.TH "larrv" 3 "Wed Feb 7 2024 11:30:40" "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
larrv \- larrv: eig tridiagonal, step in stemr & stegr
.SH SYNOPSIS
.br
.PP
.SS "Functions"

.in +1c
.ti -1c
.RI "subroutine \fBclarrv\fP (n, vl, vu, d, l, pivmin, isplit, m, dol, dou, minrgp, rtol1, rtol2, w, werr, wgap, iblock, indexw, gers, z, ldz, isuppz, work, iwork, info)"
.br
.RI "\fBCLARRV\fP computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT\&. "
.ti -1c
.RI "subroutine \fBdlarrv\fP (n, vl, vu, d, l, pivmin, isplit, m, dol, dou, minrgp, rtol1, rtol2, w, werr, wgap, iblock, indexw, gers, z, ldz, isuppz, work, iwork, info)"
.br
.RI "\fBDLARRV\fP computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT\&. "
.ti -1c
.RI "subroutine \fBslarrv\fP (n, vl, vu, d, l, pivmin, isplit, m, dol, dou, minrgp, rtol1, rtol2, w, werr, wgap, iblock, indexw, gers, z, ldz, isuppz, work, iwork, info)"
.br
.RI "\fBSLARRV\fP computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT\&. "
.ti -1c
.RI "subroutine \fBzlarrv\fP (n, vl, vu, d, l, pivmin, isplit, m, dol, dou, minrgp, rtol1, rtol2, w, werr, wgap, iblock, indexw, gers, z, ldz, isuppz, work, iwork, info)"
.br
.RI "\fBZLARRV\fP computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT\&. "
.in -1c
.SH "Detailed Description"
.PP 

.SH "Function Documentation"
.PP 
.SS "subroutine clarrv (integer n, real vl, real vu, real, dimension( * ) d, real, dimension( * ) l, real pivmin, integer, dimension( * ) isplit, integer m, integer dol, integer dou, real minrgp, real rtol1, real rtol2, real, dimension( * ) w, real, dimension( * ) werr, real, dimension( * ) wgap, integer, dimension( * ) iblock, integer, dimension( * ) indexw, real, dimension( * ) gers, complex, dimension( ldz, * ) z, integer ldz, integer, dimension( * ) isuppz, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)"

