DLARRD(3)

💖 NAME

okay so bestie, this is the manual page for DLARRD(3) — it lives in the Library Functions Manual, and the file it documents is literally this whole journey of a path:

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/dlarrd.f

like, that's the actual source file, dlarrd.f, out here with the longest address ever. 🏡✨

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✨ SYNOPSIS ✨

🧚 Functions/Subroutines

• subroutine DLARRD (range, order, n, vl, vu, il, iu, gers, reltol, d, e, e2, pivmin, nsplit, isplit, m, w, werr, wl, wu, iblock, indexw, work, iwork, info) — DLARRD computes the eigenvalues of a symmetric tridiagonal matrix to suitable accuracy.

subroutine DLARRD (range, order, n, vl, vu, il, iu, gers, reltol, d, e,
    e2, pivmin, nsplit, isplit, m, w, werr, wl, wu, iblock, indexw,
    work, iwork, info)

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🎀 Function/Subroutine Documentation 🎀

💅 subroutine DLARRD

here's the full signature in all its glory, character for character, no cap:

subroutine DLARRD (character range, character order, integer n, double
    precision vl, double precision vu, integer il, integer iu, double
    precision, dimension( * ) gers, double precision reltol, double
    precision, dimension( * ) d, double precision, dimension( * ) e, double
    precision, dimension( * ) e2, double precision pivmin, integer nsplit,
    integer, dimension( * ) isplit, integer m, double precision, dimension(
    * ) w, double precision, dimension( * ) werr, double precision wl,
    double precision wu, integer, dimension( * ) iblock, integer, dimension(
    * ) indexw, double precision, dimension( * ) work, integer, dimension( *
    ) iwork, integer info)

DLARRD computes the eigenvalues of a symmetric tridiagonal matrix to suitable accuracy. ✨

🌸 Purpose

DLARRD computes the eigenvalues of a symmetric tridiagonal matrix T to suitable accuracy. This is an auxiliary code to be called from DSTEMR. The user may ask for all eigenvalues, all eigenvalues in the half-open interval (VL, VU], or the IL-th through IU-th eigenvalues.

To avoid overflow, the matrix must be scaled so that its largest element is no greater than overflow(1/2) * underflow(1/4) in absolute value, and for greatest accuracy, it should not be much smaller than that.

See W. Kahan , Report CS41, Computer Science Dept., Stanford University, July 21, 1966.

so basically 💀 it's the helper routine — it gets called from DSTEMR, it does not fly solo. And that overflow warning? that's a real one, bestie: scale your matrix or your numbers will simply evaporate. 🫶

🦋 Parameters

RANGE — it's a CHARACTER*1, and honestly it's the main character here, it decides which eigenvalues you get:

• = '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.

ORDER — also a CHARACTER*1, and this one is the vibes-orderer:

• = 'B': () — the eigenvalues will be grouped by split-off block (see IBLOCK, ISPLIT) and ordered from smallest to largest within the block.
• = 'E': () — the eigenvalues for the entire matrix will be ordered from smallest to largest.

N — N is INTEGER. The order of the tridiagonal matrix T. N >= 0.

VL — VL is DOUBLE PRECISION. If RANGE='V', the lower bound of the interval to be searched for eigenvalues. Eigenvalues less than or equal to VL, or greater than VU, will not be returned. VL < VU. Not referenced if RANGE = 'A' or 'I'.

VU — VU is DOUBLE PRECISION. If RANGE='V', the upper bound of the interval to be searched for eigenvalues. Eigenvalues less than or equal to VL, or greater than VU, will not be returned. VL < VU. Not referenced if RANGE = 'A' or 'I'.

IL — IL is INTEGER. If RANGE='I', the index of the smallest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.

IU — IU is INTEGER. If RANGE='I', the index of the largest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.

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))).

RELTOL — RELTOL is DOUBLE PRECISION. The minimum relative width of an interval. When an interval is narrower than RELTOL times the larger (in magnitude) endpoint, then it is considered to be sufficiently small, i.e., converged. Note: this should always be at least radix*machine epsilon.

D — D is DOUBLE PRECISION array, dimension (N). The n diagonal elements of the tridiagonal matrix T.

E — E is DOUBLE PRECISION array, dimension (N-1). The (n-1) off-diagonal elements of the tridiagonal matrix T.

E2 — E2 is DOUBLE PRECISION array, dimension (N-1). The (n-1) squared off-diagonal elements of the tridiagonal matrix T. (yes, pre-squared, she's efficient 💅)

PIVMIN — PIVMIN is DOUBLE PRECISION. The minimum pivot allowed in the Sturm sequence for T.

