larrv(3)

📦

version 3.12.1 · 9 versions

3.12.1 · arch rolling (latest)
3.12.0 · opensuse-leap 16.0
3.12.1 · ubuntu stonking
3.12.1 · ubuntu questing
3.12.0 · ubuntu noble
3.12.0 · ubuntu noble
3.12.0 · centos-stream 10, fedora 43, fedora 44
3.12.1 · ubuntu resolute
3.12.1 · debian trixie

🌸 NAME 🌸

larrv(3) — larrv - larrv: eig tridiagonal, step in stemr & stegr. Welcome to the Library Functions Manual, bestie — this is the larrv(3) page and it's lowkey all about that tridiagonal eigenvector life. 💅

✨ SYNOPSIS ✨

💖 Functions 💖

subroutine clarrv (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)

CLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT.

subroutine dlarrv (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)

DLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT.

subroutine slarrv (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)

SLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT.

subroutine zlarrv (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)

ZLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT.

💫 Detailed Description 💫

Okay so this section is literally just a heading in the source, no extra prose bestie — the real main character energy is in the function docs below. 🫶

🔥 Function Documentation 🔥

🧚 subroutine clarrv

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)

CLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT.

🎯 Purpose

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. No cap, that's the whole vibe of the purpose block. ✨

🧸 Parameters

N — N is INTEGER. The order of the matrix. N >= 0.
VL — 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.
VU — 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.
D — D is REAL array, dimension (N). On entry, the N diagonal elements of the diagonal matrix D. On exit, D may be overwritten.
L — 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.
PIVMIN — PIVMIN is REAL. The minimum pivot allowed in the Sturm sequence.
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.
M — M is INTEGER. The total number of input eigenvalues. 0 <= M <= N.
DOL — DOL is INTEGER.
DOU — DOU is INTEGER. If the user wants to compute only selected eigenvectors from all the eigenvalues supplied, they 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.
MINRGP — MINRGP is REAL.
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|) ).
W — 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.
WERR — WERR is REAL array, dimension (N). The first M elements contain the semiwidth of the uncertainty interval of the corresponding eigenvalue in W.
WGAP — WGAP is REAL array, dimension (N). The separation from the right neighbor eigenvalue in W.
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 the second block.
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)). The Gerschgorin intervals should be computed from the original UNshifted matrix.
Z — 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.
LDZ — LDZ is INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
ISUPPZ — ISUPPZ is INTEGER array, dimension ( 2*max(1,M) ). The support of the eigenvectors in Z, i.e., the indices indicating the nonzero elements in Z. The I-th eigenvector is nonzero only in elements ISUPPZ( 2*I-1 ) through ISUPPZ( 2*I ).
WORK — WORK is REAL array, dimension (12*N).
IWORK — IWORK is INTEGER array, dimension (7*N).
INFO — 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 they might be trading in precision when they decrease 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.

🌟 Author

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

💜 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 281 of file clarrv.f.

🌈 `dlarrv` — double precision, double the slay

   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)

okay so DLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT. That's the whole assignment, bestie. 💅

✨ Purpose ✨

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. iykyk ✨

💅 Parameters 💅

N — INTEGER. The order of the matrix. N >= 0.
VL — 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.
VU — 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 to 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 by forcing a small right gap. And so it turns out that VU is currently not used by this implementation of DLARRV. not me explaining the whole lore but here we are 🙈
D — DOUBLE PRECISION array, dimension (N). On entry, the N diagonal elements of the diagonal matrix D. On exit, D may be overwritten.
L — 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.
PIVMIN — DOUBLE PRECISION. The minimum pivot allowed in the Sturm sequence.
ISPLIT — 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.
M — INTEGER. The total number of input eigenvalues. 0 <= M <= N.
DOL — INTEGER. That's the whole entry, bestie.
DOU — INTEGER. If the user wants to compute only selected eigenvectors from all the eigenvalues supplied, they 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.
MINRGP — DOUBLE PRECISION. No extra deets here.
RTOL1 — DOUBLE PRECISION. Also no extra deets here.
RTOL2 — DOUBLE PRECISION. Parameters for bisection. An interval [LEFT,RIGHT] has converged if RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) ).
W — 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.
WERR — DOUBLE PRECISION array, dimension (N). The first M elements contain the semiwidth of the uncertainty interval of the corresponding eigenvalue in W.
WGAP — DOUBLE PRECISION array, dimension (N). The separation from the right neighbor eigenvalue in W.
IBLOCK — 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 — 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.
GERS — 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.
Z — 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.
LDZ — INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
ISUPPZ — 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 ).
WORK — DOUBLE PRECISION array, dimension (12*N).
IWORK — INTEGER array, dimension (7*N).
INFO — 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 they might be trading in precision when they decrease 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.

👑 Author

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

🌟 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 287 of file dlarrv.f.

🦋 `slarrv` — same energy, `REAL` parameters

   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)

SLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT.