.PP
\fBCLARRV\fP computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 CLARRV computes the eigenvectors of the tridiagonal matrix
 T = L D L**T given L, D and APPROXIMATIONS to the eigenvalues of L D L**T\&.
 The input eigenvalues should have been computed by SLARRE\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrix\&.  N >= 0\&.
.fi
.PP
.br
\fIVL\fP 
.PP
.nf
          VL is REAL
          Lower bound of the interval that contains the desired
          eigenvalues\&. VL < VU\&. Needed to compute gaps on the left or right
          end of the extremal eigenvalues in the desired RANGE\&.
.fi
.PP
.br
\fIVU\fP 
.PP
.nf
          VU is REAL
          Upper bound of the interval that contains the desired
          eigenvalues\&. VL < VU\&. Needed to compute gaps on the left or right
          end of the extremal eigenvalues in the desired RANGE\&.
.fi
.PP
.br
\fID\fP 
.PP
.nf
          D is REAL array, dimension (N)
          On entry, the N diagonal elements of the diagonal matrix D\&.
          On exit, D may be overwritten\&.
.fi
.PP
.br
\fIL\fP 
.PP
.nf
          L is REAL array, dimension (N)
          On entry, the (N-1) subdiagonal elements of the unit
          bidiagonal matrix L are in elements 1 to N-1 of L
          (if the matrix is not split\&.) At the end of each block
          is stored the corresponding shift as given by SLARRE\&.
          On exit, L is overwritten\&.
.fi
.PP
.br
\fIPIVMIN\fP 
.PP
.nf
          PIVMIN is REAL
          The minimum pivot allowed in the Sturm sequence\&.
.fi
.PP
.br
\fIISPLIT\fP 
.PP
.nf
          ISPLIT is INTEGER array, dimension (N)
          The splitting points, at which T breaks up into blocks\&.
          The first block consists of rows/columns 1 to
          ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
          through ISPLIT( 2 ), etc\&.
.fi
.PP
.br
\fIM\fP 
.PP
.nf
          M is INTEGER
          The total number of input eigenvalues\&.  0 <= M <= N\&.
.fi
.PP
.br
\fIDOL\fP 
.PP
.nf
          DOL is INTEGER
.fi
.PP
.br
\fIDOU\fP 
.PP
.nf
          DOU is INTEGER
          If the user wants to compute only selected eigenvectors from all
          the eigenvalues supplied, he can specify an index range DOL:DOU\&.
          Or else the setting DOL=1, DOU=M should be applied\&.
          Note that DOL and DOU refer to the order in which the eigenvalues
          are stored in W\&.
          If the user wants to compute only selected eigenpairs, then
          the columns DOL-1 to DOU+1 of the eigenvector space Z contain the
          computed eigenvectors\&. All other columns of Z are set to zero\&.
.fi
.PP
.br
\fIMINRGP\fP 
.PP
.nf
          MINRGP is REAL
.fi
.PP
.br
\fIRTOL1\fP 
.PP
.nf
          RTOL1 is REAL
.fi
.PP
.br
\fIRTOL2\fP 
.PP
.nf
          RTOL2 is REAL
           Parameters for bisection\&.
           An interval [LEFT,RIGHT] has converged if
           RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
.fi
.PP
.br
\fIW\fP 
.PP
.nf
          W is REAL array, dimension (N)
          The first M elements of W contain the APPROXIMATE eigenvalues for
          which eigenvectors are to be computed\&.  The eigenvalues
          should be grouped by split-off block and ordered from
          smallest to largest within the block ( The output array
          W from SLARRE is expected here )\&. Furthermore, they are with
          respect to the shift of the corresponding root representation
          for their block\&. On exit, W holds the eigenvalues of the
          UNshifted matrix\&.
.fi
.PP
.br
\fIWERR\fP 
.PP
.nf
          WERR is REAL array, dimension (N)
          The first M elements contain the semiwidth of the uncertainty
          interval of the corresponding eigenvalue in W
.fi
.PP
.br
\fIWGAP\fP 
.PP
.nf
          WGAP is REAL array, dimension (N)
          The separation from the right neighbor eigenvalue in W\&.
.fi
.PP
.br
\fIIBLOCK\fP 
.PP
.nf
          IBLOCK is INTEGER array, dimension (N)
          The indices of the blocks (submatrices) associated with the
          corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue
          W(i) belongs to the first block from the top, =2 if W(i)
          belongs to the second block, etc\&.
.fi
.PP
.br
\fIINDEXW\fP 
.PP
.nf
          INDEXW is INTEGER array, dimension (N)
          The indices of the eigenvalues within each block (submatrix);
          for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the
          i-th eigenvalue W(i) is the 10-th eigenvalue in the second block\&.
.fi
.PP
.br
\fIGERS\fP 
.PP
.nf
          GERS is REAL array, dimension (2*N)
          The N Gerschgorin intervals (the i-th Gerschgorin interval
          is (GERS(2*i-1), GERS(2*i))\&. The Gerschgorin intervals should
          be computed from the original UNshifted matrix\&.
.fi
.PP
.br
\fIZ\fP 
.PP
.nf
          Z is COMPLEX array, dimension (LDZ, max(1,M) )
          If INFO = 0, the first M columns of Z contain the
          orthonormal eigenvectors of the matrix T
          corresponding to the input 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\&.
.fi
.PP
.br
\fILDZ\fP 
.PP
.nf
          LDZ is INTEGER
          The leading dimension of the array Z\&.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N)\&.
.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 eigenvector
          is nonzero only in elements ISUPPZ( 2*I-1 ) through
          ISUPPZ( 2*I )\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is REAL array, dimension (12*N)
.fi
.PP
.br
\fIIWORK\fP 
.PP
.nf
          IWORK is INTEGER array, dimension (7*N)
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          INFO is INTEGER
          = 0:  successful exit

          > 0:  A problem occurred in CLARRV\&.
          < 0:  One of the called subroutines signaled an internal problem\&.
                Needs inspection of the corresponding parameter IINFO
                for further information\&.

