SRC/dlaebz.f(3)

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💖 SRC/dlaebz.f(3) 💖

okay bestie, this is the SRC/dlaebz.f(3) page from the Library Functions Manual — part 1, and it's literally the dlaebz routine's whole personality. No cap. ✨

🌸 NAME 🌸

SRC/dlaebz.f — that's the file, and the star of the show is the dlaebz subroutine. Periodt. 💅

✨ SYNOPSIS ✨

🧚 Functions/Subroutines 🧚

       subroutine dlaebz (ijob, nitmax, n, mmax, minp, nbmin, abstol, reltol,
           pivmin, d, e, e2, nval, ab, c, mout, nab, work, iwork, info)

DLAEBZ computes the number of eigenvalues of a real symmetric tridiagonal matrix which are less than or equal to a given value, and performs other tasks required by the routine sstebz.

🦋 Function/Subroutine Documentation 🦋

   subroutine dlaebz (integer ijob, integer nitmax, integer n, integer mmax,
       integer minp, integer nbmin, double precision abstol, double precision
       reltol, double precision pivmin, double precision, dimension( * ) d,
       double precision, dimension( * ) e, double precision, dimension( * ) e2,
       integer, dimension( * ) nval, double precision, dimension( mmax, * ) ab,
       double precision, dimension( * ) c, integer mout, integer, dimension(
       mmax, * ) nab, double precision, dimension( * ) work, integer,
       dimension( * ) iwork, integer info)

DLAEBZ computes the number of eigenvalues of a real symmetric tridiagonal matrix which are less than or equal to a given value, and performs other tasks required by the routine sstebz.

💜 Purpose 💜

okay so here's the tea: DLAEBZ contains the iteration loops which compute and use the function N(w), which is the count of eigenvalues of a symmetric tridiagonal matrix T less than or equal to its argument w. She performs a choice of two types of loops:

  • IJOB=1, followed by IJOB=2: It takes as input a list of intervals and returns a list of sufficiently small intervals whose union contains the same eigenvalues as the union of the original intervals. The input intervals are (AB(j,1),AB(j,2)], j=1,...,MINP. The output interval (AB(j,1),AB(j,2)] will contain eigenvalues NAB(j,1)+1,...,NAB(j,2), where 1 <= j <= MOUT.
  • IJOB=3: It performs a binary search in each input interval (AB(j,1),AB(j,2)] for a point w(j) such that N(w(j))=NVAL(j), and uses C(j) as the starting point of the search. If such a w(j) is found, then on output AB(j,1)=AB(j,2)=w. If no such w(j) is found, then on output (AB(j,1),AB(j,2)] will be a small interval containing the point where N(w) jumps through NVAL(j), unless that point lies outside the initial interval.

Note that the intervals are in all cases half-open intervals, i.e., of the form (a,b], which includes b but not a. 🎀

To avoid underflow, the matrix should be scaled so that its largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value. To assure the most accurate computation of small eigenvalues, the matrix should be scaled to be not much smaller than that, either.

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

⚠️ Note: the arguments are, in general, not checked for unreasonable values. Like, she's trusting you, bestie — don't hand her chaos and expect a cute little vibe check. 💀

🌈 Parameters 🌈

okay bestie, parameter time — every single one, no skipping, because we don't do that here. 🫶

