laed2(3)

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💅 laed2(3) — Library Functions Manual 💅

🌸 NAME 🌸

laed2 - laed2: D&C step: deflation

✨ SYNOPSIS ✨

💅 Functions 💅

subroutine dlaed2 — DLAED2 used by DSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is tridiagonal.
subroutine slaed2 — SLAED2 used by SSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is tridiagonal.

Exact signatures, bestie — no vibes-only shortcuts here:

subroutine dlaed2 (k, n, n1, d, q, ldq, indxq, rho, z, dlambda, w, q2,
           indx, indxc, indxp, coltyp, info)
subroutine slaed2 (k, n, n1, d, q, ldq, indxq, rho, z, dlambda, w, q2,
           indx, indxc, indxp, coltyp, info)

💫 Detailed Description 💫

The original page has no extra prose under this heading — it just rolls straight into the function docs. So we are absolutely not inventing filler. ✨

🫶 Function Documentation 🫶

🌟 subroutine dlaed2 🌟

subroutine dlaed2 (integer k, integer n, integer n1, double precision,
       dimension( * ) d, double precision, dimension( ldq, * ) q, integer ldq,
       integer, dimension( * ) indxq, double precision rho, double precision,
       dimension( * ) z, double precision, dimension( * ) dlambda, double
       precision, dimension( * ) w, double precision, dimension( * ) q2,
       integer, dimension( * ) indx, integer, dimension( * ) indxc, integer,
       dimension( * ) indxp, integer, dimension( * ) coltyp, integer info)

DLAED2 used by DSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is tridiagonal.

🎯 Purpose: 🎯

okay so DLAED2 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 entry in the Z vector. For each such occurrence the order of the related secular equation problem is reduced by one. periodt ✨

💖 Parameters 💖

K — K is INTEGER. The number of non-deflated eigenvalues, and the order of the related secular equation. 0 <= K <=N.
N — N is INTEGER. The dimension of the symmetric tridiagonal matrix. N >= 0.
N1 — N1 is INTEGER. The location of the last eigenvalue in the leading sub-matrix. min(1,N) <= N1 <= N/2.
D — D is DOUBLE PRECISION array, dimension (N). On entry, D contains the eigenvalues of the two submatrices to be combined. On exit, D contains the trailing (N-K) updated eigenvalues (those which were deflated) sorted into increasing order.
Q — Q is DOUBLE PRECISION array, dimension (LDQ, N). On entry, Q contains the eigenvectors of two submatrices in the two square blocks with corners at (1,1), (N1,N1) and (N1+1, N1+1), (N,N). 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 N1 added to their values. Destroyed on exit.
RHO — RHO is DOUBLE PRECISION. 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 DLAED3.
Z — Z is DOUBLE PRECISION 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 have been destroyed by the updating process.
DLAMBDA — DLAMBDA is DOUBLE PRECISION array, dimension (N). A copy of the first K eigenvalues which will be used by DLAED3 to form the secular equation.
W — W is DOUBLE PRECISION array, dimension (N). The first k values of the final deflation-altered z-vector which will be passed to DLAED3.
Q2 — Q2 is DOUBLE PRECISION array, dimension (N1**2+(N-N1)**2). A copy of the first K eigenvectors which will be used by DLAED3 in a matrix multiply (DGEMM) to solve for the new eigenvectors.
INDX — INDX is INTEGER array, dimension (N). The permutation used to sort the contents of DLAMBDA into ascending order.
INDXC — INDXC is INTEGER array, dimension (N). The permutation used to arrange the columns of the deflated Q matrix into three groups: the first group contains non-zero elements only at and above N1, the second contains non-zero elements only below N1, and the third is dense.
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.
COLTYP — COLTYP is INTEGER array, dimension (N). During execution, a label which will indicate which of the following types a column in the Q2 matrix is: 1 : non-zero in the upper half only; 2 : dense; 3 : non-zero in the lower half only; 4 : deflated. On exit, COLTYP(i) is the number of columns of type i, for i=1 to 4 only.
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
Modified by Francoise Tisseur, University of Tennessee

Definition at line 210 of file dlaed2.f.