✨ Purpose ✨

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. periodt 💖

💅 Parameters 💅

N — INTEGER. The order of the matrix. N >= 0.
VL — 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.
VU
VU — a REAL. Upper bound of the interval that contains the eigenvalues you actually want. VL < VU. 💖 And here's the tea: VU is currently not used by this implementation of SLARRV. She gets passed to SLARRV anyway because in theory she could be used to compute gaps on the right end of the extremal eigenvalues — but with not much initial accuracy in LAMBDA and VU, that formula can lead to an overestimation of the right gap, which then leads to inadequately early RQI "convergence" 🙈. Right now that's prevented by forcing a small right gap. And so, plot twist, it turns out that VU is currently not used by this implementation of SLARRV. Periodt.
D — REAL array, dimension (N). On entry, the N diagonal elements of the diagonal matrix D. On exit, D may be overwritten. (she gets a little makeover, nbd)
L — 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 too 💅
PIVMIN — REAL. The minimum pivot allowed in the Sturm sequence.
ISPLIT — 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. Basically the group-chat admin breaking y'all into breakout rooms 🫶
M — INTEGER. The total number of input eigenvalues. 0 <= M <= N.
DOL — INTEGER. That's it. That's the whole description. Iconic minimalism ✨
DOU — INTEGER. If you want to compute only selected eigenvectors out of all the eigenvalues supplied, you 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. And if you want 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. 🎀
MINRGP — REAL. No further notes here, bestie. That's what the page gives us. 💀
RTOL1 — REAL. Same situation, no extra description in the source.
RTOL2 — REAL. Parameters for bisection. An interval [LEFT,RIGHT] has converged if

RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )

(that's the whole vibe check for "are we done yet")

W — 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. 🔥
WERR — REAL array, dimension (N). The first M elements contain the semiwidth of the uncertainty interval of the corresponding eigenvalue in W.
WGAP — REAL array, dimension (N). The separation from the right neighbor eigenvalue in W.
IBLOCK — 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. Lowkey a seating chart 🪑
INDEXW — 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.
GERS — 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. 👑
Z — 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: you (yes, you, bestie) must ensure that at least max(1,M) columns are supplied in the array Z. 🙈
LDZ — INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
ISUPPZ — 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 ). It's giving "here's exactly where the personality lives" 🌈
WORK — REAL array, dimension (12*N).
IWORK — INTEGER array, dimension (7*N).
INFO — INTEGER. The receipts:
- = 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, you should be aware that you might be trading in precision when you decrease 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.

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🫶 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 287 of file slarrv.f.

💜 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, complex16, dimension( ldz, ) z, integer ldz, integer, dimension( * ) isuppz, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

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)

ZLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT. (yes, the complex*16 sibling of SLARRV — same energy, bigger numbers 💅✨)

🦋 Purpose:

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.

🌸 Parameters

N — INTEGER. The order of the matrix. N >= 0.
VL — 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. 🎯
VU — 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. (yep, mirror image of VL, she said what she said 🪞)
D — DOUBLE PRECISION array, dimension (N). On entry, the N diagonal elements of the diagonal matrix D. On exit, D may be overwritten.
L — 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.
PIVMIN — DOUBLE PRECISION. The minimum pivot allowed in the Sturm sequence.
ISPLIT — 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.
M

okay, picking the parameter list back up right here, no cap:

M — INTEGER. The total number of input eigenvalues. 0 <= M <= N.
DOL — INTEGER — and that’s literally all the docs say, bestie 😭
DOU — INTEGER. If the user wants to compute only selected eigenvectors from all the eigenvalues supplied, they 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.
MINRGP — DOUBLE PRECISION — docs just say that and dip 🫠
RTOL1 — DOUBLE PRECISION — again, that’s all we get, no extra tea 💀
RTOL2 — DOUBLE PRECISION. Parameters for bisection. An interval [LEFT,RIGHT] has converged if RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) ).
W — 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.
WERR — DOUBLE PRECISION array, dimension (N). The first M elements contain the semiwidth of the uncertainty interval of the corresponding eigenvalue in W.
WGAP — DOUBLE PRECISION array, dimension (N). The separation from the right neighbor eigenvalue in W.
IBLOCK — 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 — 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.
GERS — 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.
Z — 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.
LDZ — INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
ISUPPZ — 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 ).
WORK — DOUBLE PRECISION array, dimension (12*N).
IWORK — INTEGER array, dimension (7*N).
INFO — INTEGER; here’s the full breakdown, bestie:
- = 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 they might be trading in precision when they decrease 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.

👑 Author

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

💖 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 281 of file zlarrv.f.

📜 Author (the auto-generated one)

Generated automatically by Doxygen for LAPACK from the source code.

This whole page came from the LAPACK source, Version 3.12.0, and you’re reading larrv(3) — catch you later, bestie 💖✨

📦 where this page comes from

arch rolling: lapack-doc 3.12.1-2 · custom
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms

larrv(3)

🌸 NAME 🌸

✨ SYNOPSIS ✨

💖 Functions 💖

💫 Detailed Description 💫

🔥 Function Documentation 🔥

🧚 subroutine clarrv

🎯 Purpose

🧸 Parameters

🌟 Author

💜 Contributors

🌈 dlarrv — double precision, double the slay

✨ Purpose ✨

💅 Parameters 💅

👑 Author

🌟 Contributors

🦋 slarrv — same energy, REAL parameters

✨ Purpose ✨

💅 Parameters 💅

👑 Author

🫶 Contributors:

🦋 Purpose:

🌸 Parameters

👑 Author

💖 Contributors:

📜 Author (the auto-generated one)

🌈 `dlarrv` — double precision, double the slay

🦋 `slarrv` — same energy, `REAL` parameters