DSTEBZ(3)

🎀 DSTEBZ 🎀

okay bestie, welcome to the Library Functions Manual ✨ this one's a real one — it lives at

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

and it's about to become your whole personality. no cap. 💅

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✨ SYNOPSIS ✨

Functions/Subroutines — yeah that's the category, we're in the function/subroutine neighborhood 🌸

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

DSTEBZ 🔥 (yes, it's giving main character energy, it gets its own name line)

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🫶 Function/Subroutine Documentation 🫶

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 💜 (again, iconic, we respect the name)

🌈 Purpose 🌈

okay so here's the vibe: DSTEBZ computes the eigenvalues of a symmetric tridiagonal matrix T. that's it, that's the whole thing. 🦋

you, the user, get to ask for:

• literally all of the eigenvalues,
• all the eigenvalues in the half-open interval (VL, VU], or
• the IL-th through IU-th eigenvalues.

ngl, there's a scaling moment you need to know about: 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. 🫶

and if you want the receipts, see W. Kahan , Report CS41, Computer Science Dept., Stanford University, July 21, 1966. 💀 that's a deep cut, respect.

🍓 Parameters 🍓

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

real talk: eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*DLAMCH('S'), not zero. not me warning you about the zero case. 😭

• D — is DOUBLE PRECISION array, dimension (N). The n diagonal elements of the tridiagonal matrix T.
• E — is DOUBLE PRECISION array, dimension (N-1). The (n-1) off-diagonal elements of the tridiagonal matrix T.
• M — is INTEGER. The actual number of eigenvalues found. 0 <= M <= N. (See also the description of INFO=2,3.) 👑
• NSPLIT — is INTEGER. The number of diagonal blocks in the matrix T. 1 <= NSPLIT <= N.
• 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 — 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 — 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 — is DOUBLE PRECISION array, dimension (4*N). 🫶
• IWORK — is INTEGER array, dimension (3*N). 🫶
• 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.

ok the PARAMETER blanks above are literally blank in the source, i'm not gaslighting you, that's just how it be 🙈

🔥 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. 🌈

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💖 Author 💖

Generated automatically by Doxygen for LAPACK from the source code. (yes the robots wrote this one, we're just vibing on top 🫶)

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and that's the whole page bestie!! 💅 LAPACK, Version 3.12.0, and the date stamp on this bad boy is the path up top — go forth and bisect your eigenvalues, you absolute icon 🎀✨💖