stebz(3)

📦

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💅 stebz(3) — Library Functions Manual 💅

okay so this is stebz(3) from the Library Functions Manual, and it's the eigenvalue era of your life, bestie. It's giving "two nearly identical Fortran subroutines wearing the same outfit" and honestly? Obsessed. Let's get into it ✨

🌸 NAME 🌸

stebz — stebz: eig, Kahan

🦋 SYNOPSIS 🦋

✨ Functions ✨

dstebz — the double-precision one, DSTEBZ, she's that girl 💅

subroutine dstebz (range, order, n, vl, vu, il, iu, abstol, d, e, m,
    nsplit, w, iblock, isplit, work, iwork, info)

sstebz — the single-precision one, SSTEBZ, same energy, less bytes 🎀

subroutine sstebz (range, order, n, vl, vu, il, iu, abstol, d, e, m,
    nsplit, w, iblock, isplit, work, iwork, info)

👑 DETAILED DESCRIPTION 👑

Function Documentation — literally just vibes and then it throws you straight into the function docs, no cap. Both routines live here: dstebz first, then sstebz, and they are fr fr twins separated only by DOUBLE PRECISION vs real 🫶

🌈 FUNCTION DOCUMENTATION 🌈

💖 subroutine dstebz 💖

subroutine dstebz (character range, character order, integer n, double
    precision vl, double precision vu, integer il, integer iu, double
    precision abstol, double precision, dimension( * ) d, double precision,
    dimension( * ) e, integer m, integer nsplit, double precision,
    dimension( * ) w, integer, dimension( * ) iblock, integer, dimension( *
    ) isplit, double precision, dimension( * ) work, integer, dimension( * )
    iwork, integer info)

🌟 DSTEBZ 🌟

✨ Purpose ✨

okay so DSTEBZ computes the eigenvalues of a symmetric tridiagonal matrix T. That's the whole assignment, bestie. 👑

You can ask her 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 has to 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. Don't be shy with your scaling, but don't be too tiny either 😌

See W. Kahan, Report CS41, Computer Science Dept., Stanford University, July 21, 1966. Citation icon, we respect her 📚

🎀 Parameters 🎀

RANGE — RANGE is CHARACTER*1
- = '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 — ORDER is CHARACTER*1
- = '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'.

ABSTOL — ABSTOL is DOUBLE PRECISION. The absolute tolerance for the eigenvalues. An eigenvalue (or cluster) is considered to be located if it has been determined to lie in an interval whose width is ABSTOL or less. If ABSTOL is less than or equal to zero, then ULP*|T| will be used, where |T| means the 1-norm of T. 🧚

Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*DLAMCH('S'), not zero. (Not me setting it to zero and crying later 😭)

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.
M — M is INTEGER. The actual number of eigenvalues found. 0 <= M <= N. (See also the description of INFO=2,3.)
NSPLIT — NSPLIT is INTEGER. The number of diagonal blocks in the matrix T. 1 <= NSPLIT <= N.
W — W is DOUBLE PRECISION array, dimension (N). On exit, the first M elements of W will contain the eigenvalues. (DSTEBZ may use the remaining N-M elements as workspace.) ✨
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. (DSTEBZ may use the remaining N-M elements as workspace.)
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.) 🫶
WORK — WORK is DOUBLE PRECISION array, dimension (4*N) 🍓
IWORK — IWORK is INTEGER array, dimension (3*N) 🌸
INFO — INFO is INTEGER
- = 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. 💀

🔥 Internal Parameters 🔥

RELFAC — DOUBLE PRECISION, default = 2.0e0. The relative tolerance. An interval (a,b] lies within if b-a < RELFAC*ulp*max(|a|,|b|), where is the machine precision (distance from 1 to the next larger floating point number.)
FUDGE — DOUBLE PRECISION, default = 2. A to widen the Gershgorin intervals. Ideally, a value of 1 should work, but on machines with sloppy arithmetic, this 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. 👑

👑 Author 👑

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

Definition at line 270 of file dstebz.f. 🎀

💜 subroutine sstebz 💜

subroutine sstebz (character range, character order, integer n, real vl,
    real vu, integer il, integer iu, real abstol, real, dimension( * ) d,
    real, dimension( * ) e, integer m, integer nsplit, real, dimension( * )
    w, integer, dimension( * ) iblock, integer, dimension( * ) isplit, real,
    dimension( * ) work, integer, dimension( * ) iwork, integer info)

⭐ SSTEBZ ⭐

✨ Purpose ✨

SSTEBZ computes the eigenvalues of a symmetric tridiagonal matrix T. Same assignment, single precision, still iconic 💅

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. 🌟

🎀 Parameters 🎀

RANGE — …and that's literally where this page stops, bestie, no cap 😭
= 'A' — () all eigenvalues will be found, every single one, no filter, no skip. 💖
= 'V' — () all eigenvalues in the half-open interval (VL, VU] will be found. Note the paren-bracket situation: VL is not included, VU is. Half-open, bestie, respect the bracket. 🎀
= 'I' — () the IL-th through IU-th eigenvalues (of the entire matrix, not of some cute little subset) will be found.

(those empty () after each = are literally just sitting there in the source, no cap — formatting ghost, ignore it, the info is all in the words 💀)

🎀 `ORDER`

ORDER is CHARACTER*1

= 'B' — () the eigenvalues will be grouped by split-off block (see IBLOCK, ISPLIT) and ordered from smallest to largest within the block. So it's small-to-big inside each block, but the blocks stay separate. Organized chaos. ✨
= 'E' — () the eigenvalues for the entire matrix will be ordered from smallest to largest. One big ascending list, no blocks, just vibes. 👑

🌸 `N`

N is INTEGER

The order of the tridiagonal matrix T. N >= 0. That's it, that's the whole thing. 💅

🦋 `VL`

VL is REAL

If RANGE='V', this is 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'. (So don't stress about filling it in if you're not in 'V' mode, she's off duty. 🫶)

🦋 `VU`

VU is REAL

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. Constraint: 1 <= IL <= IU <= N, if N > 0; and 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. Constraints again: 1 <= IL <= IU <= N, if N > 0; and IL = 1 and IU = 0 if N = 0.

