNL/NERSC, USA
Christof Voemel, University of California, Berkeley, USA

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.