lasd4(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

💖 NAME 💖

lasd4 — lasd4: D&C step: secular equation nonlinear solver ✨

This is the lasd4(3) page from the LAPACK Library Functions Manual, bestie 🫶. It's the divide-and-conquer step's secular-equation nonlinear solver, and there are two flavors: double precision (dlasd4) and single precision (slasd4) 💅.

🦋 SYNOPSIS 🦋

Functions

  • subroutine dlasd4 (n, i, d, z, delta, rho, sigma, work, info) — DLASD4 computes the square root of the i-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix. Used by dbdsdc.
  • subroutine slasd4 (n, i, d, z, delta, rho, sigma, work, info) — SLASD4 computes the square root of the i-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix. Used by sbdsdc.

✨ DETAILED DESCRIPTION ✨

Okay so this is the detailed-description zone 🎀. The source page rolls straight into the function docs from here — no extra side quests, just the two subroutines below.

🎀 FUNCTION DOCUMENTATION 🎀

💜 dlasd4 💜

Full signature, exactly as it appears:

subroutine dlasd4 (integer n, integer i, double precision, dimension( * ) d, double precision, dimension( * ) z, double precision, dimension( * ) delta, double precision rho, double precision sigma, double precision, dimension( * ) work, integer info)

DLASD4 computes the square root of the i-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix. Used by dbdsdc.

🌸 Purpose 🌸

Bestie, what this subroutine does is compute the square root of the I-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix. The diagonal entries are the squares of the matching entries in the array d, and the setup guarantees

0 <= D(i) < D(j)  for  i < j

plus RHO > 0. The calling routine arranges that, and it's no loss in generality. So the rank-one modified system is

diag( D ) * diag( D ) +  RHO * Z * Z_transpose.

with the Euclidean norm of Z assumed to be 1 🦋.

The method? It approximates the rational functions in the secular equation with simpler interpolating rational functions. Iconic behavior, honestly 💅.

🧚 Parameters 🧚

  • N — N is INTEGER. The length of all arrays.
  • I — I is INTEGER. The index of the eigenvalue to be computed. 1 <= I <= N.
  • D — D is DOUBLE PRECISION array, dimension ( N ). The original eigenvalues. It is assumed that they are in order, 0 <= D(I) < D(J) for I < J.
  • Z — Z is DOUBLE PRECISION array, dimension ( N ). The components of the updating vector.
  • DELTA — DELTA is DOUBLE PRECISION array, dimension ( N ). If N .ne. 1, DELTA contains (D(j) - sigma_I) in its j-th component. If N = 1, then DELTA(1) = 1. The vector DELTA contains the information necessary to construct the (singular) eigenvectors.
  • RHO — RHO is DOUBLE PRECISION. The scalar in the symmetric updating formula.
  • SIGMA — SIGMA is DOUBLE PRECISION. The computed sigma_I, the I-th updated eigenvalue.
  • WORK — WORK is DOUBLE PRECISION array, dimension ( N ). If N .ne. 1, WORK contains (D(j) + sigma_I) in its j-th component. If N = 1, then WORK( 1 ) = 1.
  • INFO — INFO is INTEGER.
    • = 0: successful exit
    • > 0: if INFO = 1, the updating process failed.

🦋 Internal Parameters 🦋

  • 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.

👑 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 at line 152 of file dlasd4.f.

💜 slasd4 💜

Full signature, exactly as it appears:

subroutine slasd4 (integer n, integer i, real, dimension( * ) d, real, dimension( * ) z, real, dimension( * ) delta, real rho, real sigma, real, dimension( * ) work, integer info)

SLASD4 computes the square root of the i-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix. Used by sbdsdc.

🌸 Purpose 🌸

Bestie, this one computes the square root of the I-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix. The diagonal entries are the squares of the matching entries in the array d, and the setup guarantees

0 <= D(i) < D(j)  for  i < j

plus RHO > 0. The calling routine arranges that, and it's no loss in generality. So the rank-one modified system is

diag( D ) * diag( D ) +  RHO * Z * Z_transpose.

with the Euclidean norm of Z assumed to be 1 🦋.

The method? It approximates the rational functions in the secular equation with simpler interpolating rational functions. Iconic behavior, honestly 💅.

🧚 Parameters 🧚

  • N — N is INTEGER. The length of all arrays.
  • I — I is INTEGER. The index of the eigenvalue to be computed. 1 <= I <= N.
  • D — D is REAL array, dimension ( N ). The original eigenvalues. It is assumed that they are in order, 0 <= D(I) < D(J) for I < J.
  • Z — Z is REAL array, dimension ( N ). The components of the updating vector.
  • DELTA — DELTA is REAL array, dimension ( N ). If N .ne. 1, DELTA contains (D(j) - sigma_I) in its j-th component. If N = 1, then DELTA(1) = 1. The vector DELTA contains the information necessary to construct the (singular) eigenvectors.
  • RHO — RHO is REAL. The scalar in the symmetric updating formula.
  • SIGMA — SIGMA is REAL. The computed sigma_I, the I-th updated eigenvalue.
  • WORK — WORK is REAL array, dimension ( N ). If N .ne. 1, WORK contains (D(j) + sigma_I) in its j-th component. If N = 1, then WORK( 1 ) = 1.
  • INFO — INFO is INTEGER.
    • = 0: successful exit
    • > 0: if INFO = 1, the updating process failed.

🦋 Internal Parameters 🦋

  • 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.

👑 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 at line 152 of file slasd4.f.

🌈 AUTHOR 🌈

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