          =-1:  Problem in SLARRB when refining a child's eigenvalues\&.
          =-2:  Problem in SLARRF when computing the RRR of a child\&.
                When a child is inside a tight cluster, it can be difficult
                to find an RRR\&. A partial remedy from the user's point of
                view is to make the parameter MINRGP smaller and recompile\&.
                However, as the orthogonality of the computed vectors is
                proportional to 1/MINRGP, the user should be aware that
                he might be trading in precision when he decreases MINRGP\&.
          =-3:  Problem in SLARRB when refining a single eigenvalue
                after the Rayleigh correction was rejected\&.
          = 5:  The Rayleigh Quotient Iteration failed to converge to
                full accuracy in MAXITR steps\&.
.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 
.RE
.PP

.SS "subroutine dlarrv (integer n, double precision vl, double precision vu, double precision, dimension( * ) d, double precision, dimension( * ) l, double precision pivmin, integer, dimension( * ) isplit, integer m, integer dol, integer dou, double precision minrgp, double precision rtol1, double precision rtol2, double precision, dimension( * ) w, double precision, dimension( * ) werr, double precision, dimension( * ) wgap, integer, dimension( * ) iblock, integer, dimension( * ) indexw, double precision, dimension( * ) gers, double precision, dimension( ldz, * ) z, integer ldz, integer, dimension( * ) isuppz, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)"

