.TH "stemr" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
stemr \- stemr: eig, relatively robust representation (RRR)
.SH SYNOPSIS
.br
.PP
.SS "Functions"

.in +1c
.ti -1c
.RI "subroutine \fBcstemr\fP (jobz, range, n, d, e, vl, vu, il, iu, m, w, z, ldz, nzc, isuppz, tryrac, work, lwork, iwork, liwork, info)"
.br
.RI "\fBCSTEMR\fP "
.ti -1c
.RI "subroutine \fBdstemr\fP (jobz, range, n, d, e, vl, vu, il, iu, m, w, z, ldz, nzc, isuppz, tryrac, work, lwork, iwork, liwork, info)"
.br
.RI "\fBDSTEMR\fP "
.ti -1c
.RI "subroutine \fBsstemr\fP (jobz, range, n, d, e, vl, vu, il, iu, m, w, z, ldz, nzc, isuppz, tryrac, work, lwork, iwork, liwork, info)"
.br
.RI "\fBSSTEMR\fP "
.ti -1c
.RI "subroutine \fBzstemr\fP (jobz, range, n, d, e, vl, vu, il, iu, m, w, z, ldz, nzc, isuppz, tryrac, work, lwork, iwork, liwork, info)"
.br
.RI "\fBZSTEMR\fP "
.in -1c
.SH "Detailed Description"
.PP 

.SH "Function Documentation"
.PP 
.SS "subroutine cstemr (character jobz, character range, integer n, real, dimension( * ) d, real, dimension( * ) e, real vl, real vu, integer il, integer iu, integer m, real, dimension( * ) w, complex, dimension( ldz, * ) z, integer ldz, integer nzc, integer, dimension( * ) isuppz, logical tryrac, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)"