Not referenced if RANGE = 'A' or 'V'.

⭐ `ABSTOL`

ABSTOL is REAL

The absolute tolerance for the eigenvalues. An eigenvalue (or a whole cluster of them) is considered located if it's been determined to lie in an interval whose width is ABSTOL or less. Tight.

If ABSTOL is less than or equal to zero, then ULP*|T| gets used instead, where |T| means the 1-norm of T.

Real talk: eigenvalues get computed most accurately when ABSTOL is set to twice the underflow threshold 2*SLAMCH('S') — not zero. Zero is not the move here, bestie. Set it properly. 😭

🍓 `D`

D is REAL array, dimension (N)

The n diagonal elements of the tridiagonal matrix T. The main-character diagonal. 💅

🍓 `E`

E is REAL array, dimension (N-1)

The n-1 off-diagonal elements of the tridiagonal matrix T. The sidekicks, but make it diagonal-adjacent. ✨

💜 `M`

M is INTEGER

The actual number of eigenvalues found. 0 <= M <= N.

(See also the description of INFO=2,3 — yes, it's foreshadowing. 🎀)

💜 `NSPLIT`

NSPLIT is INTEGER

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

🧚 `W`

W is REAL array, dimension (N)

On exit, the first M elements of W will contain the eigenvalues.

(SSTEBZ may use the remaining N-M elements as workspace — so like, she's borrowing your couch, don't be weird about it. 🫶)

🧚 `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 which block (from 1 to the number of blocks) the eigenvalue W(i) belongs to.

(SSTEBZ may use the remaining N-M elements as workspace. Same couch situation. 💀)

💖 `ISPLIT`

ISPLIT is INTEGER array, dimension (N)

The splitting points, at which T breaks up into submatrices. First submatrix = rows/columns 1 to ISPLIT(1), second = rows/columns ISPLIT(1)+1 through ISPLIT(2), etc., and the NSPLIT-th = rows/columns ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.

Only the first NSPLIT elements will actually be used, but since you can't know a priori what value NSPLIT will have, N words must be reserved for ISPLIT. So yeah, allocate the whole thing. 🙈

🔥 `WORK`

WORK is REAL array, dimension (4*N)

Scratch space. That's it. That's her job. ✨

🔥 `IWORK`

IWORK is INTEGER array, dimension (3*N)

Integer scratch space. Also just doing her little job. 🎀

👑 `INFO`

INFO is INTEGER

= 0 — successful exit. Slay. 🎉
< 0 — if INFO = -i, the i-th argument had an illegal value. You passed something cursed, bestie. 💀
> 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. (Not your fault. Probably. 🙈)
- = 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. (the parameter name is literally blank in the source, iykyk 🫠)
- = 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. (blank parameter name again — she left it as an exercise for the reader 💀)

⚙️ Internal Parameters

RELFAC — REAL, default = 2.0e0. The relative tolerance. An interval (a,b] lies within if b-a < RELFAC*ulp*max(|a|,|b|), where ulp is the machine precision (distance from 1 to the next larger floating point number). ✨
FUDGE — REAL, default = 2. A fudge factor to widen the Gershgorin intervals. Ideally a value of 1 should work, but on machines with sloppy arithmetic, this 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. So she's just here to help you find them, not to make them prettier. 🫶

👑 Author

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

Definition at line 270 of file sstebz.f.

👑 Author

Generated automatically by Doxygen for LAPACK from the source code.

Okay that's the whole thing, bestie — you made it to the end of the stebz saga 🎀 This little beauty comes from LAPACK, Version 3.12.0, and she lives at stebz(3). Go forth and find those eigenvalues. 💖✨

📦 where this page comes from

arch rolling: lapack-doc 3.12.1-2 · custom
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms

stebz(3)

💅 stebz(3) — Library Functions Manual 💅

🌸 NAME 🌸

🦋 SYNOPSIS 🦋

✨ Functions ✨

👑 DETAILED DESCRIPTION 👑

🌈 FUNCTION DOCUMENTATION 🌈

💖 subroutine dstebz 💖

🌟 DSTEBZ 🌟

✨ Purpose ✨

🎀 Parameters 🎀

🔥 Internal Parameters 🔥

👑 Author 👑

💜 subroutine sstebz 💜

⭐ SSTEBZ ⭐

✨ Purpose ✨

🎀 Parameters 🎀

🎀 ORDER

🌸 N

🦋 VL

🦋 VU

🌈 IL

🌈 IU

⭐ ABSTOL

🍓 D

🍓 E

💜 M

💜 NSPLIT

🧚 W

🧚 IBLOCK

💖 ISPLIT

🔥 WORK

🔥 IWORK

👑 INFO

⚙️ Internal Parameters

👑 Author

👑 Author

🎀 `ORDER`

🌸 `N`

🦋 `VL`

🦋 `VU`

🌈 `IL`

🌈 `IU`

⭐ `ABSTOL`

🍓 `D`

🍓 `E`

💜 `M`

💜 `NSPLIT`

🧚 `W`

🧚 `IBLOCK`

💖 `ISPLIT`

🔥 `WORK`

🔥 `IWORK`

👑 `INFO`