DLARRV(3)

📦 version 3.12.0, in opensuse-leap 16.0

💖 DLARRV(3) — Library Functions Manual 💖

okay bestie, welcome to the Library Functions Manual page for DLARRV. This is the source-file page for /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/dlarrv.f, and it is lowkey a whole lot of tridiagonal eigenvector tea ☕✨

📛 NAME 📛

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/dlarrv.f — that's the file this manual page is serving, no cap. 💅

🦋 SYNOPSIS 🦋

🧩 Functions/Subroutines

There is one subroutine on the menu, and it is giving main-character energy:

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. slay. 🦋

✨ Function/Subroutine Documentation ✨

Here's the full signature in all its glory:

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)

And yes, it says it again because repetition is iconic: DLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT.

🎯 Purpose

okay so the actual purpose block, bestie:

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. ✨

🧸 Parameters

This is the part where every single parameter gets its moment. No skipping, no summarizing, no "you get the vibe" — we're doing the whole list. 💅

  • N — N is INTEGER. The order of the matrix. N >= 0.
  • VL — VL is DOUBLE PRECISION. Lower bound of the interval that contains the desired eigenvalues. VL < VU. This is 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.

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. 🙈

  • 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.
  • 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 literally the whole description in the source, bestie. 🫶
  • 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 DOUBLE PRECISION. Again, that's all the source says here. No extra lore. 🤷‍♀️
  • 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|) ).
  • 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 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 — 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 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.

💌 Author

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