PDSTEBZ(l)

📦 version 1.5, in ubuntu jammy

✨ PDSTEBZ — bestie edition ✨

okay so. PDSTEBZ is a LAPACK routine (from the LAPACK routine manual, the l one, version 1.5, if you were wondering) and it's literally the girlboss of tridiagonal eigenvalue problems. pull up a chair. 🎀

💖 NAME 💖

PDSTEBZ — compute the eigenvalues of a symmetric tridiagonal matrix in parallel ✨ (yes bestie, multiple processes, we're doing this TOGETHER)

🌸 SYNOPSIS 🌸

SUBROUTINE PDSTEBZ( ICTXT, RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL, D,
                    E, M, NSPLIT, W, IBLOCK, ISPLIT, WORK, LWORK, IWORK,
                    LIWORK, INFO )

    CHARACTER       ORDER, RANGE

    INTEGER         ICTXT, IL, INFO, IU, LIWORK, LWORK, M, N, NSPLIT

    DOUBLE          PRECISION ABSTOL, VL, VU

    INTEGER         IBLOCK( * ), ISPLIT( * ), IWORK( * )

    DOUBLE          PRECISION D( * ), E( * ), W( * ), WORK( * )

🦋 PURPOSE 🦋

So what does she actually do. PDSTEBZ computes the eigenvalues of a symmetric tridiagonal matrix in parallel, no cap. You, the user, get to ask for:

  • all the eigenvalues,
  • all eigenvalues in the interval [VL, VU],
  • or the eigenvalues indexed IL through IU.

A static partitioning of work is done at the beginning of PDSTEBZ which results in all processes finding an (almost) equal number of eigenvalues. balanced. equitable. iconic. 💅

⚠️ NOTE, and this one matters: it is assumed that the user is on an IEEE machine. If the user is not on an IEEE machine, set the compile time flag NO_IEEE to 1 (in SLmake.inc). The features of IEEE arithmetic that are needed for the "fast" Sturm Count are: - (a) infinity arithmetic - (b) the sign bit of a single precision floating point number is assumed to be in the 32nd bit position - (c) the sign of negative zero.

See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal Matrix", Report CS41, Computer Science Dept., Stanford University, July 21, 1966. 📚 (yes she's citing her sources, she's academically unwell)

💜 ARGUMENTS 💜

the cast 🎭

  • ICTXT — (global input) INTEGER. The BLACS context handle. it's giving "who am I talking to" energy.
  • RANGE — (global input) CHARACTER. Specifies which eigenvalues are to be found.
    • = 'A': ("All") all eigenvalues will be found.
    • = 'V': ("Value") all eigenvalues in the interval [VL, VU] will be found.
    • = 'I': ("Index") the IL-th through IU-th eigenvalues (of the entire matrix) will be found.
  • ORDER — (global input) CHARACTER. Specifies the order in which the eigenvalues and their block numbers are stored in W and IBLOCK.
    • = 'B': ("By Block") the eigenvalues will be grouped by split-off block (see IBLOCK, ISPLIT) and ordered from smallest to largest within the block.
    • = 'E': ("Entire matrix") the eigenvalues for the entire matrix will be ordered from smallest to largest.
  • N — (global input) INTEGER. The order of the tridiagonal matrix T. N >= 0. 📏
  • VL — (global input) DOUBLE PRECISION. If RANGE='V', the lower bound of the interval to be searched for eigenvalues. Eigenvalues less than VL will not be returned. Not referenced if RANGE='A' or 'I'.
  • VU — (global input) DOUBLE PRECISION. If RANGE='V', the upper bound of the interval to be searched for eigenvalues. Eigenvalues greater than VU will not be returned. VU must be greater than VL. Not referenced if RANGE='A' or 'I'.
  • IL — (global input) INTEGER. If RANGE='I', the index (from smallest to largest) of the smallest eigenvalue to be returned. IL must be at least 1. Not referenced if RANGE='A' or 'V'.
  • IU — (global input) INTEGER. If RANGE='I', the index (from smallest to largest) of the largest eigenvalue to be returned. IU must be at least IL and no greater than N. Not referenced if RANGE='A' or 'V'.
  • ABSTOL — (global input) 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 the underflow threshold DLAMCH('U'), not zero. 💀 (yes, zero is the trap answer)
📌 Note: if eigenvectors are desired later by inverse iteration (PDSTEIN), ABSTOL should be set to 2*PDLAMCH('S').
  • D — (global input) DOUBLE PRECISION array, dimension (N). The n diagonal elements of the tridiagonal matrix T. To avoid overflow, the matrix must be scaled so that its largest entry 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.
  • E — (global input) DOUBLE PRECISION array, dimension (N-1). The (n-1) off-diagonal elements of the tridiagonal matrix T. To avoid overflow, the matrix must be scaled so that its largest entry 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. (same deal as D, they're a matching set 🫶)
  • M — (global output) INTEGER. The actual number of eigenvalues found. 0 <= M <= N. (See also the description of INFO=2.)
  • NSPLIT — (global output) INTEGER. The number of diagonal blocks in the matrix T. 1 <= NSPLIT <= N.
  • W — (global output) DOUBLE PRECISION array, dimension (N). On exit, the first M elements of W contain the eigenvalues on all processes. 🎉
  • IBLOCK — (global output) 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 IBLOCK(i) specifies which block (from 1 to the number of blocks) the eigenvalue W(i) belongs to.
⚠️ NOTE: in the (theoretically impossible) event that bisection does not converge for some or all eigenvalues, INFO is set to 1 and the ones for which it did not are identified by a negative block number.
  • ISPLIT — (global output) 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 — (local workspace) DOUBLE PRECISION array, dimension ( MAX( 5*N, 7 ) )
  • LWORK — (local input) INTEGER. size of array WORK must be >= MAX( 5*N, 7 ). If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA.
  • IWORK — (local workspace) INTEGER array, dimension ( MAX( 4*N, 14 ) )
  • LIWORK — (local input) INTEGER. size of array IWORK must be >= MAX( 4*N, 14, NPROCS ). If LIWORK = -1, then LIWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA.
  • INFO — (global output) INTEGER. the vibe check report 📋
    • = 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 : 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 arithmetic which is less accurate than PDLAMCH says.
      • = 2 : There is a mismatch between the number of eigenvalues output and the number desired.
      • = 3 : RANGE='i', and the Gershgorin interval initially used was incorrect. No eigenvalues were computed. Probable cause: your machine has sloppy floating point arithmetic. Cure: Increase the PARAMETER "FUDGE", recompile, and try again.

🧚 PARAMETERS 🧚

  • RELFAC — DOUBLE PRECISION, default = 2.0. The relative tolerance. An interval [a,b] lies within "relative tolerance" 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 — DOUBLE PRECISION, default = 2.0. 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 the accuracy of the solution. 🍬

and that's the whole page bestie, you're so welcome, go forth and compute eigenvalues in parallel 💖✨🎀

this little love letter was brought to you by LAPACK version 1.5, dated 12 May 1997 — yes she's vintage, she's a classic, she's still slaying. 👑

📦 where this page comes from