NSPLIT — NSPLIT is INTEGER. The number of diagonal blocks in the matrix T. 1 <= NSPLIT <= N.

ISPLIT — ISPLIT is INTEGER array, dimension (N). The splitting points, at which T breaks up into submatrices. The first submatrix 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. (Only the first NSPLIT elements will actually be used, but since the user cannot know a priori what value NSPLIT will have, N words must be reserved for ISPLIT.)

M — M is INTEGER. The actual number of eigenvalues found. 0 <= M <= N. (See also the description of INFO=2,3.)

W — W is DOUBLE PRECISION array, dimension (N). On exit, the first M elements of W will contain the eigenvalue approximations. DLARRD computes an interval I_j = (a_j, b_j] that includes eigenvalue j. The eigenvalue approximation is given as the interval midpoint W(j)= ( a_j + b_j)/2. The corresponding error is bounded by WERR(j) = abs( a_j - b_j)/2.

WERR — WERR is DOUBLE PRECISION array, dimension (N). The error bound on the corresponding eigenvalue approximation in W.

WL — WL is DOUBLE PRECISION. (that's it, that's the whole description, she's mysterious 🌈)

WU — WU is DOUBLE PRECISION. The interval (WL, WU] contains all the wanted eigenvalues.

• If RANGE='V', then WL=VL and WU=VU.
• If RANGE='A', then WL and WU are the global Gerschgorin bounds on the spectrum.
• If RANGE='I', then WL and WU are computed by DLAEBZ from the index range specified.

IBLOCK — IBLOCK is INTEGER array, dimension (N). At each row/column j where E(j) is zero or small, the matrix T is considered to split into a block diagonal matrix. On exit, if INFO = 0, IBLOCK(i) specifies to which block (from 1 to the number of blocks) the eigenvalue W(i) belongs. (DLARRD may use the remaining N-M elements as workspace.)

INDEXW — INDEXW is INTEGER array, dimension (N). The indices of the eigenvalues within each block (submatrix); for example, INDEXW(i)= j and IBLOCK(i)=k imply that the i-th eigenvalue W(i) is the j-th eigenvalue in block k.

WORK — WORK is DOUBLE PRECISION array, dimension (4*N). (pure workspace energy 🫶)

IWORK — IWORK is INTEGER array, dimension (3*N). (same, but integers)

INFO — INFO is INTEGER, and this is the one you actually check, bestie:

• = 0: successful exit 🎉
• < 0: if INFO = -i, the i-th argument had an illegal value
• > 0: some or all of the eigenvalues failed to converge or were not computed:
  • =1 or 3: Bisection failed to converge for some eigenvalues; these eigenvalues are flagged by a negative block number. The effect is that the eigenvalues may not be as accurate as the absolute and relative tolerances. This is generally caused by unexpectedly inaccurate arithmetic.
  • =2 or 3: RANGE='I' only: Not all of the eigenvalues IL:IU were found.
    • Effect: M < IU+1-IL
    • Cause: non-monotonic arithmetic, causing the Sturm sequence to be non-monotonic.
    • Cure: recalculate, using RANGE='A', and pick out eigenvalues IL:IU. In some cases, increasing the PARAMETER may make things work.
  • = 4: RANGE='I', and the Gershgorin interval initially used was too small. No eigenvalues were computed.
    • Probable cause: your machine has sloppy floating-point arithmetic.
    • Cure: Increase the PARAMETER , recompile, and try again.

💀 real talk: INFO > 0 is not a "whatever" situation. Negative block numbers mean your answers might not hit the tolerances you asked for, and = 2 or 3 on RANGE='I' literally means some eigenvalues you wanted just… weren't found. The fixes above are the fixes above — don't improvise.

Internal Parameters

(okay so this is where part 1 of the page runs out — the Internal Parameters section is next, and there's more coming 💖)

• FUDGE — DOUBLE PRECISION, default = 2. This is the knob that widens the Gershgorin intervals. Ideally a value of 1 should work, but on machines with sloppy arithmetic it needs to be larger — the default for publicly released versions should be large enough to handle the worst machine around. Note that this has no effect on accuracy of the solution. 🫶✨

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💖 Contributors

• W. Kahan, University of California, Berkeley, USA
• 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

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👑 Author

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

Definition at line 325 of file dlarrd.f.

🌈 Author

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Source/version/date footer tea: LAPACK Version 3.12.0, auto-generated by Doxygen for LAPACK from the source code, with the /home/abui...RC/dlarrd.f(3) path and its footer date energy. Periodt. 💅🎀