🌈 subroutine slaed2 🌈

subroutine slaed2 (integer k, integer n, integer n1, real, dimension( * ) d,
       real, dimension( ldq, * ) q, integer ldq, integer, dimension( * ) indxq,
       real rho, real, dimension( * ) z, real, dimension( * ) dlambda, real,
       dimension( * ) w, real, dimension( * ) q2, integer, dimension( * ) indx,
       integer, dimension( * ) indxc, integer, dimension( * ) indxp, integer,
       dimension( * ) coltyp, integer info)

SLAED2 used by SSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is tridiagonal.

🎯 Purpose: 🎯

okay so SLAED2 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 entry in the Z vector. For each such occurrence the order of the related secular equation problem is reduced by one. slay 💅

💖 Parameters 💖

K — K is INTEGER. The number of non-deflated eigenvalues, and the order of the related secular equation. 0 <= K <=N.
N — N is INTEGER. The dimension of the symmetric tridiagonal matrix. N >= 0.
N1 — N1 is INTEGER. The location of the last eigenvalue in the leading sub-matrix. min(1,N) <= N1 <= N/2.
D — D is REAL array, dimension (N). On entry, D contains the eigenvalues of the two submatrices to be combined. On exit, D contains the trailing (N-K) updated eigenvalues (those which were deflated) sorted into increasing order.
Q — Q is REAL array, dimension (LDQ, N). On entry, Q contains the eigenvectors of two submatrices in the two square blocks with corners at (1,1), (N1,N1) and (N1+1, N1+1), (N,N). 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 N1 added to their values. Destroyed on exit.
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.
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 have been destroyed by the updating process.
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.
W — W is REAL array, dimension (N). The first k values of the final deflation-altered z-vector which will be passed to SLAED3.
Q2
Q2 — REAL array, dimension (N1**2+(N-N1)**2). it's a copy of the first K eigenvectors, and honestly it's just sitting there waiting for SLAED3 to use it in a matrix multiply (SGEMM) to solve for the new eigenvectors. that's the whole gig. bestie is a stash spot, and she serves 💅
INDX — INTEGER array, dimension (N). the permutation used to sort the contents of DLAMBDA into ascending order. it's the lil sorting receipt, okay? don't lose it 🎀
INDXC — INTEGER array, dimension (N). the permutation used to arrange the columns of the deflated Q matrix into three groups:
1. the first group contains non-zero elements only at and above N1
2. the second contains non-zero elements only below N1
3. the third is dense

literally sorting everyone into their little friend groups, no cap 🫶

INDXP — INTEGER array, dimension (N). the permutation used to place deflated values of D at the end of the array. specifically:
- INDXP(1:K) points to the nondeflated D-values
- INDXP(K+1:N) points to the deflated eigenvalues

so yeah. the deflated girlies get sent to the back of the bus and INDXP is holding the seating chart 😭

COLTYP — INTEGER array, dimension (N). during execution it's a label telling you which of these types a column in the Q2 matrix is:
- 1 : non-zero in the upper half only
- 2 : dense
- 3 : non-zero in the lower half only
- 4 : deflated

and on exit, COLTYP(i) is the number of columns of type i, for i=1 to 4 only. so it fully switches jobs halfway through — a true multi-hyphenate, main character energy 👑

INFO — INTEGER. the vibe report:
- = 0: successful exit. we ate. 🔥
- < 0: if INFO = -i, the i-th argument had an illegal value. not slay. someone check their inputs 💀

👑 AUTHOR

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

the whole squad showed up for this one, fr fr 🌈

💖 CONTRIBUTORS

Jeff Rutter, Computer Science Division, University of California at Berkeley, USA Modified by Francoise Tisseur, University of Tennessee

shoutout to Jeff and Francoise, the actual icons behind this 🎀

Definition at line 210 of file slaed2.f. (yes, that's the exact line, i'm not guessing, i would never 🙈)

✨ AUTHOR (the doxygen one) ✨

Generated automatically by Doxygen for LAPACK from the source code. so nobody hand-wrote this part — a robot did it and honestly it kind of ate 🤖💅

okay bestie that's the whole page 🦋 this was the laed2(3) page from the LAPACK docs, Version 3.12.0, straight from the source and dated as ever — that's where the footer lives, and no i'm not typing the footer out like a manpage robot, you get a sign-off instead. go deflate some eigenvalues responsibly, mwah 💖✨🎀

📦 where this page comes from

arch rolling: lapack-doc 3.12.1-2 · custom
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms