--- name: "larre" section: "3" platform: "linux" package: "lapack-doc" version: "3.12.1-2" release: "Arch Linux 2026-11-01" platforms: ["linux"] url: "https://new.carta.tech/larre" --- # larre(3) larre: step in stemr ## NAME larre - larre: step in stemr ## SYNOPSIS ### Functions subroutine **dlarre** (range, n, vl, vu, il, iu, d, e, e2, rtol1, rtol2, spltol, nsplit, isplit, m, w, werr, wgap, iblock, indexw, gers, pivmin, work, iwork, info)\ **DLARRE** given the tridiagonal matrix T, sets small off-diagonal elements to zero and for each unreduced block Ti, finds base representations and eigenvalues. subroutine **slarre** (range, n, vl, vu, il, iu, d, e, e2, rtol1, rtol2, spltol, nsplit, isplit, m, w, werr, wgap, iblock, indexw, gers, pivmin, work, iwork, info)\ **SLARRE** given the tridiagonal matrix T, sets small off-diagonal elements to zero and for each unreduced block Ti, finds base representations and eigenvalues. ## Detailed Description ## Function Documentation ### subroutine dlarre (character range, integer n, double precision vl, double precision vu, integer il, integer iu, double precision, dimension( \* ) d, double precision, dimension( \* ) e, double precision, dimension( \* ) e2, double precision rtol1, double precision rtol2, double precision spltol, integer nsplit, integer, dimension( \* ) isplit, integer m, double precision, dimension( \* ) w, double precision, dimension( \* ) werr, double precision, dimension( \* ) wgap, integer, dimension( \* ) iblock, integer, dimension( \* ) indexw, double precision, dimension( \* ) gers, double precision pivmin, double precision, dimension( \* ) work, integer, dimension( \* ) iwork, integer info) **DLARRE** given the tridiagonal matrix T, sets small off-diagonal elements to zero and for each unreduced block Ti, finds base representations and eigenvalues. **Purpose:** ``` !> !> To find the desired eigenvalues of a given real symmetric !> tridiagonal matrix T, DLARRE sets any off-diagonal !> elements to zero, and for each unreduced block T_i, it finds !> (a) a suitable shift at one end of the block's spectrum, !> (b) the base representation, T_i - sigma_i I = L_i D_i L_i^T, and !> (c) eigenvalues of each L_i D_i L_i^T. !> The representations and eigenvalues found are then used by !> DSTEMR to compute the eigenvectors of T. !> The accuracy varies depending on whether bisection is used to !> find a few eigenvalues or the dqds algorithm (subroutine DLASQ2) to !> compute all and then discard any unwanted one. !> As an added benefit, DLARRE also outputs the n !> Gerschgorin intervals for the matrices L_i D_i L_i^T. !> ``` **Parameters** *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 (of the !> entire matrix) will be found. !> ``` *N* ``` !> N is INTEGER !> The order of the matrix. N > 0. !> ``` *VL* ``` !> VL is DOUBLE PRECISION !> If RANGE='V', the lower bound for the eigenvalues. !> Eigenvalues less than or equal to VL, or greater than VU, !> will not be returned. VL < VU. !> If RANGE='I' or ='A', DLARRE computes bounds on the desired !> part of the spectrum. !> ``` *VU* ``` !> VU is DOUBLE PRECISION !> If RANGE='V', the upper bound for the eigenvalues. !> Eigenvalues less than or equal to VL, or greater than VU, !> will not be returned. VL < VU. !> If RANGE='I' or ='A', DLARRE computes bounds on the desired !> part of the spectrum. !> ``` *IL* ``` !> IL is INTEGER !> If RANGE='I', the index of the !> smallest eigenvalue to be returned. !> 1 <= IL <= IU <= N. !> ``` *IU* ``` !> IU is INTEGER !> If RANGE='I', the index of the !> largest eigenvalue to be returned. !> 1 <= IL <= IU <= N. !> ``` *D* ``` !> D is DOUBLE PRECISION array, dimension (N) !> On entry, the N diagonal elements of the tridiagonal !> matrix T. !> On exit, the N diagonal elements of the diagonal !> matrices D_i. !> ``` *E* ``` !> E is DOUBLE PRECISION array, dimension (N) !> On entry, the first (N-1) entries contain the subdiagonal !> elements of the tridiagonal matrix T; E(N) need not be set. !> On exit, E contains the subdiagonal elements of the unit !> bidiagonal matrices L_i. The entries E( ISPLIT( I ) ), !> 1 <= I <= NSPLIT, contain the base points sigma_i on output. !> ``` *E2* ``` !> E2 is DOUBLE PRECISION array, dimension (N) !> On entry, the first (N-1) entries contain the SQUARES of the !> subdiagonal elements of the tridiagonal matrix T; !> E2(N) need not be set. !> On exit, the entries E2( ISPLIT( I ) ), !> 1 <= I <= NSPLIT, have been set to zero !> ``` *RTOL1* ``` !> RTOL1 is DOUBLE PRECISION !> ``` *RTOL2* ``` !> RTOL2 is DOUBLE PRECISION !> Parameters for bisection. !> An interval [LEFT,RIGHT] has converged if !> RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) ) !