.PP
\fBDLARRV\fP computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 DLARRV computes the eigenvectors of the tridiagonal matrix
 T = L D L**T given L, D and APPROXIMATIONS to the eigenvalues of L D L**T\&.
 The input eigenvalues should have been computed by DLARRE\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrix\&.  N >= 0\&.
.fi
.PP
.br
\fIVL\fP 
.PP
.nf
          VL is DOUBLE PRECISION
          Lower bound of the interval that contains the desired
          eigenvalues\&. VL < VU\&. Needed to compute gaps on the left or right
          end of the extremal eigenvalues in the desired RANGE\&.
.fi
.PP
.br
\fIVU\fP 
.PP
.nf
          VU is DOUBLE PRECISION
          Upper bound of the interval that contains the desired
          eigenvalues\&. VL < VU\&. 
          Note: VU is currently not used by this implementation of DLARRV, VU is
          passed to DLARRV because it could be used compute gaps on the right end
          of the extremal eigenvalues\&. However, with not much initial accuracy in
          LAMBDA and VU, the formula can lead to an overestimation of the right gap
          and thus to inadequately early RQI 'convergence'\&. This is currently
          prevented this by forcing a small right gap\&. And so it turns out that VU
          is currently not used by this implementation of DLARRV\&.
.fi
.PP
.br
\fID\fP 
.PP
.nf
          D is DOUBLE PRECISION array, dimension (N)
          On entry, the N diagonal elements of the diagonal matrix D\&.
          On exit, D may be overwritten\&.
.fi
.PP
.br
\fIL\fP 
.PP
.nf
          L is DOUBLE PRECISION array, dimension (N)
          On entry, the (N-1) subdiagonal elements of the unit
          bidiagonal matrix L are in elements 1 to N-1 of L
          (if the matrix is not split\&.) At the end of each block
          is stored the corresponding shift as given by DLARRE\&.
          On exit, L is overwritten\&.
.fi
.PP
.br
\fIPIVMIN\fP 
.PP
.nf
          PIVMIN is DOUBLE PRECISION
          The minimum pivot allowed in the Sturm sequence\&.
.fi
.PP
.br
\fIISPLIT\fP 
.PP
.nf
          ISPLIT is INTEGER array, dimension (N)
          The splitting points, at which T breaks up into blocks\&.
          The first block consists of rows/columns 1 to
          ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
          through ISPLIT( 2 ), etc\&.
.fi
.PP
.br
\fIM\fP 
.PP
.nf
          M is INTEGER
          The total number of input eigenvalues\&.  0 <= M <= N\&.
.fi
.PP
.br
\fIDOL\fP 
.PP
.nf
          DOL is INTEGER
.fi
.PP
.br
\fIDOU\fP 
.PP
.nf
          DOU is INTEGER
          If the user wants to compute only selected eigenvectors from all
          the eigenvalues supplied, he can specify an index range DOL:DOU\&.
          Or else the setting DOL=1, DOU=M should be applied\&.
          Note that DOL and DOU refer to the order in which the eigenvalues
          are stored in W\&.
          If the user wants to compute only selected eigenpairs, then
          the columns DOL-1 to DOU+1 of the eigenvector space Z contain the
          computed eigenvectors\&. All other columns of Z are set to zero\&.
.fi
.PP
.br
\fIMINRGP\fP 
.PP
.nf
          MINRGP is DOUBLE PRECISION
.fi
.PP
.br
\fIRTOL1\fP 
.PP
.nf
          RTOL1 is DOUBLE PRECISION
.fi
.PP
.br
\fIRTOL2\fP 
.PP
.nf
          RTOL2 is DOUBLE PRECISION
           Parameters for bisection\&.
           An interval [LEFT,RIGHT] has converged if
           RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
.fi
.PP
.br
\fIW\fP 
.PP
.nf
          W is DOUBLE PRECISION array, dimension (N)
          The first M elements of W contain the APPROXIMATE eigenvalues for
          which eigenvectors are to be computed\&.  The eigenvalues
          should be grouped by split-off block and ordered from
          smallest to largest within the block ( The output array
          W from DLARRE is expected here )\&. Furthermore, they are with
          respect to the shift of the corresponding root representation
          for their block\&. On exit, W holds the eigenvalues of the
          UNshifted matrix\&.
.fi
.PP
.br
\fIWERR\fP 
.PP
.nf
          WERR is DOUBLE PRECISION array, dimension (N)
          The first M elements contain the semiwidth of the uncertainty
          interval of the corresponding eigenvalue in W
.fi
.PP
.br
\fIWGAP\fP 
.PP
.nf
          WGAP is DOUBLE PRECISION array, dimension (N)
          The separation from the right neighbor eigenvalue in W\&.
.fi
.PP
.br
\fIIBLOCK\fP 
.PP
.nf
          IBLOCK is INTEGER array, dimension (N)
          The indices of the blocks (submatrices) associated with the
          corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue
          W(i) belongs to the first block from the top, =2 if W(i)
          belongs to the second block, etc\&.
.fi
.PP
.br
\fIINDEXW\fP 
.PP
.nf
          INDEXW is INTEGER array, dimension (N)
          The indices of the eigenvalues within each block (submatrix);
          for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the
          i-th eigenvalue W(i) is the 10-th eigenvalue in the second block\&.
.fi
.PP
.br
\fIGERS\fP 
.PP
.nf
          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))\&. The Gerschgorin intervals should
          be computed from the original UNshifted matrix\&.
.fi
.PP
.br
\fIZ\fP 
.PP
.nf
          Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
          If INFO = 0, the first M columns of Z contain the
          orthonormal eigenvectors of the matrix T
          corresponding to the input 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\&.
.fi
.PP
.br
\fILDZ\fP 
.PP
.nf
          LDZ is INTEGER
          The leading dimension of the array Z\&.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N)\&.
.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 eigenvector
          is nonzero only in elements ISUPPZ( 2*I-1 ) through
          ISUPPZ( 2*I )\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is DOUBLE PRECISION array, dimension (12*N)
.fi
.PP
.br
\fIIWORK\fP 
.PP
.nf
          IWORK is INTEGER array, dimension (7*N)
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          INFO is INTEGER
          = 0:  successful exit

          > 0:  A problem occurred in DLARRV\&.
          < 0:  One of the called subroutines signaled an internal problem\&.
                Needs inspection of the corresponding parameter IINFO
                for further information\&.