.PP
\fBCSTEMR\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!> CSTEMR computes selected eigenvalues and, optionally, eigenvectors
!> of a real symmetric tridiagonal matrix T\&. Any such unreduced matrix has
!> a well defined set of pairwise different real eigenvalues, the corresponding
!> real eigenvectors are pairwise orthogonal\&.
!>
!> The spectrum may be computed either completely or partially by specifying
!> either an interval (VL,VU] or a range of indices IL:IU for the desired
!> eigenvalues\&.
!>
!> Depending on the number of desired eigenvalues, these are computed either
!> by bisection or the dqds algorithm\&. Numerically orthogonal eigenvectors are
!> computed by the use of various suitable L D L^T factorizations near clusters
!> of close eigenvalues (referred to as RRRs, Relatively Robust
!> Representations)\&. An informal sketch of the algorithm follows\&.
!>
!> For each unreduced block (submatrix) of T,
!>    (a) Compute T - sigma I  = L D L^T, so that L and D
!>        define all the wanted eigenvalues to high relative accuracy\&.
!>        This means that small relative changes in the entries of D and L
!>        cause only small relative changes in the eigenvalues and
!>        eigenvectors\&. The standard (unfactored) representation of the
!>        tridiagonal matrix T does not have this property in general\&.
!>    (b) Compute the eigenvalues to suitable accuracy\&.
!>        If the eigenvectors are desired, the algorithm attains full
!>        accuracy of the computed eigenvalues only right before
!>        the corresponding vectors have to be computed, see steps c) and d)\&.
!>    (c) For each cluster of close eigenvalues, select a new
!>        shift close to the cluster, find a new factorization, and refine
!>        the shifted eigenvalues to suitable accuracy\&.
!>    (d) For each eigenvalue with a large enough relative separation compute
!>        the corresponding eigenvector by forming a rank revealing twisted
!>        factorization\&. Go back to (c) for any clusters that remain\&.
!>
!> For more details, see:
!> - Inderjit S\&. Dhillon and Beresford N\&. Parlett: 
!>   Linear Algebra and its Applications, 387(1), pp\&. 1-28, August 2004\&.
!> - Inderjit Dhillon and Beresford Parlett:  SIAM Journal on Matrix Analysis and Applications, Vol\&. 25,
!>   2004\&.  Also LAPACK Working Note 154\&.
!> - Inderjit Dhillon: ,
!>   Computer Science Division Technical Report No\&. UCB/CSD-97-971,
!>   UC Berkeley, May 1997\&.
!>
!> Further Details
!> 1\&.CSTEMR works only on machines which follow IEEE-754
!> floating-point standard in their handling of infinities and NaNs\&.
!> This permits the use of efficient inner loops avoiding a check for
!> zero divisors\&.
!>
!> 2\&. LAPACK routines can be used to reduce a complex Hermitean matrix to
!> real symmetric tridiagonal form\&.
!>
!> (Any complex Hermitean tridiagonal matrix has real values on its diagonal
!> and potentially complex numbers on its off-diagonals\&. By applying a
!> similarity transform with an appropriate diagonal matrix
!> diag(1,e^{i \\phy_1}, \&.\&.\&. , e^{i \\phy_{n-1}}), the complex Hermitean
!> matrix can be transformed into a real symmetric matrix and complex
!> arithmetic can be entirely avoided\&.)
!>
!> While the eigenvectors of the real symmetric tridiagonal matrix are real,
!> the eigenvectors of original complex Hermitean matrix have complex entries
!> in general\&.
!> Since LAPACK drivers overwrite the matrix data with the eigenvectors,
!> CSTEMR accepts complex workspace to facilitate interoperability
!> with CUNMTR or CUPMTR\&.
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOBZ\fP 
.PP
.nf
!>          JOBZ is CHARACTER*1
!>          = 'N':  Compute eigenvalues only;
!>          = 'V':  Compute eigenvalues and eigenvectors\&.
!> 
.fi
.PP
.br
\fIRANGE\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIN\fP 
.PP
.nf
!>          N is INTEGER
!>          The order of the matrix\&.  N >= 0\&.
!> 
.fi
.PP
.br
\fID\fP 
.PP
.nf
!>          D is REAL array, dimension (N)
!>          On entry, the N diagonal elements of the tridiagonal matrix
!>          T\&. On exit, D is overwritten\&.
!> 
.fi
.PP
.br
\fIE\fP 
.PP
.nf
!>          E is REAL array, dimension (N)
!>          On entry, the (N-1) subdiagonal elements of the tridiagonal
!>          matrix T in elements 1 to N-1 of E\&. E(N) need not be set on
!>          input, but is used internally as workspace\&.
!>          On exit, E is overwritten\&.
!> 
.fi
.PP
.br
\fIVL\fP 
.PP
.nf
!>          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'\&.
!> 
.fi
.PP
.br
\fIVU\fP 
.PP
.nf
!>          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'\&.
!> 
.fi
.PP
.br
\fIIL\fP 
.PP
.nf
!>          IL is INTEGER
!>
!>          If RANGE='I', the index of the
!>          smallest eigenvalue to be returned\&.
!>          1 <= IL <= IU <= N, if N > 0\&.
!>          Not referenced if RANGE = 'A' or 'V'\&.
!> 
.fi
.PP
.br
\fIIU\fP 
.PP
.nf
!>          IU is INTEGER
!>
!>          If RANGE='I', the index of the
!>          largest eigenvalue to be returned\&.
!>          1 <= IL <= IU <= N, if N > 0\&.
!>          Not referenced if RANGE = 'A' or 'V'\&.
!> 
.fi
.PP
.br
\fIM\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIW\fP 
.PP
.nf
!>          W is REAL array, dimension (N)
!>          The first M elements contain the selected eigenvalues in
!>          ascending order\&.
!> 
.fi
.PP
.br
\fIZ\fP 
.PP
.nf
!>          Z is COMPLEX array, dimension (LDZ, max(1,M) )
!>          If JOBZ = 'V', and if INFO = 0, then the first M columns of Z
!>          contain the orthonormal eigenvectors of the matrix T
!>          corresponding to the selected eigenvalues, with the i-th
!>          column of Z holding the eigenvector associated with W(i)\&.
!>          If JOBZ = 'N', then Z is not referenced\&.
!>          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 can be computed with a workspace
!>          query by setting NZC = -1, see below\&.
!> 
.fi
.PP
.br
\fILDZ\fP 
.PP
.nf
!>          LDZ is INTEGER
!>          The leading dimension of the array Z\&.  LDZ >= 1, and if
!>          JOBZ = 'V', then LDZ >= max(1,N)\&.
!> 
.fi
.PP
.br
\fINZC\fP 
.PP
.nf
!>          NZC is INTEGER
!>          The number of eigenvectors to be held in the array Z\&.
!>          If RANGE = 'A', then NZC >= max(1,N)\&.
!>          If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU]\&.
!>          If RANGE = 'I', then NZC >= IU-IL+1\&.
!>          If NZC = -1, then a workspace query is assumed; the
!>          routine calculates the number of columns of the array Z that
!>          are needed to hold the eigenvectors\&.
!>          This value is returned as the first entry of the Z array, and
!>          no error message related to NZC is issued by XERBLA\&.
!> 
.fi
.PP
.br
\fIISUPPZ\fP 
.PP
.nf
!>          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 computed eigenvector
!>          is nonzero only in elements ISUPPZ( 2*i-1 ) through
!>          ISUPPZ( 2*i )\&. This is relevant in the case when the matrix
!>          is split\&. ISUPPZ is only accessed when JOBZ is 'V' and N > 0\&.
!> 
.fi
.PP
.br
\fITRYRAC\fP 
.PP
.nf
!>          TRYRAC is LOGICAL
!>          If TRYRAC = \&.TRUE\&., indicates that the code should check whether
!>          the tridiagonal matrix defines its eigenvalues to high relative
!>          accuracy\&.  If so, the code uses relative-accuracy preserving
!>          algorithms that might be (a bit) slower depending on the matrix\&.
!>          If the matrix does not define its eigenvalues to high relative
!>          accuracy, the code can uses possibly faster algorithms\&.
!>          If TRYRAC = \&.FALSE\&., the code is not required to guarantee
!>          relatively accurate eigenvalues and can use the fastest possible
!>          techniques\&.
!>          On exit, a \&.TRUE\&. TRYRAC will be set to \&.FALSE\&. if the matrix
!>          does not define its eigenvalues to high relative accuracy\&.
!> 
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
!>          WORK is REAL array, dimension (LWORK)
!>          On exit, if INFO = 0, WORK(1) returns the optimal
!>          (and minimal) LWORK\&.
!> 
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
!>          LWORK is INTEGER
!>          The dimension of the array WORK\&. LWORK >= max(1,18*N)
!>          if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'\&.
!>          If LWORK = -1, then a workspace query is assumed; the routine
!>          only calculates the optimal size of the WORK array, returns
!>          this value as the first entry of the WORK array, and no error
!>          message related to LWORK is issued by XERBLA\&.
!> 
.fi
.PP
.br
\fIIWORK\fP 
.PP
.nf
!>          IWORK is INTEGER array, dimension (LIWORK)
!>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK\&.
!> 
.fi
.PP
.br
\fILIWORK\fP 
.PP
.nf
!>          LIWORK is INTEGER
!>          The dimension of the array IWORK\&.  LIWORK >= max(1,10*N)
!>          if the eigenvectors are desired, and LIWORK >= max(1,8*N)
!>          if only the eigenvalues are to be computed\&.
!>          If LIWORK = -1, then a workspace query is assumed; the
!>          routine only calculates the optimal size of the IWORK array,
!>          returns this value as the first entry of the IWORK array, and
!>          no error message related to LIWORK is issued by XERBLA\&.
!> 
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
!>          INFO is INTEGER
!>          On exit, INFO
!>          = 0:  successful exit
!>          < 0:  if INFO = -i, the i-th argument had an illegal value
!>          > 0:  if INFO = 1X, internal error in SLARRE,
!>                if INFO = 2X, internal error in CLARRV\&.
!>                Here, the digit X = ABS( IINFO ) < 10, where IINFO is
!>                the nonzero error code returned by SLARRE or
!>                CLARRV, respectively\&.
!> 
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

