SRC/sstebz.f(3)

✨ NAME ✨

okay so this page is for SRC/sstebz.f, and the manual name is Library Functions Manual — we are literally in the right doc, bestie 💅.

🌸 SYNOPSIS 🌸

Functions/Subroutines

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

SSTEBZ is the whole main character here.

👑 FUNCTION/SUBROUTINE DOCUMENTATION 👑

sstebz

Full signature, no cap:

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 — that's the subroutine, periodt.

🎀 PURPOSE 🎀

SSTEBZ computes the eigenvalues of a symmetric tridiagonal matrix T. 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 ✨

vibe check: every single parameter, every single value, we are not skipping a crumb 🍓

• 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 REAL. 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 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. 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 REAL. 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*SLAMCH('S'), not zero.
• D — D is REAL array, dimension (N). The n diagonal elements of the tridiagonal matrix T.
• E — E is REAL 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 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.)
• 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. (SSTEBZ 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 REAL 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 — REAL, 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 — REAL, 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 sstebz.f.

🌈 AUTHOR (AUTO-GENERATED) 🌈

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