          =-1:  Problem in DLARRB when refining a child's eigenvalues\&.
          =-2:  Problem in DLARRF when computing the RRR of a child\&.
                When a child is inside a tight cluster, it can be difficult
                to find an RRR\&. A partial remedy from the user's point of
                view is to make the parameter MINRGP smaller and recompile\&.
                However, as the orthogonality of the computed vectors is
                proportional to 1/MINRGP, the user should be aware that
                he might be trading in precision when he decreases MINRGP\&.
          =-3:  Problem in DLARRB when refining a single eigenvalue
                after the Rayleigh correction was rejected\&.
          = 5:  The Rayleigh Quotient Iteration failed to converge to
                full accuracy in MAXITR steps\&.
.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 
.RE
.PP

.SS "subroutine slarrv (integer n, real vl, real vu, real, dimension( * ) d, real, dimension( * ) l, real pivmin, integer, dimension( * ) isplit, integer m, integer dol, integer dou, real minrgp, real rtol1, real rtol2, real, dimension( * ) w, real, dimension( * ) werr, real, dimension( * ) wgap, integer, dimension( * ) iblock, integer, dimension( * ) indexw, real, dimension( * ) gers, real, dimension( ldz, * ) z, integer ldz, integer, dimension( * ) isuppz, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)"

