ZLAED7(3)

💖 ZLAED7(3) — the eigenproblem glow-up routine 💖

Okay bestie, so this is a page straight out of LAPACK (that's the Linear Algebra PACKage, the blueprint of iconic numeric tea), and the routine lives at SRC/zlaed7.f inside the source tree (zlaed7.f, definition at line 245). LAPACK itself is chilling at Version 3.12.0. It got auto-generated by Doxygen from the source code, so ngl the original is giving ✨machine-made✨ energy — but we're about to make it make sense fr fr. 🫶

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✨ SYNOPSIS ✨

Like, here's the whole main-character lineup of the subroutine — every argument, in order, no skips:

subroutine ZLAED7 (n, cutpnt, qsiz, tlvls, curlvl, curpbm, d, q, ldq,
    rho, indxq, qstore, qptr, prmptr, perm, givptr, givcol, givnum,
    work, rwork, iwork, info)

The full signature, typed out with all the declarations, is literally:

subroutine ZLAED7 (integer n, integer cutpnt, integer qsiz, integer tlvls,
    integer curlvl, integer curpbm, double precision, dimension( * ) d,
    complex*16, dimension( ldq, * ) q, integer ldq, double precision rho,
    integer, dimension( * ) indxq, double precision, dimension( * ) qstore,
    integer, dimension( * ) qptr, integer, dimension( * ) prmptr, integer,
    dimension( * ) perm, integer, dimension( * ) givptr, integer, dimension(
    2, * ) givcol, double precision, dimension( 2, * ) givnum, complex*16,
    dimension( * ) work, double precision, dimension( * ) rwork, integer,
    dimension( * ) iwork, integer info)

So what's the vibe? ZLAED7 is used by ZSTEDC. It computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix, and it gets called when the original matrix is dense. Basically it's the glowing-up eigensystem machine. 💅

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🦋 Function/Subroutine Documentation 🦋

Bestie. Real talk. Here's what this icon actually does. 👑

💜 Purpose 💜

ZLAED7 computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. This routine is used only for the eigenproblem which requires all eigenvalues and optionally eigenvectors of a dense or banded Hermitian matrix that has been reduced to tridiagonal form.

And here's the equation doing its whole main character moment:

T = Q(in) ( D(in) + RHO * Z*Z**H ) Q**H(in) = Q(out) * D(out) * Q**H(out)

where Z = Q**Hu, and u is a vector of length N with ones in the CUTPNT and CUTPNT + 1 th elements and zeros elsewhere. Slay. ✨

The eigenvectors of the original matrix are stored in Q, and the eigenvalues are in D. The algorithm consists of three stages — and yes, each one is iconic:

1. The first stage consists of deflating the size of the problem when there are multiple eigenvalues or if there is a zero in the Z vector. For each such occurrence the dimension of the secular equation problem is reduced by one. This stage is performed by the routine DLAED2.
2. The second stage consists of calculating the updated eigenvalues. This is done by finding the roots of the secular equation via the routine DLAED4 (as called by SLAED3). This routine also calculates the eigenvectors of the current problem.
3. The final stage consists of computing the updated eigenvectors directly using the updated eigenvalues. The eigenvectors for the current problem are multiplied with the eigenvectors from the overall problem.

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🌸 Parameters 🌸

Every single one, bestie, with the specs untouched. 🎀

• N — INTEGER. The dimension of the symmetric tridiagonal matrix. N >= 0.
• CUTPNT — INTEGER. Contains the location of the last eigenvalue in the leading sub-matrix. min(1,N) <= CUTPNT <= N.
• QSIZ — INTEGER. The dimension of the unitary matrix used to reduce the full matrix to tridiagonal form. QSIZ >= N.
• TLVLS — INTEGER. The total number of merging levels in the overall divide and conquer tree.
• CURLVL — INTEGER. The current level in the overall merge routine, 0 <= curlvl <= tlvls.
• CURPBM — INTEGER. The current problem in the current level in the overall merge routine (counting from upper left to lower right).
• D — DOUBLE PRECISION array, dimension (N). On entry, the eigenvalues of the rank-1-perturbed matrix. On exit, the eigenvalues of the repaired matrix.
• Q — COMPLEX*16 array, dimension (LDQ,N). On entry, the eigenvectors of the rank-1-perturbed matrix. On exit, the eigenvectors of the repaired tridiagonal matrix.
• LDQ — INTEGER. The leading dimension of the array Q. LDQ >= max(1,N).
• RHO — DOUBLE PRECISION. Contains the subdiagonal element used to create the rank-1 modification.
• INDXQ — INTEGER array, dimension (N). This contains the permutation which will reintegrate the subproblem just solved back into sorted order, ie. D( INDXQ( I = 1, N ) ) will be in ascending order.
• IWORK — INTEGER array, dimension (4*N).
• RWORK — DOUBLE PRECISION array, dimension (3*N+2*QSIZ*N).
• WORK — COMPLEX*16 array, dimension (QSIZ*N).
• QSTORE — DOUBLE PRECISION array, dimension (N**2+1). Stores eigenvectors of submatrices encountered during divide and conquer, packed together. QPTR points to beginning of the submatrices.
• QPTR — INTEGER array, dimension (N+2). List of indices pointing to beginning of submatrices stored in QSTORE. The submatrices are numbered starting at the bottom left of the divide and conquer tree, from left to right and bottom to top.
• PRMPTR — INTEGER array, dimension (N lg N). Contains a list of pointers which indicate where in PERM a level's permutation is stored. PRMPTR(i+1) - PRMPTR(i) indicates the size of the permutation and also the size of the full, non-deflated problem.
• PERM — INTEGER array, dimension (N lg N). Contains the permutations (from deflation and sorting) to be applied to each eigenblock.
• GIVPTR — INTEGER array, dimension (N lg N). Contains a list of pointers which indicate where in GIVCOL a level's Givens rotations are stored. GIVPTR(i+1) - GIVPTR(i) indicates the number of Givens rotations.
• GIVCOL — INTEGER array, dimension (2, N lg N). Each pair of numbers indicates a pair of columns to take place in a Givens rotation.
• GIVNUM — DOUBLE PRECISION array, dimension (2, N lg N). Each number indicates the S value to be used in the corresponding Givens rotation.
• INFO — INTEGER. Okay so this one's important, bestie:
  • = 0: successful exit. 🎉
  • < 0: if INFO = -i, the i-th argument had an illegal value. 🙈
  • > 0: if INFO = 1, an eigenvalue did not converge. 😭

Lowkey watch out for that INFO > 0 case — INFO = 1 means an eigenvalue did not converge, and that's the "it's giving unbothered, but we are in fact bothered" moment. 💀

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👑 Author 👑

The absolute legends behind this routine:

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

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🌈 Sign-off 🌈

Definition at line 245 of file zlaed7.f, and this page itself was generated automatically by Doxygen for LAPACK from the source code — LAPACK Version 3.12.0, dated /home/abui...RC/zlaed7.f(3) (that truncated path is just the build machine's business, don't stress it 🙈). Periodt. 💅✨