DSTEVR computes selected eigenvalues and, optionally, eigenvectors


 of a real symmetric tridiagonal matrix T.  Eigenvalues and


 eigenvectors can be selected by specifying either a range of values


 or a range of indices for the desired eigenvalues.


 Whenever possible, DSTEVR calls DSTEMR to compute the


 eigenspectrum using Relatively Robust Representations.  DSTEMR


 computes eigenvalues by the dqds algorithm, while orthogonal


 eigenvectors are computed from various 'good' L D L^T representations


 (also known as Relatively Robust Representations). Gram-Schmidt


 orthogonalization is avoided as far as possible. More specifically,


 the various steps of the algorithm are as follows. For the i-th


 unreduced block of T,


    (a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T


         is a relatively robust representation,


    (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high


        relative accuracy by the dqds algorithm,


    (c) If there is a cluster of close eigenvalues, 'choose' sigma_i


        close to the cluster, and go to step (a),


    (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T,


        compute the corresponding eigenvector by forming a


        rank-revealing twisted factorization.


 The desired accuracy of the output can be specified by the input


 parameter ABSTOL.


 For more details, see 'A new O(n^2) algorithm for the symmetric


 tridiagonal eigenvalue/eigenvector problem', by Inderjit Dhillon,


 Computer Science Division Technical Report No. UCB//CSD-97-971,


 UC Berkeley, May 1997.


 Note 1 : DSTEVR calls DSTEMR when the full spectrum is requested


 on machines which conform to the ieee-754 floating point standard.


 DSTEVR calls DSTEBZ and DSTEIN on non-ieee machines and


 when partial spectrum requests are made.


 Normal execution of DSTEMR may create NaNs and infinities and


 hence may abort due to a floating point exception in environments


 which do not handle NaNs and infinities in the ieee standard default


 manner.

JOBZ

          JOBZ is CHARACTER*1


          = 'N':  Compute eigenvalues only;


          = 'V':  Compute eigenvalues and eigenvectors.

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 will be found.


          For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and


          DSTEIN are called

N

          N is INTEGER


          The order of the matrix.  N >= 0.

D

          D is DOUBLE PRECISION array, dimension (N)


          On entry, the n diagonal elements of the tridiagonal matrix


          A.


          On exit, D may be multiplied by a constant factor chosen


          to avoid over/underflow in computing the eigenvalues.

E

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


          On entry, the (n-1) subdiagonal elements of the tridiagonal


          matrix A in elements 1 to N-1 of E.


          On exit, E may be multiplied by a constant factor chosen


          to avoid over/underflow in computing the eigenvalues.

VL

          VL is DOUBLE PRECISION


          If RANGE='V', the lower bound of the interval to


          be searched for eigenvalues. 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. 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 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 A to tridiagonal form.


          See 'Computing Small Singular Values of Bidiagonal Matrices


          with Guaranteed High Relative Accuracy,' by Demmel and


          Kahan, LAPACK Working Note #3.


          If high relative accuracy is important, set ABSTOL to


          DLAMCH( 'Safe minimum' ).  Doing so will guarantee that


          eigenvalues are computed to high relative accuracy when


          possible in future releases.  The current code does not


          make any guarantees about high relative accuracy, but


          future releases will. See J. Barlow and J. Demmel,


          'Computing Accurate Eigensystems of Scaled Diagonally


          Dominant Matrices', LAPACK Working Note #7, for a discussion


          of which matrices define their eigenvalues to high relative


          accuracy.

M

          M is INTEGER


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


          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

W

          W is DOUBLE PRECISION array, dimension (N)


          The first M elements contain the selected eigenvalues in


          ascending order.

Z

          Z is DOUBLE PRECISION 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).


          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

          LDZ is INTEGER


          The leading dimension of the array Z.  LDZ >= 1, and if


          JOBZ = 'V', LDZ >= max(1,N).

ISUPPZ

          ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )


          The support of the eigenvectors in Z, i.e., the indices


          indicating the nonzero elements in Z. The i-th eigenvector


          is nonzero only in elements ISUPPZ( 2*i-1 ) through


          ISUPPZ( 2*i ).


          Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) returns the optimal (and


          minimal) LWORK.

LWORK

          LWORK is INTEGER


          The dimension of the array WORK.  LWORK >= max(1,20*N).


          If LWORK = -1, then a workspace query is assumed; the routine


          only calculates the optimal sizes of the WORK and IWORK


          arrays, returns these values as the first entries of the WORK


          and IWORK arrays, and no error message related to LWORK or


          LIWORK is issued by XERBLA.

IWORK

          IWORK is INTEGER array, dimension (MAX(1,LIWORK))


          On exit, if INFO = 0, IWORK(1) returns the optimal (and


          minimal) LIWORK.

LIWORK

          LIWORK is INTEGER


          The dimension of the array IWORK.  LIWORK >= max(1,10*N).


          If LIWORK = -1, then a workspace query is assumed; the


          routine only calculates the optimal sizes of the WORK and


          IWORK arrays, returns these values as the first entries of


          the WORK and IWORK arrays, and no error message related to


          LWORK or LIWORK is issued by XERBLA.

INFO

          INFO is INTEGER


          = 0:  successful exit


          < 0:  if INFO = -i, the i-th argument had an illegal value


          > 0:  Internal error

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Inderjit Dhillon, IBM Almaden, USA
Osni Marques, LBNL/NERSC, USA
Ken Stanley, Computer Science Division, University of California at Berkeley, USA

 SSTEVR computes selected eigenvalues and, optionally, eigenvectors


 of a real symmetric tridiagonal matrix T.  Eigenvalues and


 eigenvectors can be selected by specifying either a range of values


 or a range of indices for the desired eigenvalues.


