SLAR1V(3)

✨ NAME ✨

okay bestie, welcome to the Library Functions Manual moment for header path /home/abui...RC/slar1v.f(3) — this whole page is about this file:

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/slar1v.f

it's giving very specific linear algebra energy and I am lowkey obsessed 🦋

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🦋 SYNOPSIS 🦋

🌸 Functions/Subroutines 🌸

   subroutine SLAR1V (n, b1, bn, lambda, d, l, ld, lld, pivmin, gaptol, z,
       wantnc, negcnt, ztz, mingma, r, isuppz, nrminv, resid, rqcorr, work)
       SLAR1V computes the (scaled) r-th column of the inverse of the
       submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI.

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💅 Function/Subroutine Documentation 💅

   subroutine SLAR1V (integer n, integer b1, integer bn, real lambda, real,
       dimension( * ) d, real, dimension( * ) l, real, dimension( * ) ld, real,
       dimension( * ) lld, real pivmin, real gaptol, real, dimension( * ) z,
       logical wantnc, integer negcnt, real ztz, real mingma, integer r,
       integer, dimension( * ) isuppz, real nrminv, real resid, real rqcorr,
       real, dimension( * ) work)

SLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI. iconic behavior, honestly 👑

🌸 Purpose 🌸

okay so here's the vibe: SLAR1V computes the (scaled) r-th column of the inverse of the sumbmatrix in rows B1 through BN of the tridiagonal matrix L D L**T - sigma I. yes, the source says “sumbmatrix” — I'm keeping it, it's canon 💀

When sigma is close to an eigenvalue, the computed vector is an accurate eigenvector. Usually, r corresponds to the index where the eigenvector is largest in magnitude.

The following steps accomplish this computation:

• (a) Stationary qd transform, L D L**T - sigma I = L(+) D(+) L(+)**T,
• (b) Progressive qd transform, L D L**T - sigma I = U(-) D(-) U(-)**T,
• (c) Computation of the diagonal elements of the inverse of L D L**T - sigma I by combining the above transforms, and choosing r as the index where the diagonal of the inverse is (one of the) largest in magnitude.
• (d) Computation of the (scaled) r-th column of the inverse using the twisted factorization obtained by combining the top part of the stationary and the bottom part of the progressive transform.

🎀 Parameters 🎀

⭐ N ⭐

N is INTEGER. The order of the matrix L D L**T.

⭐ B1 ⭐

B1 is INTEGER. First index of the submatrix of L D L**T.

⭐ BN ⭐

BN is INTEGER. Last index of the submatrix of L D L**T.

⭐ LAMBDA ⭐

LAMBDA is REAL. The shift. In order to compute an accurate eigenvector, LAMBDA should be a good approximation to an eigenvalue of L D L**T.

⭐ L ⭐

L is REAL array, dimension (N-1). The (n-1) subdiagonal elements of the unit bidiagonal matrix L, in elements 1 to N-1.

⭐ D ⭐

D is REAL array, dimension (N). The n diagonal elements of the diagonal matrix D.

⭐ LD ⭐

LD is REAL array, dimension (N-1). The n-1 elements L(i)*D(i).

⭐ LLD ⭐

LLD is REAL array, dimension (N-1). The n-1 elements L(i)*L(i)*D(i).

⭐ PIVMIN ⭐

PIVMIN is REAL. The minimum pivot in the Sturm sequence.

⭐ GAPTOL ⭐

GAPTOL is REAL. Tolerance that indicates when eigenvector entries are negligible w.r.t. their contribution to the residual.

⭐ Z ⭐

Z is REAL array, dimension (N). On input, all entries of Z must be set to 0. On output, Z contains the (scaled) r-th column of the inverse. The scaling is such that Z(R) equals 1.

⭐ WANTNC ⭐

WANTNC is LOGICAL. Specifies whether NEGCNT has to be computed.

⭐ NEGCNT ⭐

NEGCNT is INTEGER. If WANTNC is .TRUE. then NEGCNT = the number of pivots < pivmin in the matrix factorization L D L**T, and NEGCNT = -1 otherwise.

⭐ ZTZ ⭐

ZTZ is REAL. The square of the 2-norm of Z.

⭐ MINGMA ⭐

MINGMA is REAL. The reciprocal of the largest (in magnitude) diagonal element of the inverse of L D L**T - sigma I.

⭐ R ⭐

R is INTEGER. The twist index for the twisted factorization used to compute Z.

On input, 0 <= R <= N. If R is input as 0, R is set to the index where (L D L**T - sigma I)^{-1} is largest in magnitude. If 1 <= R <= N, R is unchanged.

On output, R contains the twist index used to compute Z. Ideally, R designates the position of the maximum entry in the eigenvector.

⭐ ISUPPZ ⭐

ISUPPZ is INTEGER array, dimension (2). The support of the vector in Z, i.e., the vector Z is nonzero only in elements ISUPPZ(1) through ISUPPZ( 2 ).

⭐ NRMINV ⭐

NRMINV is REAL. NRMINV = 1/SQRT( ZTZ )

⭐ RESID ⭐

RESID is REAL. The residual of the FP vector. RESID = ABS( MINGMA )/SQRT( ZTZ )

⭐ RQCORR ⭐

RQCORR is REAL. The Rayleigh Quotient correction to LAMBDA. RQCORR = MINGMA*TMP

⭐ WORK ⭐

WORK is REAL array, dimension (4*N).

👑 Author 👑

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

🫶 Contributors: 🫶

• Beresford Parlett, University of California, Berkeley, USA
• Jim Demmel, University of California, Berkeley, USA
• Inderjit Dhillon, University of Texas, Austin, USA
• Osni Marques, LBNL/NERSC, USA
• Christof Voemel, University of California, Berkeley, USA

🧚 Definition 🧚

Definition at line 227 of file slar1v.f.

👑 Author 👑

Generated automatically by Doxygen for LAPACK from the source code.

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okay bestie, that's the whole page — LAPACK Version 3.12.0, file slar1v.f(3), from the Library Functions Manual. no date listed on this footer, so I'm not inventing one 💅 now go compute that (scaled) r-th column with main-character energy 💖✨