.PP
Univ\&. of Colorado Denver 

.PP
NAG Ltd\&. 
.RE
.PP
\fBContributors:\fP
.RS 4
Beresford Parlett, University of California, Berkeley, USA 
.br
Jim Demmel, University of California, Berkeley, USA 
.br
Inderjit Dhillon, University of Texas, Austin, USA 
.br
Osni Marques, LBNL/NERSC, USA 
.br
Christof Voemel, University of California, Berkeley, USA 
.br
Aravindh Krishnamoorthy, FAU, Erlangen, Germany 
.br
.RE
.PP

.PP
Definition at line \fB336\fP of file \fBcstemr\&.f\fP\&.
.SS "subroutine dstemr (character jobz, character range, integer n, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision vl, double precision vu, integer il, integer iu, integer m, double precision, dimension( * ) w, double precision, dimension( ldz, * ) z, integer ldz, integer nzc, integer, dimension( * ) isuppz, logical tryrac, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)"

.PP
\fBDSTEMR\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!> DSTEMR computes selected eigenvalues and, optionally, eigenvectors
!> of a real symmetric tridiagonal matrix T\&. Any such unreduced matrix has
!> a well defined set of pairwise different real eigenvalues, the corresponding
!> real eigenvectors are pairwise orthogonal\&.
!>
!> The spectrum may be computed either completely or partially by specifying
!> either an interval (VL,VU] or a range of indices IL:IU for the desired
!> eigenvalues\&.
!>
!> Depending on the number of desired eigenvalues, these are computed either
!> by bisection or the dqds algorithm\&. Numerically orthogonal eigenvectors are
!> computed by the use of various suitable L D L^T factorizations near clusters
!> of close eigenvalues (referred to as RRRs, Relatively Robust
!> Representations)\&. An informal sketch of the algorithm follows\&.
!>
!> For each unreduced block (submatrix) of T,
!>    (a) Compute T - sigma I  = L D L^T, so that L and D
!>        define all the wanted eigenvalues to high relative accuracy\&.
!>        This means that small relative changes in the entries of D and L
!>        cause only small relative changes in the eigenvalues and
!>        eigenvectors\&. The standard (unfactored) representation of the
!>        tridiagonal matrix T does not have this property in general\&.
!>    (b) Compute the eigenvalues to suitable accuracy\&.
!>        If the eigenvectors are desired, the algorithm attains full
!>        accuracy of the computed eigenvalues only right before
!>        the corresponding vectors have to be computed, see steps c) and d)\&.
!>    (c) For each cluster of close eigenvalues, select a new
!>        shift close to the cluster, find a new factorization, and refine
!>        the shifted eigenvalues to suitable accuracy\&.
!>    (d) For each eigenvalue with a large enough relative separation compute
!>        the corresponding eigenvector by forming a rank revealing twisted
!>        factorization\&. Go back to (c) for any clusters that remain\&.
!>
!> For more details, see:
!> - Inderjit S\&. Dhillon and Beresford N\&. Parlett: 
!>   Linear Algebra and its Applications, 387(1), pp\&. 1-28, August 2004\&.
!> - Inderjit Dhillon and Beresford Parlett:  SIAM Journal on Matrix Analysis and Applications, Vol\&. 25,
!>   2004\&.  Also LAPACK Working Note 154\&.
!> - Inderjit Dhillon: ,
!>   Computer Science Division Technical Report No\&. UCB/CSD-97-971,
!>   UC Berkeley, May 1997\&.
!>
!> Further Details
!> 1\&.DSTEMR works only on machines which follow IEEE-754
!> floating-point standard in their handling of infinities and NaNs\&.
!> This permits the use of efficient inner loops avoiding a check for
!> zero divisors\&.
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOBZ\fP 
.PP
.nf
!>          JOBZ is CHARACTER*1
!>          = 'N':  Compute eigenvalues only;
!>          = 'V':  Compute eigenvalues and eigenvectors\&.
!> 
.fi
.PP
.br
\fIRANGE\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIN\fP 
.PP
.nf
!>          N is INTEGER
!>          The order of the matrix\&.  N >= 0\&.
!> 
.fi
.PP
.br
\fID\fP 
.PP
.nf
!>          D is DOUBLE PRECISION array, dimension (N)
!>          On entry, the N diagonal elements of the tridiagonal matrix
!>          T\&. On exit, D is overwritten\&.
!> 
.fi
.PP
.br
\fIE\fP 
.PP
.nf
!>          E is DOUBLE PRECISION array, dimension (N)
!>          On entry, the (N-1) subdiagonal elements of the tridiagonal
!>          matrix T in elements 1 to N-1 of E\&. E(N) need not be set on
!>          input, but is used internally as workspace\&.
!>          On exit, E is overwritten\&.
!> 
.fi
.PP
.br
\fIVL\fP 
.PP
.nf
!>          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'\&.
!> 
.fi
.PP
.br
\fIVU\fP 
.PP
.nf
!>          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'\&.
!> 
.fi
.PP
.br
\fIIL\fP 
.PP
.nf
!>          IL is INTEGER
!>
!>          If RANGE='I', the index of the
!>          smallest eigenvalue to be returned\&.
!>          1 <= IL <= IU <= N, if N > 0\&.
!>          Not referenced if RANGE = 'A' or 'V'\&.
!> 
.fi
.PP
.br
\fIIU\fP 
.PP
.nf
!>          IU is INTEGER