> ``` *SPLTOL* ``` !> SPLTOL is DOUBLE PRECISION !> The threshold for splitting. !> ``` *NSPLIT* ``` !> NSPLIT is INTEGER !> The number of blocks T splits into. 1 <= NSPLIT <= N. !> ``` *ISPLIT* ``` !> 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., and the NSPLIT-th consists of rows/columns !> ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N. !> ``` *M* ``` !> M is INTEGER !> The total number of eigenvalues (of all L_i D_i L_i^T) !> found. !> ``` *W* ``` !> W is DOUBLE PRECISION array, dimension (N) !> The first M elements contain the eigenvalues. The !> eigenvalues of each of the blocks, L_i D_i L_i^T, are !> sorted in ascending order ( DLARRE may use the !> remaining N-M elements as workspace). !> ``` *WERR* ``` !> WERR is DOUBLE PRECISION array, dimension (N) !> The error bound on the corresponding eigenvalue in W. !> ``` *WGAP* ``` !> WGAP is DOUBLE PRECISION array, dimension (N) !> The separation from the right neighbor eigenvalue in W. !> The gap is only with respect to the eigenvalues of the same block !> as each block has its own representation tree. !> Exception: at the right end of a block we store the left gap !> ``` *IBLOCK* ``` !> 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. !> ``` *INDEXW* ``` !> 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 block 2 !> ``` *GERS* ``` !> GERS is DOUBLE PRECISION array, dimension (2*N) !> The N Gerschgorin intervals (the i-th Gerschgorin interval !> is (GERS(2*i-1), GERS(2*i)). !> ``` *PIVMIN* ``` !> PIVMIN is DOUBLE PRECISION !> The minimum pivot in the Sturm sequence for T. !> ``` *WORK* ``` !> WORK is DOUBLE PRECISION array, dimension (6*N) !> Workspace. !> ``` *IWORK* ``` !> IWORK is INTEGER array, dimension (5*N) !> Workspace. !> ``` *INFO* ``` !> INFO is INTEGER !> = 0: successful exit !> > 0: A problem occurred in DLARRE. !> < 0: One of the called subroutines signaled an internal problem. !> Needs inspection of the corresponding parameter IINFO !> for further information. !> !> =-1: Problem in DLARRD. !> = 2: No base representation could be found in MAXTRY iterations. !> Increasing MAXTRY and recompilation might be a remedy. !> =-3: Problem in DLARRB when computing the refined root !> representation for DLASQ2. !> =-4: Problem in DLARRB when preforming bisection on the !> desired part of the spectrum. !> =-5: Problem in DLASQ2. !> =-6: Problem in DLASQ2. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. **Further Details:** ``` !> !> The base representations are required to suffer very little !> element growth and consequently define all their eigenvalues to !> high relative accuracy. !> ``` **Contributors:** Beresford Parlett, University of California, Berkeley, USA \ Jim Demmel, University of California, Berkeley, USA \ Inderjit Dhillon, University of Texas, Austin, USA \ Osni Marques, LBNL/NERSC, USA \ Christof Voemel, University of California, Berkeley, USA \ Definition at line **301** of file **dlarre.f**. ### subroutine slarre (character range, integer n, real vl, real vu, integer il, integer iu, real, dimension( \* ) d, real, dimension( \* ) e, real, dimension( \* ) e2, real rtol1, real rtol2, real spltol, integer nsplit, integer, dimension( \* ) isplit, integer m, real, dimension( \* ) w, real, dimension( \* ) werr, real, dimension( \* ) wgap, integer, dimension( \* ) iblock, integer, dimension( \* ) indexw, real, dimension( \* ) gers, real pivmin, real, dimension( \* ) work, integer, dimension( \* ) iwork, integer info) **SLARRE** given the tridiagonal matrix T, sets small off-diagonal elements to zero and for each unreduced block Ti, finds base representations and eigenvalues. **Purpose:** ``` !> !> To find the desired eigenvalues of a given real symmetric !> tridiagonal matrix T, SLARRE sets any off-diagonal !> elements to zero, and for each unreduced block T_i, it finds !> (a) a suitable shift at one end of the block's spectrum, !> (b) the base representation, T_i - sigma_i I = L_i D_i L_i^T, and !> (c) eigenvalues of each L_i D_i L_i^T. !> The representations and eigenvalues found are then used by !> SSTEMR to compute the eigenvectors of T. !> The accuracy varies depending on whether bisection is used to !> find a few eigenvalues or the dqds algorithm (subroutine SLASQ2) to !> compute all and then discard any unwanted one. !> As an added benefit, SLARRE also outputs the n !> Gerschgorin intervals for the matrices L_i D_i L_i^T. !> ``` **Parameters** *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 (of the !> entire matrix) will be found. !> ``` *N* ``` !> N is INTEGER !> The order of the matrix. N > 0. !> ``` *VL* ``` !> VL is REAL !> If RANGE='V', the lower bound for the eigenvalues. !