SLAEBZ(3)

💅 SLAEBZ(3) 💅

Okay bestie, this is from the Library Functions Manual, and the header vibe is /home/abui...RC/slaebz.f(3) (yes, truncated like that — iykyk). The subroutine is about to be the main character. 🫶

🌸 NAME 🌸

The NAME here is literally the source path, no cute nickname. It is:

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

✨ SYNOPSIS ✨

Under Functions/Subroutines, we have:

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

And the one-line mood is: SLAEBZ 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 🦋

Here's the full typed-out signature, bestie:

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

SLAEBZ 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 💖

SLAEBZ contains the iteration loops which compute and use the function N(w) — that's 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. 💀 So like, she trusts you. Be normal.

🎀 Parameters 🎀

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 SLAEBZ to return with INFO=-1.

NITMAX

NITMAX is INTEGER. The maximum number of bisection (the original has a doubled of — we keep the vibe) 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 SLAEBZ 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 REAL. 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 REAL. 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 REAL. 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 REAL array, dimension (N). The diagonal elements of the tridiagonal matrix T.

E

E is REAL array, dimension (N). The offdiagonal elements of the tridiagonal matrix T in positions 1 through N-1. E(N) is arbitrary.

E2

E2 is REAL 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 REAL 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 REAL 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 SLAEBZ 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 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 SLAEBZ is called.

WORK

WORK is REAL 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 original leaves this section empty — a true minimalist icon. 💅

okay so real talk bestie — before we get into the actual recipe, SLAEBZ wants you to know who she's for 👀

This routine is intended to be called only by other LAPACK routines, which is why the interface is lowkey less user-friendly than your usual fave 😭. She's got exactly two purposes, and that's it, no cap:

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🅰️ Purpose one: finding eigenvalues ✨

You set up one or more initial intervals in AB, then call SLAEBZ with IJOB=1. That call sets up NAB and also counts the eigenvalues 🎀. Intervals with no eigenvalues would usually get thrown out at this point — like, bye, not needed 💅.

Also, if not all the eigenvalues in an interval i are desired, NAB(i,1) can be increased or NAB(i,2) decreased. Classic little trim 🫶. For example, set NAB(i,1)=NAB(i,2)-1 to get the largest eigenvalue. Obsessed.

Then SLAEBZ gets called again with IJOB=2, and MMAX must be no smaller than the value of MOUT that came back from the IJOB=1 call. 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 🔥. Chef's kiss. Periodt.

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🅱️ Purpose two: finding an interval (a',b'] containing eigenvalues w(f),...,w(l) 🦋

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. 📍

Then NAB(i,1) should be -1 and NAB(i,2) should be N+1 — that's the tripwire, so it flags an error if the desired interval does not lie in (a,b) 💀. Very main character of her to set a trap.

SLAEBZ 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
• if, to the specified tolerance, w(f-k)=...=w(f+r) with 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 🌸. Like, same energy, different index, you know? iykyk.

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So where does this baddie actually live? Right here bestie:

Definition at line 316 of file slaebz.f.

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👑 AUTHOR

Generated automatically by Doxygen for LAPACK from the source code.

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and that's the whole vibe check 💖 this queen is LAPACK Version 3.12.0, straight out of /home/abui...RC/slaebz.f(3) — go forth and call her only from inside other LAPACK routines, okay? no rogue solo missions 🙈🫶 slay 🎀✨