!>
!>          If RANGE='I', the index of the
!>          largest eigenvalue to be returned\&.
!>          1 <= IL <= IU <= N, if N > 0\&.
!>          Not referenced if RANGE = 'A' or 'V'\&.
!> 
.fi
.PP
.br
\fIM\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIW\fP 
.PP
.nf
!>          W is DOUBLE PRECISION array, dimension (N)
!>          The first M elements contain the selected eigenvalues in
!>          ascending order\&.
!> 
.fi
.PP
.br
\fIZ\fP 
.PP
.nf
!>          Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
!>          If JOBZ = 'V', and if INFO = 0, then the first M columns of Z
!>          contain the orthonormal eigenvectors of the matrix T
!>          corresponding to the selected eigenvalues, with the i-th
!>          column of Z holding the eigenvector associated with W(i)\&.
!>          If JOBZ = 'N', then Z is not referenced\&.
!>          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 can be computed with a workspace
!>          query by setting NZC = -1, see below\&.
!> 
.fi
.PP
.br
\fILDZ\fP 
.PP
.nf
!>          LDZ is INTEGER
!>          The leading dimension of the array Z\&.  LDZ >= 1, and if
!>          JOBZ = 'V', then LDZ >= max(1,N)\&.
!> 
.fi
.PP
.br
\fINZC\fP 
.PP
.nf
!>          NZC is INTEGER
!>          The number of eigenvectors to be held in the array Z\&.
!>          If RANGE = 'A', then NZC >= max(1,N)\&.
!>          If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU]\&.
!>          If RANGE = 'I', then NZC >= IU-IL+1\&.
!>          If NZC = -1, then a workspace query is assumed; the
!>          routine calculates the number of columns of the array Z that
!>          are needed to hold the eigenvectors\&.
!>          This value is returned as the first entry of the Z array, and
!>          no error message related to NZC is issued by XERBLA\&.
!> 
.fi
.PP
.br
\fIISUPPZ\fP 
.PP
.nf
!>          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 computed eigenvector
!>          is nonzero only in elements ISUPPZ( 2*i-1 ) through
!>          ISUPPZ( 2*i )\&. This is relevant in the case when the matrix
!>          is split\&. ISUPPZ is only accessed when JOBZ is 'V' and N > 0\&.
!> 
.fi
.PP
.br
\fITRYRAC\fP 
.PP
.nf
!>          TRYRAC is LOGICAL
!>          If TRYRAC = \&.TRUE\&., indicates that the code should check whether
!>          the tridiagonal matrix defines its eigenvalues to high relative
!>          accuracy\&.  If so, the code uses relative-accuracy preserving
!>          algorithms that might be (a bit) slower depending on the matrix\&.
!>          If the matrix does not define its eigenvalues to high relative
!>          accuracy, the code can uses possibly faster algorithms\&.
!>          If TRYRAC = \&.FALSE\&., the code is not required to guarantee
!>          relatively accurate eigenvalues and can use the fastest possible
!>          techniques\&.
!>          On exit, a \&.TRUE\&. TRYRAC will be set to \&.FALSE\&. if the matrix
!>          does not define its eigenvalues to high relative accuracy\&.
!> 
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
!>          WORK is DOUBLE PRECISION array, dimension (LWORK)
!>          On exit, if INFO = 0, WORK(1) returns the optimal
!>          (and minimal) LWORK\&.
!> 
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
!>          LWORK is INTEGER
!>          The dimension of the array WORK\&. LWORK >= max(1,18*N)
!>          if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'\&.
!>          If LWORK = -1, then a workspace query is assumed; the routine
!>          only calculates the optimal size of the WORK array, returns
!>          this value as the first entry of the WORK array, and no error
!>          message related to LWORK is issued by XERBLA\&.
!> 
.fi
.PP
.br
\fIIWORK\fP 
.PP
.nf
!>          IWORK is INTEGER array, dimension (LIWORK)
!>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK\&.
!> 
.fi
.PP
.br
\fILIWORK\fP 
.PP
.nf
!>          LIWORK is INTEGER
!>          The dimension of the array IWORK\&.  LIWORK >= max(1,10*N)
!>          if the eigenvectors are desired, and LIWORK >= max(1,8*N)
!>          if only the eigenvalues are to be computed\&.
!>          If LIWORK = -1, then a workspace query is assumed; the
!>          routine only calculates the optimal size of the IWORK array,
!>          returns this value as the first entry of the IWORK array, and
!>          no error message related to LIWORK is issued by XERBLA\&.
!> 
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
!>          INFO is INTEGER
!>          On exit, INFO
!>          = 0:  successful exit
!>          < 0:  if INFO = -i, the i-th argument had an illegal value
!>          > 0:  if INFO = 1X, internal error in DLARRE,
!>                if INFO = 2X, internal error in DLARRV\&.
!>                Here, the digit X = ABS( IINFO ) < 10, where IINFO is
!>                the nonzero error code returned by DLARRE or
!>                DLARRV, respectively\&.
!> 
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

.PP
Univ\&. of Colorado Denver 

.PP
NAG Ltd\&. 
.RE
.PP
\fBContributors:\fP
.RS 4
Beresford Parlett, University of California, Berkeley, USA 
.br
Jim Demmel, University of California, Berkeley, USA 
.br
Inderjit Dhillon, University of Texas, Austin, USA 
.br
Osni Marques, LBNL/NERSC, USA 
.br
Christof Voemel, University of California, Berkeley, USA 
.br
Aravindh Krishnamoorthy, FAU, Erlangen, Germany 
.br
.RE
.PP

.PP
Definition at line \fB319\fP of file \fBdstemr\&.f\fP\&.
.SS "subroutine sstemr (character jobz, character range, integer n, real, dimension( * ) d, real, dimension( * ) e, real vl, real vu, integer il, integer iu, integer m, real, dimension( * ) w, real, dimension( ldz, * ) z, integer ldz, integer nzc, integer, dimension( * ) isuppz, logical tryrac, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)"

.PP
\fBSSTEMR\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!> SSTEMR computes selected eigenvalues and, optionally, eigenvectors
!> of a real symmetric tridiagonal matrix T\&. Any such unreduced matrix has
!> a well defined set of pairwise different real eigenvalues, the corresponding
!> real eigenvectors are pairwise orthogonal\&.
!>
!> The spectrum may be computed either completely or partially by specifying
!> either an interval (VL,VU] or a range of indices IL:IU for the desired
!> eigenvalues\&.
!>
!> Depending on the number of desired eigenvalues, these are computed either
!> by bisection or the dqds algorithm\&. Numerically orthogonal eigenvectors are
!> computed by the use of various suitable L D L^T factorizations near clusters
!> of close eigenvalues (referred to as RRRs, Relatively Robust
!> Representations)\&. An informal sketch of the algorithm follows\&.
!>
!> For each unreduced block (submatrix) of T,
!>    (a) Compute T - sigma I  = L D L^T, so that L and D
!>        define all the wanted eigenvalues to high relative accuracy\&.
!>        This means that small relative changes in the entries of D and L
!>        cause only small relative changes in the eigenvalues and
!>        eigenvectors\&. The standard (unfactored) representation of the
!>        tridiagonal matrix T does not have this property in general\&.
!>    (b) Compute the eigenvalues to suitable accuracy\&.
!>        If the eigenvectors are desired, the algorithm attains full
!>        accuracy of the computed eigenvalues only right before
!>        the corresponding vectors have to be computed, see steps c) and d)\&.
!>    (c) For each cluster of close eigenvalues, select a new
!>        shift close to the cluster, find a new factorization, and refine
!>        the shifted eigenvalues to suitable accuracy\&.
!>    (d) For each eigenvalue with a large enough relative separation compute
!>        the corresponding eigenvector by forming a rank revealing twisted
!>        factorization\&. Go back to (c) for any clusters that remain\&.
!>
!> For more details, see:
!> - Inderjit S\&. Dhillon and Beresford N\&. Parlett: 
!>   Linear Algebra and its Applications, 387(1), pp\&. 1-28, August 2004\&.
!> - Inderjit Dhillon and Beresford Parlett:  SIAM Journal on Matrix Analysis and Applications, Vol\&. 25,
!>   2004\&.  Also LAPACK Working Note 154\&.
!> - Inderjit Dhillon: ,
!>   Computer Science Division Technical Report No\&. UCB/CSD-97-971,
!>   UC Berkeley, May 1997\&.
!>
!> Further Details
!> 1\&.SSTEMR works only on machines which follow IEEE-754
!> floating-point standard in their handling of infinities and NaNs\&.
!> This permits the use of efficient inner loops avoiding a check for
!> zero divisors\&.
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOBZ\fP 
.PP
.nf
!>          JOBZ is CHARACTER*1
!>          = 'N':  Compute eigenvalues only;
!>          = 'V':  Compute eigenvalues and eigenvectors\&.
!> 
.fi
.PP
.br
\fIRANGE\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIN\fP 
.PP
.nf
!>          N is INTEGER
!>          The order of the matrix\&.  N >= 0\&.
!> 
.fi
.PP
.br
\fID\fP 
.PP
.nf
!>          D is REAL array, dimension (N)
!>          On entry, the N diagonal elements of the tridiagonal matrix
!>          T\&. On exit, D is overwritten\&.
!> 
.fi
.PP
.br
\fIE\fP 
.PP
.nf
!>          E is REAL array, dimension (N)
!>          On entry, the (N-1) subdiagonal elements of the tridiagonal
!>          matrix T in elements 1 to N-1 of E\&. E(N) need not be set on
!>          input, but is used internally as workspace\&.
!>          On exit, E is overwritten\&.
!> 
.fi
.PP
.br
\fIVL\fP 
.PP
.nf
!>          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'\&.
!> 
.fi
.PP
.br
\fIVU\fP 
.PP
.nf
!>          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'\&.
!> 
.fi
.PP
.br
\fIIL\fP 
.PP
.nf
!>          IL is INTEGER
!>
!>          If RANGE='I', the index of the
!>          smallest eigenvalue to be returned\&.
!>          1 <= IL <= IU <= N, if N > 0\&.
!>          Not referenced if RANGE = 'A' or 'V'\&.
!> 
.fi
.PP
.br
\fIIU\fP 
.PP
.nf
!>          IU is INTEGER
!>
!>          If RANGE='I', the index of the
!>          largest eigenvalue to be returned\&.
!>          1 <= IL <= IU <= N, if N > 0\&.
!>          Not referenced if RANGE = 'A' or 'V'\&.
!> 
.fi
.PP
.br
\fIM\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIW\fP 
.PP
.nf
!>          W is REAL array, dimension (N)
!>          The first M elements contain the selected eigenvalues in
!>          ascending order\&.
!> 
.fi
.PP
.br
\fIZ\fP 
.PP
.nf
!>          Z is REAL array, dimension (LDZ, max(1,M) )
!>          If JOBZ = 'V', and if INFO = 0, then the first M columns of Z
!>          contain the orthonormal eigenvectors of the matrix T
!>          corresponding to the selected eigenvalues, with the i-th
!>          column of Z holding the eigenvector associated with W(i)\&.
!>          If JOBZ = 'N', then Z is not referenced\&.
!>          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 can be computed with a workspace
!>          query by setting NZC = -1, see below\&.
!> 
.fi
.PP
.br
\fILDZ\fP 
.PP
.nf
!>          LDZ is INTEGER
!>          The leading dimension of the array Z\&.  LDZ >= 1, and if
!>          JOBZ = 'V', then LDZ >= max(1,N)\&.
!> 
.fi
.PP
.br
\fINZC\fP 
.PP
.nf
!>          NZC is INTEGER
!>          The number of eigenvectors to be held in the array Z\&.
!>          If RANGE = 'A', then NZC >= max(1,N)\&.
!>          If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU]\&.
!>          If RANGE = 'I', then NZC >= IU-IL+1\&.
!>          If NZC = -1, then a workspace query is assumed; the
!>          routine calculates the number of columns of the array Z that
!>          are needed to hold the eigenvectors\&.
!>          This value is returned as the first entry of the Z array, and
!>          no error message related to NZC is issued by XERBLA\&.
!> 
.fi
.PP
.br
\fIISUPPZ\fP 
.PP
.nf
!>          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 computed eigenvector
!>          is nonzero only in elements ISUPPZ( 2*i-1 ) through
!>          ISUPPZ( 2*i )\&. This is relevant in the case when the matrix
!>          is split\&. ISUPPZ is only accessed when JOBZ is 'V' and N > 0\&.
!> 
.fi
.PP
.br
\fITRYRAC\fP 
.PP
.nf
!>          TRYRAC is LOGICAL
!>          If TRYRAC = \&.TRUE\&., indicates that the code should check whether
!>          the tridiagonal matrix defines its eigenvalues to high relative
!>          accuracy\&.  If so, the code uses relative-accuracy preserving
!>          algorithms that might be (a bit) slower depending on the matrix\&.
!>          If the matrix does not define its eigenvalues to high relative
!>          accuracy, the code can uses possibly faster algorithms\&.
!>          If TRYRAC = \&.FALSE\&., the code is not required to guarantee
!>          relatively accurate eigenvalues and can use the fastest possible
!>          techniques\&.
!>          On exit, a \&.TRUE\&. TRYRAC will be set to \&.FALSE\&. if the matrix
!>          does not define its eigenvalues to high relative accuracy\&.
!> 
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
!>          WORK is REAL array, dimension (LWORK)
!>          On exit, if INFO = 0, WORK(1) returns the optimal
!>          (and minimal) LWORK\&.
!> 
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
!>          LWORK is INTEGER
!>          The dimension of the array WORK\&. LWORK >= max(1,18*N)
!>          if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'\&.
!>          If LWORK = -1, then a workspace query is assumed; the routine
!>          only calculates the optimal size of the WORK array, returns
!>          this value as the first entry of the WORK array, and no error
!>          message related to LWORK is issued by XERBLA\&.
!> 
.fi
.PP
.br
\fIIWORK\fP 
.PP
.nf
!>          IWORK is INTEGER array, dimension (LIWORK)
!>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK\&.
!> 
.fi
.PP
.br
\fILIWORK\fP 
.PP
.nf
!>          LIWORK is INTEGER
!>          The dimension of the array IWORK\&.  LIWORK >= max(1,10*N)
!>          if the eigenvectors are desired, and LIWORK >= max(1,8*N)
!>          if only the eigenvalues are to be computed\&.
!>          If LIWORK = -1, then a workspace query is assumed; the
!>          routine only calculates the optimal size of the IWORK array,
!>          returns this value as the first entry of the IWORK array, and
!>          no error message related to LIWORK is issued by XERBLA\&.
!> 
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
!>          INFO is INTEGER
!>          On exit, INFO
!>          = 0:  successful exit
!>          < 0:  if INFO = -i, the i-th argument had an illegal value
!>          > 0:  if INFO = 1X, internal error in SLARRE,
!>                if INFO = 2X, internal error in SLARRV\&.
!>                Here, the digit X = ABS( IINFO ) < 10, where IINFO is
!>                the nonzero error code returned by SLARRE or
!>                SLARRV, respectively\&.
!> 
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

.PP
Univ\&. of Colorado Denver 

.PP
NAG Ltd\&. 
.RE
.PP
\fBContributors:\fP
.RS 4
Beresford Parlett, University of California, Berkeley, USA 
.br
Jim Demmel, University of California, Berkeley, USA 
.br
Inderjit Dhillon, University of Texas, Austin, USA 
.br
Osni Marques, LBNL/NERSC, USA 
.br
Christof Voemel, University of California, Berkeley, USA 
.br
Aravindh Krishnamoorthy, FAU, Erlangen, Germany 
.br
.RE
.PP

.PP
Definition at line \fB319\fP of file \fBsstemr\&.f\fP\&.
.SS "subroutine zstemr (character jobz, character range, integer n, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision vl, double precision vu, integer il, integer iu, integer m, double precision, dimension( * ) w, complex*16, dimension( ldz, * ) z, integer ldz, integer nzc, integer, dimension( * ) isuppz, logical tryrac, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)"

.PP
\fBZSTEMR\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!> ZSTEMR computes selected eigenvalues and, optionally, eigenvectors
!> of a real symmetric tridiagonal matrix T\&. Any such unreduced matrix has
!> a well defined set of pairwise different real eigenvalues, the corresponding
!> real eigenvectors are pairwise orthogonal\&.
!>
!> The spectrum may be computed either completely or partially by specifying
!> either an interval (VL,VU] or a range of indices IL:IU for the desired
!> eigenvalues\&.
!>
!> Depending on the number of desired eigenvalues, these are computed either
!> by bisection or the dqds algorithm\&. Numerically orthogonal eigenvectors are
!> computed by the use of various suitable L D L^T factorizations near clusters
!> of close eigenvalues (referred to as RRRs, Relatively Robust
!> Representations)\&. An informal sketch of the algorithm follows\&.
!>
!> For each unreduced block (submatrix) of T,
!>    (a) Compute T - sigma I  = L D L^T, so that L and D
!>        define all the wanted eigenvalues to high relative accuracy\&.
!>        This means that small relative changes in the entries of D and L
!>        cause only small relative changes in the eigenvalues and
!>        eigenvectors\&. The standard (unfactored) representation of the
!>        tridiagonal matrix T does not have this property in general\&.
!>    (b) Compute the eigenvalues to suitable accuracy\&.
!>        If the eigenvectors are desired, the algorithm attains full
!>        accuracy of the computed eigenvalues only right before
!>        the corresponding vectors have to be computed, see steps c) and d)\&.
!>    (c) For each cluster of close eigenvalues, select a new
!>        shift close to the cluster, find a new factorization, and refine
!>        the shifted eigenvalues to suitable accuracy\&.
!>    (d) For each eigenvalue with a large enough relative separation compute
!>        the corresponding eigenvector by forming a rank revealing twisted
!>        factorization\&. Go back to (c) for any clusters that remain\&.
!>
!> For more details, see:
!> - Inderjit S\&. Dhillon and Beresford N\&. Parlett: 
!>   Linear Algebra and its Applications, 387(1), pp\&. 1-28, August 2004\&.
!> - Inderjit Dhillon and Beresford Parlett:  SIAM Journal on Matrix Analysis and Applications, Vol\&. 25,
!>   2004\&.  Also LAPACK Working Note 154\&.
!> - Inderjit Dhillon: ,
!>   Computer Science Division Technical Report No\&. UCB/CSD-97-971,
!>   UC Berkeley, May 1997\&.
!>
!> Further Details
!> 1\&.ZSTEMR works only on machines which follow IEEE-754
!> floating-point standard in their handling of infinities and NaNs\&.
!> This permits the use of efficient inner loops avoiding a check for
!> zero divisors\&.
!>
!> 2\&. LAPACK routines can be used to reduce a complex Hermitean matrix to
!> real symmetric tridiagonal form\&.
!>
!> (Any complex Hermitean tridiagonal matrix has real values on its diagonal
!> and potentially complex numbers on its off-diagonals\&. By applying a
!> similarity transform with an appropriate diagonal matrix
!> diag(1,e^{i \\phy_1}, \&.\&.\&. , e^{i \\phy_{n-1}}), the complex Hermitean
!> matrix can be transformed into a real symmetric matrix and complex
!> arithmetic can be entirely avoided\&.)
!>
!> While the eigenvectors of the real symmetric tridiagonal matrix are real,
!> the eigenvectors of original complex Hermitean matrix have complex entries
!> in general\&.
!> Since LAPACK drivers overwrite the matrix data with the eigenvectors,
!> ZSTEMR accepts complex workspace to facilitate interoperability
!> with ZUNMTR or ZUPMTR\&.
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOBZ\fP 
.PP
.nf
!>          JOBZ is CHARACTER*1
!>          = 'N':  Compute eigenvalues only;
!>          = 'V':  Compute eigenvalues and eigenvectors\&.
!> 
.fi
.PP
.br
\fIRANGE\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIN\fP 
.PP
.nf
!>          N is INTEGER
!>          The order of the matrix\&.  N >= 0\&.
!> 
.fi
.PP
.br
\fID\fP 
.PP
.nf
!>          D is DOUBLE PRECISION array, dimension (N)
!>          On entry, the N diagonal elements of the tridiagonal matrix
!>          T\&. On exit, D is overwritten\&.
!> 
.fi
.PP
.br
\fIE\fP 
.PP
.nf
!>          E is DOUBLE PRECISION array, dimension (N)
!>          On entry, the (N-1) subdiagonal elements of the tridiagonal
!>          matrix T in elements 1 to N-1 of E\&. E(N) need not be set on
!>          input, but is used internally as workspace\&.
!>          On exit, E is overwritten\&.
!> 
.fi
.PP
.br
\fIVL\fP 
.PP
.nf
!>          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'\&.
!> 
.fi
.PP
.br
\fIVU\fP 
.PP
.nf
!>          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'\&.
!> 
.fi
.PP
.br
\fIIL\fP 
.PP
.nf
!>          IL is INTEGER
!>
!>          If RANGE='I', the index of the
!>          smallest eigenvalue to be returned\&.
!>          1 <= IL <= IU <= N, if N > 0\&.
!>          Not referenced if RANGE = 'A' or 'V'\&.
!> 
.fi
.PP
.br
\fIIU\fP 
.PP
.nf
!>          IU is INTEGER
!>
!>          If RANGE='I', the index of the
!>          largest eigenvalue to be returned\&.
!>          1 <= IL <= IU <= N, if N > 0\&.
!>          Not referenced if RANGE = 'A' or 'V'\&.
!> 
.fi
.PP
.br
\fIM\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIW\fP 
