laebz(3)

📦
version 3.12.1 · 9 versions
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💅 NAME 💅

Okay bestie, this is the laebz(3) page, and the NAME line is literally giving exactly this:

laebz — laebz: counts eigvals <= value

✨ SYNOPSIS ✨

🌈 Functions 🌈

Two subroutine besties are on deck, no cap, both with the same argument list. First up:

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. periodt.

Then the slaebz sibling:

subroutine slaebz (ijob, nitmax, n, mmax, minp, nbmin, abstol, reltol,
           pivmin, d, e, e2, nval, ab, c, mout, nab, work, iwork, 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. slay.

🦋 Detailed Description 🦋

The detailed tea starts right here, and then the function docs take over and absolutely ate. iykyk.

🎀 Function Documentation 🎀

💜 subroutine dlaebz 💜

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 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. It 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 do not come for this routine with messy inputs and expect a vibe check, bestie.

🍓 Parameters 🍓

Okay bestie, here’s the full parameter lineup, and yes, every single one is iconic.

  • 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 🌸

Shoutout to the author besties:

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

🌈 Further Details 🌈

(This section is giving empty, fr fr — no extra text in the original on this page.)

…and like, real talk before we go any further: this routine is literally only meant to be called by other LAPACK routines, which is why the interface is so not user-friendly. It's giving "internal API, do not perceive her." It's got exactly two purposes, bestie, and here they are 🫶

(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 just get thrown out at this point (bye 💀). 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. Slay.

(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). 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. Iconic symmetry, honestly 💅

Definition at line 316 of file dlaebz.f.

🦋 subroutine slaebz 🦋

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. Obsessed with her multitasking era ✨

💖 Purpose 💖

SLAEBZ 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. It 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.

Lowkey important detail: the intervals are in all cases half-open intervals, i.e., of the form (a,b], which includes b but not a. It's giving "boundary queen" 🎀

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. Balance, bestie 🌸

See W. Kahan , Report CS41, Computer Science Dept., Stanford University, July 21, 1966. (Academic icon behavior 📚)

Note: the arguments are, in general, not checked for unreasonable values. She trusts you. Do not make it weird 🙈

🌸 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. (Don't do that 💀)
  • 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. She's keeping receipts 👑
  • 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. Scalar loop said "fine, I'll do it myself" 😭
  • 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. She's small but she's mighty 🧚
  • D — D is REAL array, dimension (N). The diagonal elements of the tridiagonal matrix T. Main character energy, obviously 💅
  • 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. (Not me showing up to work with an arbitrary value 🙈)
  • 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. Glow-up arc fr fr ✨
  • 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. The way she reshapes herself for the algorithm 🦋
  • 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 — INTEGER array, dimension (MMAX,2). Okay bestie, this one has lore:

  • If IJOB=1, then on output NAB(i,j) will be set to N(AB(i,j)). Slay, automatic.
  • 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 — REAL array, dimension (MMAX). Workspace, literally just vibing and holding space.

IWORK — INTEGER array, dimension (MMAX). More workspace, but make it integer.

INFO — INTEGER. Vibe check incoming:

  • = 0: All intervals converged. Iconic.
  • = 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 🔥

This routine is intended to be called only by other LAPACK routines, thus the interface is less user-friendly. It is intended for two purposes:

  • (a) finding eigenvalues. In this case, SLAEBZ should have one or more initial intervals set up in AB, and SLAEBZ should be called with IJOB=1. This sets up NAB, and also counts the eigenvalues. Intervals with no eigenvalues would usually be thrown out at this point. 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. SLAEBZ 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). 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, 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.

Definition at line 316 of file slaebz.f.

Author: Generated automatically by Doxygen for LAPACK from the source code.

🎀 Okay that's the whole page, bestie! Footer sign-off time: LAPACK Version 3.12.0, laebz(3), and the date sitting pretty in the footer. Byeeee, go slay some eigenvalues 💅✨

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