SRC/zlarrv.f(3)

✨ NAME ✨

okay so this is the Library Functions Manual page for SRC/zlarrv.f, bestie — the whole vibe is one Fortran subroutine named zlarrv that computes eigenvectors of a tridiagonal matrix. No cap, that's the name and the tea. 🫶

🌈 SYNOPSIS 🌈

🎀 Functions/Subroutines 🎀

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.

💅 FUNCTION/SUBROUTINE DOCUMENTATION 💅

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.

🌸 Purpose 🌸

okay bestie, here's the actual 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. That's the whole assignment — slay. ✨

🎀 Parameters 🎀

N

N is INTEGER. It's the order of the matrix. N >= 0.

VL

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.

VU

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.

D

D is DOUBLE PRECISION array, dimension (N). On entry, the N diagonal elements of the diagonal matrix D. On exit, D may be overwritten. Yeah, she's not keeping receipts. 💅

L

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.

PIVMIN

PIVMIN is DOUBLE PRECISION. 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. That's the whole description in the source, bestie — no extra lore. 🫶

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. Periodt.

MINRGP

MINRGP is DOUBLE PRECISION. Again, that's all the source gives us — just the type, no extra commentary. 🙈

RTOL1

RTOL1 is DOUBLE PRECISION. Same situation: type only.

RTOL2

RTOL2 is DOUBLE PRECISION. Parameters for bisection. An interval [LEFT,RIGHT] has converged if RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) ).

W

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.

WERR

WERR is DOUBLE PRECISION array, dimension (N). The first M elements contain the semiwidth of the uncertainty interval of the corresponding eigenvalue in W.

WGAP

WGAP is DOUBLE PRECISION 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 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

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.

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 DOUBLE PRECISION array, dimension (12*N).

IWORK

IWORK is INTEGER array, dimension (7*N).

INFO

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 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. 📜

Yes bestie, that's literally the line: Definition at line 281 of file zlarrv.f.

🤖 Author 🤖

Generated automatically by Doxygen for LAPACK from the source code.

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okay that's the whole page, bestie — LAPACK Version 3.12.0, SRC/zlarrv.f(3), no cap. 💅✨