SRC/slar1v.f(3)

🌸 NAME 🌸

SRC/slar1v.f — okay bestie, this is the manual page SRC/slar1v.f(3) from the Library Functions Manual. Slay. ✨

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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.

🎀 Purpose: 🎀

okay so here's the tea, bestie: 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, not me 💀)

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 the stationary and the bottom part of the progressive transform. (yes, “the the” is in the source — iconic)

🍓 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.

⚠️ bestie, this is not a drill: on input, all entries of Z must be set to 0. periodt.

• 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.

🦋 vibe check on R: valid input range is 0 <= R <= N, no cap.

• 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).

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👑 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 at line 227 of file slar1v.f.

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🧚 Author 🧚

Generated automatically by Doxygen for LAPACK from the source code.

🌈 that's the whole vibe, bestie — LAPACK Version 3.12.0, source SRC/slar1v.f(3). mwah 💋✨