  • IJOB — IJOB is INTEGER. Specifies what is to be done:
    • = 1: Compute NAB for the initial intervals.
    • = 2: Perform bisection iteration to find eigenvalues of T.
    • = 3: Perform bisection iteration to invert N(w), i.e., to find a point which has a specified number of eigenvalues of T to its left.
    • Other values will cause DLAEBZ to return with INFO=-1.
  • NITMAX — NITMAX is INTEGER. The maximum number of of bisection to be performed, i.e., an interval of width W will not be made smaller than 2^(-NITMAX) * W. If not all intervals have converged after NITMAX iterations, then INFO is set to the number of non-converged intervals.
  • N — N is INTEGER. The dimension n of the tridiagonal matrix T. It must be at least 1.
  • MMAX — MMAX is INTEGER. The maximum number of intervals. If more than MMAX intervals are generated, then DLAEBZ will quit with INFO=MMAX+1.
  • MINP — MINP is INTEGER. The initial number of intervals. It may not be greater than MMAX.
  • NBMIN — NBMIN is INTEGER. The smallest number of intervals that should be processed using a vector loop. If zero, then only the scalar loop will be used.
  • ABSTOL — ABSTOL is DOUBLE PRECISION. The minimum (absolute) width of an interval. When an interval is narrower than ABSTOL, or than RELTOL times the larger (in magnitude) endpoint, then it is considered to be sufficiently small, i.e., converged. This must be at least zero.
  • RELTOL — RELTOL is DOUBLE PRECISION. The minimum relative width of an interval. When an interval is narrower than ABSTOL, or 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.
  • PIVMIN — PIVMIN is DOUBLE PRECISION. The minimum absolute value of a in the Sturm sequence loop. This must be at least max |e(j)**2|*safe_min and at least safe_min, where safe_min is at least the smallest number that can divide one without overflow.
  • D — D is DOUBLE PRECISION array, dimension (N). The diagonal elements of the tridiagonal matrix T.
  • E — E is DOUBLE PRECISION array, dimension (N). The offdiagonal elements of the tridiagonal matrix T in positions 1 through N-1. E(N) is arbitrary.
  • E2 — E2 is DOUBLE PRECISION array, dimension (N). The squares of the offdiagonal elements of the tridiagonal matrix T. E2(N) is ignored.
  • NVAL — NVAL is INTEGER array, dimension (MINP). If IJOB=1 or 2, not referenced. If IJOB=3, the desired values of N(w). The elements of NVAL will be reordered to correspond with the intervals in AB. Thus, NVAL(j) on output will not, in general be the same as NVAL(j) on input, but it will correspond with the interval (AB(j,1),AB(j,2)] on output.
  • AB — AB is DOUBLE PRECISION array, dimension (MMAX,2). The endpoints of the intervals. AB(j,1) is a(j), the left endpoint of the j-th interval, and AB(j,2) is b(j), the right endpoint of the j-th interval. The input intervals will, in general, be modified, split, and reordered by the calculation.
  • C — C is DOUBLE PRECISION array, dimension (MMAX). If IJOB=1, ignored. If IJOB=2, workspace. If IJOB=3, then on input C(j) should be initialized to the first search point in the binary search.
  • MOUT — MOUT is INTEGER. If IJOB=1, the number of eigenvalues in the intervals. If IJOB=2 or 3, the number of intervals output. If IJOB=3, MOUT will equal MINP.
  • NAB — NAB is INTEGER array, dimension (MMAX,2).
    • If IJOB=1, then on output NAB(i,j) will be set to N(AB(i,j)).
    • If IJOB=2, then on input, NAB(i,j) should be set. It must satisfy the condition: N(AB(i,1)) <= NAB(i,1) <= NAB(i,2) <= N(AB(i,2)), which means that in interval i only eigenvalues NAB(i,1)+1,...,NAB(i,2) will be considered. Usually, NAB(i,j)=N(AB(i,j)), from a previous call to DLAEBZ with IJOB=1. On output, NAB(i,j) will contain max(na(k),min(nb(k),N(AB(i,j)))), where k is the index of the input interval that the output interval (AB(j,1),AB(j,2)] came from, and na(k) and nb(k) are the the input values of NAB(k,1) and NAB(k,2).
    • If IJOB=3, then on output, NAB(i,j) contains N(AB(i,j)), unless N(w) > NVAL(i) for all search points w, in which case NAB(i,1) will not be modified, i.e., the output value will be the same as the input value (modulo reorderings -- see NVAL and AB), or unless N(w) < NVAL(i) for all search points w, in which case NAB(i,2) will not be modified. Normally, NAB should be set to some distinctive value(s) before DLAEBZ is called.
  • WORK — WORK is DOUBLE PRECISION array, dimension (MMAX). Workspace.
  • IWORK — IWORK is INTEGER array, dimension (MMAX). Workspace.
  • INFO — INFO is INTEGER.
    • = 0: All intervals converged.
    • = 1--MMAX: The last INFO intervals did not converge.
    • = MMAX+1: More than MMAX intervals were generated.

👑 Author 👑

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

🎀 Further Details 🎀

The source has the Further Details: label here, and nothing else in this part. 💜

okay bestie, real talk before literally anything else: this routine is not for you to call directly. It's intended to be called only by other LAPACK routines, which is exactly why the interface is less user-friendly 🙈 — she's an internal girlie, she doesn't do small talk. She has two purposes and she serves both of them flawlessly:

  • (a) finding eigenvalues 👑 — In this case, DLAEBZ should have one or more initial intervals set up in AB, and DLAEBZ should be called with IJOB=1. That call sets up NAB, and also counts the eigenvalues. Intervals with no eigenvalues would usually be thrown out at this point (no notes, just ghosted 💀). Also, if not all the eigenvalues in an interval i are desired, NAB(i,1) can be increased or NAB(i,2) decreased. For example, set NAB(i,1)=NAB(i,2)-1 to get the largest eigenvalue. DLAEBZ is then called with IJOB=2 and MMAX no smaller than the value of MOUT returned by the call with IJOB=1. After this (IJOB=2) call, eigenvalues NAB(i,1)+1 through NAB(i,2) are approximately AB(i,1) (or AB(i,2)) to the tolerance specified by ABSTOL and RELTOL. ✨
  • (b) finding an interval (a',b'] containing eigenvalues w(f),...,w(l) 🦋 — In this case, start with a Gershgorin interval (a,b). Set up AB to contain 2 search intervals, both initially (a,b). One NVAL element should contain f-1 and the other should contain l , while C should contain a and b, resp. NAB(i,1) should be -1 and NAB(i,2) should be N+1, to flag an error if the desired interval does not lie in (a,b) (yes, bestie, that's literally the point — it's the ✨error flag✨, not a typo 🎀). DLAEBZ is then called with IJOB=3.

On exit, if w(f-1) < w(f), then one of the intervals -- j -- will have AB(j,1)=AB(j,2) and NAB(j,1)=NAB(j,2)=f-1, while if, to the specified tolerance, w(f-k)=...=w(f+r), k > 0 and r >= 0, then the interval will have N(AB(j,1))=NAB(j,1)=f-k and N(AB(j,2))=NAB(j,2)=f+r. The cases w(l) < w(l+1) and w(l-r)=...=w(l+k) are handled similarly. 👑

ngl the exit conditions are the ✨main character energy✨ of this whole routine — read them twice, they're doing a lot 💅

Definition at line 316 of file dlaebz.f.

🫶 AUTHOR

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