.PP
\fBSLARRV\fP computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 SLARRV computes the eigenvectors of the tridiagonal matrix
 T = L D L**T given L, D and APPROXIMATIONS to the eigenvalues of L D L**T\&.
 The input eigenvalues should have been computed by SLARRE\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrix\&.  N >= 0\&.
.fi
.PP
.br
\fIVL\fP 
.PP
.nf
          VL is REAL
          Lower bound of the interval that contains the desired
          eigenvalues\&. VL < VU\&. Needed to compute gaps on the left or right
          end of the extremal eigenvalues in the desired RANGE\&.
.fi
.PP
.br
\fIVU\fP 
.PP
.nf
          VU is REAL
          Upper bound of the interval that contains the desired
          eigenvalues\&. VL < VU\&. 
          Note: VU is currently not used by this implementation of SLARRV, VU is
          passed to SLARRV because it could be used compute gaps on the right end
          of the extremal eigenvalues\&. However, with not much initial accuracy in
          LAMBDA and VU, the formula can lead to an overestimation of the right gap
          and thus to inadequately early RQI 'convergence'\&. This is currently
          prevented this by forcing a small right gap\&. And so it turns out that VU
          is currently not used by this implementation of SLARRV\&.
.fi
.PP
.br
\fID\fP 
.PP
.nf
          D is REAL array, dimension (N)
          On entry, the N diagonal elements of the diagonal matrix D\&.
          On exit, D may be overwritten\&.
.fi
.PP
.br
\fIL\fP 
.PP
.nf
          L is REAL array, dimension (N)
          On entry, the (N-1) subdiagonal elements of the unit
          bidiagonal matrix L are in elements 1 to N-1 of L
          (if the matrix is not split\&.) At the end of each block
          is stored the corresponding shift as given by SLARRE\&.
          On exit, L is overwritten\&.
.fi
.PP
.br
\fIPIVMIN\fP 
.PP
.nf
          PIVMIN is REAL
          The minimum pivot allowed in the Sturm sequence\&.
.fi
.PP
.br
\fIISPLIT\fP 
.PP
.nf
          ISPLIT is INTEGER array, dimension (N)
          The splitting points, at which T breaks up into blocks\&.
          The first block consists of rows/columns 1 to
          ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
          through ISPLIT( 2 ), etc\&.
.fi
.PP
.br
\fIM\fP 
.PP
.nf
          M is INTEGER
          The total number of input eigenvalues\&.  0 <= M <= N\&.
.fi
.PP
.br
\fIDOL\fP 
.PP
.nf
          DOL is INTEGER
.fi
.PP
.br
\fIDOU\fP 
.PP
.nf
          DOU is INTEGER
          If the user wants to compute only selected eigenvectors from all
          the eigenvalues supplied, he can specify an index range DOL:DOU\&.
          Or else the setting DOL=1, DOU=M should be applied\&.
          Note that DOL and DOU refer to the order in which the eigenvalues
          are stored in W\&.
          If the user wants to compute only selected eigenpairs, then
          the columns DOL-1 to DOU+1 of the eigenvector space Z contain the
          computed eigenvectors\&. All other columns of Z are set to zero\&.
.fi
.PP
.br
\fIMINRGP\fP 
.PP
.nf
          MINRGP is REAL
.fi
.PP
.br
\fIRTOL1\fP 
.PP
.nf
          RTOL1 is REAL
.fi
.PP
.br
\fIRTOL2\fP 
.PP
.nf
          RTOL2 is REAL
           Parameters for bisection\&.
           An interval [LEFT,RIGHT] has converged if
           RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
.fi
.PP
.br
\fIW\fP 
.PP
.nf
          W is REAL array, dimension (N)
          The first M elements of W contain the APPROXIMATE eigenvalues for
          which eigenvectors are to be computed\&.  The eigenvalues
          should be grouped by split-off block and ordered from
          smallest to largest within the block ( The output array
          W from SLARRE is expected here )\&. Furthermore, they are with
          respect to the shift of the corresponding root representation
          for their block\&. On exit, W holds the eigenvalues of the
          UNshifted matrix\&.
.fi
.PP
.br
\fIWERR\fP 
.PP
.nf
          WERR is REAL array, dimension (N)
          The first M elements contain the semiwidth of the uncertainty
          interval of the corresponding eigenvalue in W
.fi
.PP
.br
\fIWGAP\fP 
.PP
.nf
          WGAP is REAL array, dimension (N)
          The separation from the right neighbor eigenvalue in W\&.
.fi
.PP
.br
\fIIBLOCK\fP 
.PP
.nf
          IBLOCK is INTEGER array, dimension (N)
          The indices of the blocks (submatrices) associated with the
          corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue
          W(i) belongs to the first block from the top, =2 if W(i)
          belongs to the second block, etc\&.
.fi
.PP
.br
\fIINDEXW\fP 
.PP
.nf
          INDEXW is INTEGER array, dimension (N)
          The indices of the eigenvalues within each block (submatrix);
          for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the
          i-th eigenvalue W(i) is the 10-th eigenvalue in the second block\&.
.fi
.PP
.br
\fIGERS\fP 
.PP
.nf
          GERS is REAL array, dimension (2*N)
          The N Gerschgorin intervals (the i-th Gerschgorin interval
          is (GERS(2*i-1), GERS(2*i))\&. The Gerschgorin intervals should
          be computed from the original UNshifted matrix\&.
.fi
.PP
.br
\fIZ\fP 
.PP
.nf
          Z is REAL array, dimension (LDZ, max(1,M) )
          If INFO = 0, the first M columns of Z contain the
          orthonormal eigenvectors of the matrix T
          corresponding to the input 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\&.
.fi
.PP
.br
\fILDZ\fP 
.PP
.nf
          LDZ is INTEGER
          The leading dimension of the array Z\&.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N)\&.
.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 eigenvector
          is nonzero only in elements ISUPPZ( 2*I-1 ) through
          ISUPPZ( 2*I )\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is REAL array, dimension (12*N)
.fi
.PP
.br
\fIIWORK\fP 
.PP
.nf
          IWORK is INTEGER array, dimension (7*N)
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          INFO is INTEGER
          = 0:  successful exit

          > 0:  A problem occurred in SLARRV\&.
          < 0:  One of the called subroutines signaled an internal problem\&.
                Needs inspection of the corresponding parameter IINFO
                for further information\&.

