DLAED4(3)

🌸 NAME 🌸

ok so bestie, this page is from the Library Functions Manual, and the thing it's documenting is literally just this file:

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/dlaed4.f

that's it, that's the name section 💅 (yes, it's a path, the manpage generator was being extra)

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✨ SYNOPSIS ✨

Functions/Subroutines 🎀

• subroutine DLAED4 (n, i, d, z, delta, rho, dlam, info) — this is the icon itself. DLAED4 is used by DSTEDC, and its whole brand is finding a single root of the secular equation 🦋

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💖 Function/Subroutine Documentation 💖

subroutine DLAED4 (integer n, integer i, double precision, dimension( * ) d,
    double precision, dimension( * ) z, double precision, dimension( * )
    delta, double precision rho, double precision dlam, integer info)

DLAED4 used by DSTEDC. Finds a single root of the secular equation. yup, that's the whole vibe ✨

🌈 Purpose:

ok so here's the tea, straight from the source comments:

!>
!> This subroutine computes the I-th updated eigenvalue of a symmetric
!> rank-one modification to a diagonal matrix whose elements are
!> given in the array d, and that
!>
!>            D(i) < D(j)  for  i < j
!>
!> and that RHO > 0.  This is arranged by the calling routine, and is
!> no loss in generality.  The rank-one modified system is thus
!>
!>            diag( D )  +  RHO * Z * Z_transpose.
!>
!> where we assume the Euclidean norm of Z is 1.
!>
!> The method consists of approximating the rational functions in the
!> secular equation by simpler interpolating rational functions.
!>

so basically: you hand it a diagonal matrix, someone yeets a rank-one update at it (RHO * Z * Z_transpose, no cap), and DLAED4 finds you the I-th updated eigenvalue. the trick? approximating those rational functions in the secular equation with simpler interpolating rational functions. it's giving "math but make it efficient" 👑

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🧚 Parameters

• N — N is INTEGER. the length of all the arrays. every single one. it's the main character of lengths 💜
• I — I is INTEGER. the index of the eigenvalue you're computing, and the range is strict: 1 <= I <= N. don't get cute with it 🙈
• D — D is DOUBLE PRECISION array, dimension (N). the original eigenvalues. the routine assumes they're already in order, meaning D(I) < D(J) for I < J. if you didn't sort them, that's on you bestie 😭
• Z — Z is DOUBLE PRECISION array, dimension (N). the components of the updating vector. slay ✨
• DELTA — DELTA is DOUBLE PRECISION array, dimension (N). okay this one has lore:
  • if N > 2, then DELTA contains (D(j) - lambda_I) in its j-th component
  • if N = 1, then DELTA(1) = 1
  • if N = 2, see DLAED5 for detail (not me, go read that one)
  • the vector DELTA holds the info you need to construct the eigenvectors with DLAED3 and DLAED9 🫶
• RHO — RHO is DOUBLE PRECISION. the scalar in the symmetric updating formula. tiny but iconic 💅
• DLAM — DLAM is DOUBLE PRECISION. the computed lambda_I, aka the I-th updated eigenvalue. the whole point, honestly ⭐
• INFO — INFO is INTEGER:
  • = 0: successful exit 🎉
  • > 0: if INFO = 1, the updating process failed 💀

lowkey important: INFO = 1 means failure, not a vibe. check it, fr fr.

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👑 Internal Parameters:

these are the internals, iykyk:

!>  Logical variable ORGATI (origin-at-i?) is used for distinguishing
!>  whether D(i) or D(i+1) is treated as the origin.
!>
!>            ORGATI = .true.    origin at i
!>            ORGATI = .false.   origin at i+1
!>
!>   Logical variable SWTCH3 (switch-for-3-poles?) is for noting
!>   if we are working with THREE poles!
!>
!>   MAXIT is the maximum number of iterations allowed for each
!>   eigenvalue.
!>

so ORGATI is the "origin-at-i?" flag deciding whether D(i) or D(i+1) is the origin. SWTCH3 is the "switch-for-3-poles?" flag and yes it's for noting if we're working with THREE poles (three!! poles!!). and MAXIT is the maximum number of iterations allowed for each eigenvalue 🌸

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🦋 Author

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

🍓 Contributors:

• Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA

🔥 Definition

Definition at line 144 of file dlaed4.f. periodt 💅

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🎀 Author (the meta one)

Generated automatically by Doxygen for LAPACK from the source code. so nobody hand-wrote this part, a robot did it for us 🙈

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and that's the whole page bestie!! 💖 this was LAPACK, Version 3.12.0, dated /home/abui...RC/dlaed4.f(3) — go forth and find those roots, you're gonna slay 🫶✨🌈