SLAED8(3)

✨ SLAED8 ✨

okay so first things first: this page is from the Library Functions Manual, and the star of the show lives at /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/slaed8.f (the header up top shortens that whole journey to /home/abui...RC/slaed8.f 💅). It's a Fortran source file, and it's so slay.

---

🦋 SYNOPSIS 🦋

Functions/Subroutines

subroutine SLAED8 (icompq, k, n, qsiz, d, q, ldq, indxq, rho, cutpnt, z,
    dlambda, q2, ldq2, w, perm, givptr, givcol, givnum, indxp, indx,
    info)

SLAED8 is the bestie that SSTEDC calls. It merges eigenvalues and deflates the secular equation, and it's the one you use when the original matrix is dense. periodt.

---

🌸 Function/Subroutine Documentation 🌸

🎀 subroutine SLAED8 🎀

subroutine SLAED8 (integer icompq, integer k, integer n, integer qsiz, real,
    dimension( * ) d, real, dimension( ldq, * ) q, integer ldq, integer,
    dimension( * ) indxq, real rho, integer cutpnt, real, dimension( * ) z,
    real, dimension( * ) dlambda, real, dimension( ldq2, * ) q2, integer
    ldq2, real, dimension( * ) w, integer, dimension( * ) perm, integer
    givptr, integer, dimension( 2, * ) givcol, real, dimension( 2, * )
    givnum, integer, dimension( * ) indxp, integer, dimension( * ) indx,
    integer info)

SLAED8 used by SSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense. (yes she's that girl, she's literally the dense-matrix specialist 🫶)

---

🎯 Purpose 🎯

SLAED8 merges the two sets of eigenvalues together into a single sorted set. Then it tries to deflate the size of the problem. There are two ways in which deflation can occur: when two or more eigenvalues are close together or if there is a tiny element in the Z vector. For each such occurrence the order of the related secular equation problem is reduced by one.

so basically: merge the two eigenvalue sets, sort them, and then shrink the problem wherever it can. two ways to deflate — eigenvalues being cozy-close together, or a lil baby element showing up in the Z vector. every time that happens, the order of the secular equation problem drops by one. lowkey iconic 💅

---

🌈 Parameters 🌈

• ICOMPQ — ICOMPQ is INTEGER
  • = 0: Compute eigenvalues only.
  • = 1: Compute eigenvectors of original dense symmetric matrix also. On entry, Q contains the orthogonal matrix used to reduce the original matrix to tridiagonal form.
• K — K is INTEGER
  • The number of non-deflated eigenvalues, and the order of the related secular equation. the survivors, basically 👑
• N — N is INTEGER
  • The dimension of the symmetric tridiagonal matrix. N >= 0.
• QSIZ — QSIZ is INTEGER
  • The dimension of the orthogonal matrix used to reduce the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.
• D — D is REAL array, dimension (N)
  • On entry, the eigenvalues of the two submatrices to be combined. On exit, the trailing (N-K) updated eigenvalues (those which were deflated) sorted into increasing order. she comes in as two squads and leaves sorted ✨
• Q — Q is REAL array, dimension (LDQ,N)
  • If ICOMPQ = 0, Q is not referenced. Otherwise, on entry, Q contains the eigenvectors of the partially solved system which has been previously updated in matrix multiplies with other partially solved eigensystems. On exit, Q contains the trailing (N-K) updated eigenvectors (those which were deflated) in its last N-K columns.
• LDQ — LDQ is INTEGER
  • The leading dimension of the array Q. LDQ >= max(1,N).
• INDXQ — INDXQ is INTEGER array, dimension (N)
  • The permutation which separately sorts the two sub-problems in D into ascending order. Note that elements in the second half of this permutation must first have CUTPNT added to their values in order to be accurate. (yes, added — don't skip that step bestie, or the numbers lie 😭)
• RHO — RHO is REAL
  • On entry, the off-diagonal element associated with the rank-1 cut which originally split the two submatrices which are now being recombined. On exit, RHO has been modified to the value required by SLAED3. she gets a whole little glow-up on the way out 💖
• CUTPNT — CUTPNT is INTEGER
  • The location of the last eigenvalue in the leading sub-matrix. min(1,N) <= CUTPNT <= N.
• Z — Z is REAL array, dimension (N)
  • On entry, Z contains the updating vector (the last row of the first sub-eigenvector matrix and the first row of the second sub-eigenvector matrix). On exit, the contents of Z are destroyed by the updating process. 💀 like, fully gone. do not come looking for her afterwards.
• DLAMBDA — DLAMBDA is REAL array, dimension (N)
  • A copy of the first K eigenvalues which will be used by SLAED3 to form the secular equation.
• Q2 — Q2 is REAL array, dimension (LDQ2,N)
  • If ICOMPQ = 0, Q2 is not referenced. Otherwise, a copy of the first K eigenvectors which will be used by SLAED7 in a matrix multiply (SGEMM) to update the new eigenvectors.
• LDQ2 — LDQ2 is INTEGER
  • The leading dimension of the array Q2. LDQ2 >= max(1,N).
• W — W is REAL array, dimension (N)
  • The first k values of the final deflation-altered z-vector and will be passed to SLAED3.
• PERM — PERM is INTEGER array, dimension (N)
  • The permutations (from deflation and sorting) to be applied to each eigenblock.
• GIVPTR — GIVPTR is INTEGER
  • The number of Givens rotations which took place in this subproblem. count 'em 🔥
• GIVCOL — GIVCOL is INTEGER array, dimension (2, N)
  • Each pair of numbers indicates a pair of columns to take place in a Givens rotation.
• GIVNUM — GIVNUM is REAL array, dimension (2, N)
  • Each number indicates the S value to be used in the corresponding Givens rotation.
• INDXP — INDXP is INTEGER array, dimension (N)
  • The permutation used to place deflated values of D at the end of the array. INDXP(1:K) points to the nondeflated D-values and INDXP(K+1:N) points to the deflated eigenvalues.
• INDX — INDX is INTEGER array, dimension (N)
  • The permutation used to sort the contents of D into ascending order.
• INFO — INFO is INTEGER
  • = 0: successful exit. 🎉
  • < 0: if INFO = -i, the i-th argument had an illegal value. 🙈

---

👑 Author 👑

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

🧚 Contributors: 🧚

Jeff Rutter, Computer Science Division, University of California at Berkeley, USA — ty for your service, king 🫶

Definition at line 240 of file slaed8.f.

---

⭐ Author ⭐

Generated automatically by Doxygen for LAPACK from the source code. (no human wrote this bit, the robot did 🤖✨)

---

and that's the whole page bestie — no cap, every single parameter accounted for 💅

signing off from the Library Functions Manual: this was slaed8.f(3) from LAPACK Version 3.12.0, path vibes /home/abui...RC/slaed8.f(3). drink some water, check your LDQ, and slay the day 💜🎀