          =-1:  Problem in SLARRB when refining a child's eigenvalues\&.
          =-2:  Problem in SLARRF when computing the RRR of a child\&.
                When a child is inside a tight cluster, it can be difficult
                to find an RRR\&. A partial remedy from the user's point of
                view is to make the parameter MINRGP smaller and recompile\&.
                However, as the orthogonality of the computed vectors is
                proportional to 1/MINRGP, the user should be aware that
                he might be trading in precision when he decreases MINRGP\&.
          =-3:  Problem in SLARRB when refining a single eigenvalue
                after the Rayleigh correction was rejected\&.
          = 5:  The Rayleigh Quotient Iteration failed to converge to
                full accuracy in MAXITR steps\&.
.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 
.RE
.PP

.SS "subroutine zlarrv (integer n, double precision vl, double precision vu, double precision, dimension( * ) d, double precision, dimension( * ) l, double precision pivmin, integer, dimension( * ) isplit, integer m, integer dol, integer dou, double precision minrgp, double precision rtol1, double precision rtol2, double precision, dimension( * ) w, double precision, dimension( * ) werr, double precision, dimension( * ) wgap, integer, dimension( * ) iblock, integer, dimension( * ) indexw, double precision, dimension( * ) gers, complex*16, dimension( ldz, * ) z, integer ldz, integer, dimension( * ) isuppz, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)"