> Eigenvalues less than or equal to VL, or greater than VU, !> will not be returned. VL < VU. !> If RANGE='I' or ='A', SLARRE computes bounds on the desired !> part of the spectrum. !> ``` *VU* ``` !> VU is REAL !> If RANGE='V', the upper bound for the eigenvalues. !> Eigenvalues less than or equal to VL, or greater than VU, !> will not be returned. VL < VU. !> If RANGE='I' or ='A', SLARRE computes bounds on the desired !> part of the spectrum. !> ``` *IL* ``` !> IL is INTEGER !> If RANGE='I', the index of the !> smallest eigenvalue to be returned. !> 1 <= IL <= IU <= N. !> ``` *IU* ``` !> IU is INTEGER !> If RANGE='I', the index of the !> largest eigenvalue to be returned. !> 1 <= IL <= IU <= N. !> ``` *D* ``` !> D is REAL array, dimension (N) !> On entry, the N diagonal elements of the tridiagonal !> matrix T. !> On exit, the N diagonal elements of the diagonal !> matrices D_i. !> ``` *E* ``` !> E is REAL array, dimension (N) !> On entry, the first (N-1) entries contain the subdiagonal !> elements of the tridiagonal matrix T; E(N) need not be set. !> On exit, E contains the subdiagonal elements of the unit !> bidiagonal matrices L_i. The entries E( ISPLIT( I ) ), !> 1 <= I <= NSPLIT, contain the base points sigma_i on output. !> ``` *E2* ``` !> E2 is REAL array, dimension (N) !> On entry, the first (N-1) entries contain the SQUARES of the !> subdiagonal elements of the tridiagonal matrix T; !> E2(N) need not be set. !> On exit, the entries E2( ISPLIT( I ) ), !> 1 <= I <= NSPLIT, have been set to zero !> ``` *RTOL1* ``` !> RTOL1 is REAL !> ``` *RTOL2* ``` !> RTOL2 is REAL !> Parameters for bisection. !> An interval [LEFT,RIGHT] has converged if !> RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) ) !> ``` *SPLTOL* ``` !> SPLTOL is REAL !> The threshold for splitting. !> ``` *NSPLIT* ``` !> NSPLIT is INTEGER !> The number of blocks T splits into. 1 <= NSPLIT <= N. !> ``` *ISPLIT* ``` !> 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., and the NSPLIT-th consists of rows/columns !> ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N. !> ``` *M* ``` !> M is INTEGER !> The total number of eigenvalues (of all L_i D_i L_i^T) !> found. !> ``` *W* ``` !> W is REAL array, dimension (N) !> The first M elements contain the eigenvalues. The !> eigenvalues of each of the blocks, L_i D_i L_i^T, are !> sorted in ascending order ( SLARRE may use the !> remaining N-M elements as workspace). !> ``` *WERR* ``` !> WERR is REAL array, dimension (N) !> The error bound on the corresponding eigenvalue in W. !> ``` *WGAP* ``` !> WGAP is REAL array, dimension (N) !> The separation from the right neighbor eigenvalue in W. !> The gap is only with respect to the eigenvalues of the same block !> as each block has its own representation tree. !> Exception: at the right end of a block we store the left gap !> ``` *IBLOCK* ``` !> 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. !> ``` *INDEXW* ``` !> 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 block 2 !> ``` *GERS* ``` !> GERS is REAL array, dimension (2*N) !> The N Gerschgorin intervals (the i-th Gerschgorin interval !> is (GERS(2*i-1), GERS(2*i)). !> ``` *PIVMIN* ``` !> PIVMIN is REAL !> The minimum pivot in the Sturm sequence for T. !> ``` *WORK* ``` !> WORK is REAL array, dimension (6*N) !> Workspace. !> ``` *IWORK* ``` !> IWORK is INTEGER array, dimension (5*N) !> Workspace. !> ``` *INFO* ``` !> INFO is INTEGER !> = 0: successful exit !> > 0: A problem occurred in SLARRE. !> < 0: One of the called subroutines signaled an internal problem. !> Needs inspection of the corresponding parameter IINFO !> for further information. !> !> =-1: Problem in SLARRD. !> = 2: No base representation could be found in MAXTRY iterations. !> Increasing MAXTRY and recompilation might be a remedy. !> =-3: Problem in SLARRB when computing the refined root !> representation for SLASQ2. !> =-4: Problem in SLARRB when preforming bisection on the !> desired part of the spectrum. !> =-5: Problem in SLASQ2. !> =-6: Problem in SLASQ2. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. **Further Details:** ``` !> !> The base representations are required to suffer very little !> element growth and consequently define all their eigenvalues to !> high relative accuracy. !> ``` **Contributors:** Beresford Parlett, University of California, Berkeley, USA \ Jim Demmel, University of California, Berkeley, USA \ Inderjit Dhillon, University of Texas, Austin, USA \ Osni Marques, LBNL/NERSC, USA \ Christof Voemel, University of California, Berkeley, USA \ Definition at line **301** of file **slarre.f**. ## Author Generated automatically by Doxygen for LAPACK from the source code.