.PP
.nf
!>          W is DOUBLE PRECISION array, dimension (N)
!>          The first M elements contain the selected eigenvalues in
!>          ascending order\&.
!> 
.fi
.PP
.br
\fIZ\fP 
.PP
.nf
!>          Z is COMPLEX*16 array, dimension (LDZ, max(1,M) )
!>          If JOBZ = 'V', and if INFO = 0, then the first M columns of Z
!>          contain the orthonormal eigenvectors of the matrix T
!>          corresponding to the selected eigenvalues, with the i-th
!>          column of Z holding the eigenvector associated with W(i)\&.
!>          If JOBZ = 'N', then Z is not referenced\&.
!>          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 can be computed with a workspace
!>          query by setting NZC = -1, see below\&.
!> 
.fi
.PP
.br
\fILDZ\fP 
.PP
.nf
!>          LDZ is INTEGER
!>          The leading dimension of the array Z\&.  LDZ >= 1, and if
!>          JOBZ = 'V', then LDZ >= max(1,N)\&.
!> 
.fi
.PP
.br
\fINZC\fP 
.PP
.nf
!>          NZC is INTEGER
!>          The number of eigenvectors to be held in the array Z\&.
!>          If RANGE = 'A', then NZC >= max(1,N)\&.
!>          If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU]\&.
!>          If RANGE = 'I', then NZC >= IU-IL+1\&.
!>          If NZC = -1, then a workspace query is assumed; the
!>          routine calculates the number of columns of the array Z that
!>          are needed to hold the eigenvectors\&.
!>          This value is returned as the first entry of the Z array, and
!>          no error message related to NZC is issued by XERBLA\&.
!> 
.fi
.PP
.br
\fIISUPPZ\fP 
.PP
.nf
!>          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 computed eigenvector
!>          is nonzero only in elements ISUPPZ( 2*i-1 ) through
!>          ISUPPZ( 2*i )\&. This is relevant in the case when the matrix
!>          is split\&. ISUPPZ is only accessed when JOBZ is 'V' and N > 0\&.
!> 
.fi
.PP
.br
\fITRYRAC\fP 
.PP
.nf
!>          TRYRAC is LOGICAL
!>          If TRYRAC = \&.TRUE\&., indicates that the code should check whether
!>          the tridiagonal matrix defines its eigenvalues to high relative
!>          accuracy\&.  If so, the code uses relative-accuracy preserving
!>          algorithms that might be (a bit) slower depending on the matrix\&.
!>          If the matrix does not define its eigenvalues to high relative
!>          accuracy, the code can uses possibly faster algorithms\&.
!>          If TRYRAC = \&.FALSE\&., the code is not required to guarantee
!>          relatively accurate eigenvalues and can use the fastest possible
!>          techniques\&.
!>          On exit, a \&.TRUE\&. TRYRAC will be set to \&.FALSE\&. if the matrix
!>          does not define its eigenvalues to high relative accuracy\&.
!> 
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
!>          WORK is DOUBLE PRECISION array, dimension (LWORK)
!>          On exit, if INFO = 0, WORK(1) returns the optimal
!>          (and minimal) LWORK\&.
!> 
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
!>          LWORK is INTEGER
!>          The dimension of the array WORK\&. LWORK >= max(1,18*N)
!>          if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'\&.
!>          If LWORK = -1, then a workspace query is assumed; the routine
!>          only calculates the optimal size of the WORK array, returns
!>          this value as the first entry of the WORK array, and no error
!>          message related to LWORK is issued by XERBLA\&.
!> 
.fi
.PP
.br
\fIIWORK\fP 
.PP
.nf
!>          IWORK is INTEGER array, dimension (LIWORK)
!>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK\&.
!> 
.fi
.PP
.br
\fILIWORK\fP 
.PP
.nf
!>          LIWORK is INTEGER
!>          The dimension of the array IWORK\&.  LIWORK >= max(1,10*N)
!>          if the eigenvectors are desired, and LIWORK >= max(1,8*N)
!>          if only the eigenvalues are to be computed\&.
!>          If LIWORK = -1, then a workspace query is assumed; the
!>          routine only calculates the optimal size of the IWORK array,
!>          returns this value as the first entry of the IWORK array, and
!>          no error message related to LIWORK is issued by XERBLA\&.
!> 
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
!>          INFO is INTEGER
!>          On exit, INFO
!>          = 0:  successful exit
!>          < 0:  if INFO = -i, the i-th argument had an illegal value
!>          > 0:  if INFO = 1X, internal error in DLARRE,
!>                if INFO = 2X, internal error in ZLARRV\&.
!>                Here, the digit X = ABS( IINFO ) < 10, where IINFO is
!>                the nonzero error code returned by DLARRE or
!>                ZLARRV, respectively\&.
!> 
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

.PP
Univ\&. of Colorado Denver 

.PP
NAG Ltd\&. 
.RE
.PP
\fBContributors:\fP
.RS 4
Beresford Parlett, University of California, Berkeley, USA 
.br
Jim Demmel, University of California, Berkeley, USA 
.br
Inderjit Dhillon, University of Texas, Austin, USA 
.br
Osni Marques, LBNL/NERSC, USA 
.br
Christof Voemel, University of California, Berkeley, USA 
.br
Aravindh Krishnamoorthy, FAU, Erlangen, Germany 
.br
.RE
.PP

.PP
Definition at line \fB336\fP of file \fBzstemr\&.f\fP\&.
.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.