 Whenever possible, SSTEVR calls SSTEMR to compute the


 eigenspectrum using Relatively Robust Representations.  SSTEMR


 computes eigenvalues by the dqds algorithm, while orthogonal


 eigenvectors are computed from various 'good' L D L^T representations


 (also known as Relatively Robust Representations). Gram-Schmidt


 orthogonalization is avoided as far as possible. More specifically,


 the various steps of the algorithm are as follows. For the i-th


 unreduced block of T,


    (a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T


         is a relatively robust representation,


    (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high


        relative accuracy by the dqds algorithm,


    (c) If there is a cluster of close eigenvalues, 'choose' sigma_i


        close to the cluster, and go to step (a),


    (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T,


        compute the corresponding eigenvector by forming a


        rank-revealing twisted factorization.


 The desired accuracy of the output can be specified by the input


 parameter ABSTOL.


 For more details, see 'A new O(n^2) algorithm for the symmetric


 tridiagonal eigenvalue/eigenvector problem', by Inderjit Dhillon,


 Computer Science Division Technical Report No. UCB//CSD-97-971,


 UC Berkeley, May 1997.


 Note 1 : SSTEVR calls SSTEMR when the full spectrum is requested


 on machines which conform to the ieee-754 floating point standard.


 SSTEVR calls SSTEBZ and SSTEIN on non-ieee machines and


 when partial spectrum requests are made.


 Normal execution of SSTEMR may create NaNs and infinities and


 hence may abort due to a floating point exception in environments


 which do not handle NaNs and infinities in the ieee standard default


 manner.

JOBZ

          JOBZ is CHARACTER*1


          = 'N':  Compute eigenvalues only;


          = 'V':  Compute eigenvalues and eigenvectors.

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 will be found.


          For RANGE = 'V' or 'I' and IU - IL < N - 1, SSTEBZ and


          SSTEIN are called

N

          N is INTEGER


          The order of the matrix.  N >= 0.

D

          D is REAL array, dimension (N)


          On entry, the n diagonal elements of the tridiagonal matrix


          A.


          On exit, D may be multiplied by a constant factor chosen


          to avoid over/underflow in computing the eigenvalues.

E

          E is REAL array, dimension (max(1,N-1))


          On entry, the (n-1) subdiagonal elements of the tridiagonal


          matrix A in elements 1 to N-1 of E.


          On exit, E may be multiplied by a constant factor chosen


          to avoid over/underflow in computing the eigenvalues.

VL

          VL is REAL


          If RANGE='V', the lower bound of the interval to


          be searched for eigenvalues. 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. 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 REAL


          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 A to tridiagonal form.


          See 'Computing Small Singular Values of Bidiagonal Matrices


          with Guaranteed High Relative Accuracy,' by Demmel and


          Kahan, LAPACK Working Note #3.


          If high relative accuracy is important, set ABSTOL to


          SLAMCH( 'Safe minimum' ).  Doing so will guarantee that


          eigenvalues are computed to high relative accuracy when


          possible in future releases.  The current code does not


          make any guarantees about high relative accuracy, but


          future releases will. See J. Barlow and J. Demmel,


          'Computing Accurate Eigensystems of Scaled Diagonally


          Dominant Matrices', LAPACK Working Note #7, for a discussion


          of which matrices define their eigenvalues to high relative


          accuracy.

M

          M is INTEGER


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


          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

W

          W is REAL array, dimension (N)


          The first M elements contain the selected eigenvalues in


          ascending order.

Z

          Z is REAL 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).


          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

          LDZ is INTEGER


          The leading dimension of the array Z.  LDZ >= 1, and if


          JOBZ = 'V', LDZ >= max(1,N).

ISUPPZ

          ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )


          The support of the eigenvectors in Z, i.e., the indices


          indicating the nonzero elements in Z. The i-th eigenvector


          is nonzero only in elements ISUPPZ( 2*i-1 ) through


          ISUPPZ( 2*i ).


          Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1

WORK

          WORK is REAL array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) returns the optimal (and


          minimal) LWORK.

LWORK

          LWORK is INTEGER


          The dimension of the array WORK.  LWORK >= 20*N.


          If LWORK = -1, then a workspace query is assumed; the routine


          only calculates the optimal sizes of the WORK and IWORK


          arrays, returns these values as the first entries of the WORK


          and IWORK arrays, and no error message related to LWORK or


          LIWORK is issued by XERBLA.

IWORK

          IWORK is INTEGER array, dimension (MAX(1,LIWORK))


          On exit, if INFO = 0, IWORK(1) returns the optimal (and


          minimal) LIWORK.

LIWORK

          LIWORK is INTEGER


          The dimension of the array IWORK.  LIWORK >= 10*N.


          If LIWORK = -1, then a workspace query is assumed; the


          routine only calculates the optimal sizes of the WORK and


          IWORK arrays, returns these values as the first entries of


          the WORK and IWORK arrays, and no error message related to


          LWORK or LIWORK is issued by XERBLA.

INFO

          INFO is INTEGER


          = 0:  successful exit


          < 0:  if INFO = -i, the i-th argument had an illegal value


          > 0:  Internal error

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Inderjit Dhillon, IBM Almaden, USA
Osni Marques, LBNL/NERSC, USA
Ken Stanley, Computer Science Division, University of California at Berkeley, USA
Jason Riedy, Computer Science Division, University of California at Berkeley, USA