.PP
\fBZLARRV\fP computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 ZLARRV computes the eigenvectors of the tridiagonal matrix
 T = L D L**T given L, D and APPROXIMATIONS to the eigenvalues of L D L**T\&.
 The input eigenvalues should have been computed by DLARRE\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrix\&.  N >= 0\&.
.fi
.PP
.br
\fIVL\fP 
.PP
.nf
          VL is DOUBLE PRECISION
          Lower bound of the interval that contains the desired
          eigenvalues\&. VL < VU\&. Needed to compute gaps on the left or right
          end of the extremal eigenvalues in the desired RANGE\&.
.fi
.PP
.br
\fIVU\fP 
.PP
.nf
          VU is DOUBLE PRECISION
          Upper bound of the interval that contains the desired
          eigenvalues\&. VL < VU\&. Needed to compute gaps on the left or right
          end of the extremal eigenvalues in the desired RANGE\&.
.fi
.PP
.br
\fID\fP 
.PP
.nf
          D is DOUBLE PRECISION array, dimension (N)
          On entry, the N diagonal elements of the diagonal matrix D\&.
          On exit, D may be overwritten\&.
.fi
.PP
.br
\fIL\fP 
.PP
.nf
          L is DOUBLE PRECISION array, dimension (N)
          On entry, the (N-1) subdiagonal elements of the unit
          bidiagonal matrix L are in elements 1 to N-1 of L
          (if the matrix is not split\&.) At the end of each block
          is stored the corresponding shift as given by DLARRE\&.
          On exit, L is overwritten\&.
.fi
.PP
.br
\fIPIVMIN\fP 
.PP
.nf
          PIVMIN is DOUBLE PRECISION
          The minimum pivot allowed in the Sturm sequence\&.
.fi
.PP
.br
\fIISPLIT\fP 
.PP
.nf
          ISPLIT is INTEGER array, dimension (N)
          The splitting points, at which T breaks up into blocks\&.
          The first block consists of rows/columns 1 to
          ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
          through ISPLIT( 2 ), etc\&.
.fi
.PP
.br
\fIM\fP 
.PP
.nf
          M is INTEGER
          The total number of input eigenvalues\&.  0 <= M <= N\&.
.fi
.PP
.br
\fIDOL\fP 
.PP
.nf
          DOL is INTEGER
.fi
.PP
.br
\fIDOU\fP 
.PP
.nf
          DOU is INTEGER
          If the user wants to compute only selected eigenvectors from all
          the eigenvalues supplied, he can specify an index range DOL:DOU\&.
          Or else the setting DOL=1, DOU=M should be applied\&.
          Note that DOL and DOU refer to the order in which the eigenvalues
          are stored in W\&.
          If the user wants to compute only selected eigenpairs, then
          the columns DOL-1 to DOU+1 of the eigenvector space Z contain the
          computed eigenvectors\&. All other columns of Z are set to zero\&.
.fi
.PP
.br
\fIMINRGP\fP 
.PP
.nf
          MINRGP is DOUBLE PRECISION
.fi
.PP
.br
\fIRTOL1\fP 
.PP
.nf
          RTOL1 is DOUBLE PRECISION
.fi
.PP
.br
\fIRTOL2\fP 
.PP
.nf
          RTOL2 is DOUBLE PRECISION
           Parameters for bisection\&.
           An interval [LEFT,RIGHT] has converged if
           RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
.fi
.PP
.br
\fIW\fP 
.PP
.nf
          W is DOUBLE PRECISION array, dimension (N)
          The first M elements of W contain the APPROXIMATE eigenvalues for
          which eigenvectors are to be computed\&.  The eigenvalues
          should be grouped by split-off block and ordered from
          smallest to largest within the block ( The output array
          W from DLARRE is expected here )\&. Furthermore, they are with
          respect to the shift of the corresponding root representation
          for their block\&. On exit, W holds the eigenvalues of the
          UNshifted matrix\&.
.fi
.PP
.br
\fIWERR\fP 
.PP
.nf
          WERR is DOUBLE PRECISION array, dimension (N)
          The first M elements contain the semiwidth of the uncertainty
          interval of the corresponding eigenvalue in W
.fi
.PP
.br
\fIWGAP\fP 
.PP
.nf
          WGAP is DOUBLE PRECISION array, dimension (N)
          The separation from the right neighbor eigenvalue in W\&.
.fi
.PP
.br
\fIIBLOCK\fP 
.PP
.nf
          IBLOCK is INTEGER array, dimension (N)
          The indices of the blocks (submatrices) associated with the
          corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue
          W(i) belongs to the first block from the top, =2 if W(i)
          belongs to the second block, etc\&.
.fi
.PP
.br
\fIINDEXW\fP 
.PP
.nf
          INDEXW is INTEGER array, dimension (N)
          The indices of the eigenvalues within each block (submatrix);
          for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the
          i-th eigenvalue W(i) is the 10-th eigenvalue in the second block\&.
.fi
.PP
.br
\fIGERS\fP 
.PP
.nf
          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))\&. The Gerschgorin intervals should
          be computed from the original UNshifted matrix\&.
.fi
.PP
.br
\fIZ\fP 
.PP
.nf
          Z is COMPLEX*16 array, dimension (LDZ, max(1,M) )
          If INFO = 0, the first M columns of Z contain the
          orthonormal eigenvectors of the matrix T
          corresponding to the input 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\&.
.fi
.PP
.br
\fILDZ\fP 
.PP
.nf
          LDZ is INTEGER
          The leading dimension of the array Z\&.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N)\&.
.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 eigenvector
          is nonzero only in elements ISUPPZ( 2*I-1 ) through
          ISUPPZ( 2*I )\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is DOUBLE PRECISION array, dimension (12*N)
.fi
.PP
.br
\fIIWORK\fP 
.PP
.nf
          IWORK is INTEGER array, dimension (7*N)
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          INFO is INTEGER
          = 0:  successful exit

          > 0:  A problem occurred in ZLARRV\&.
          < 0:  One of the called subroutines signaled an internal problem\&.
                Needs inspection of the corresponding parameter IINFO
                for further information\&.

          =-1:  Problem in DLARRB when refining a child's eigenvalues\&.
          =-2:  Problem in DLARRF when computing the RRR of a child\&.
                When a child is inside a tight cluster, it can be difficult
                to find an RRR\&. A partial remedy from the user's point of
                view is to make the parameter MINRGP smaller and recompile\&.
                However, as the orthogonality of the computed vectors is
                proportional to 1/MINRGP, the user should be aware that
                he might be trading in precision when he decreases MINRGP\&.
          =-3:  Problem in DLARRB when refining a single eigenvalue
                after the Rayleigh correction was rejected\&.
          = 5:  The Rayleigh Quotient Iteration failed to converge to
                full accuracy in MAXITR steps\&.
.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 
.RE
.PP

.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.