ZLAR1V(3)

👑 ZLAR1V(3) 👑

okay so bestie, welcome to the Library Functions Manual page for ZLAR1V — yes, that’s the whole vibe: a Fortran subroutine that computes a scaled column of an inverse and is lowkey iconic about it. 💅

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🌸 NAME 🌸

The NAME section is literally this path, no cap:

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zlar1v.f

That’s the file, bestie. That’s the main character. 🎀

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✨ SYNOPSIS ✨

Functions/Subroutines

subroutine ZLAR1V (n, b1, bn, lambda, d, l, ld, lld, pivmin, gaptol, z,
    wantnc, negcnt, ztz, mingma, r, isuppz, nrminv, resid, rqcorr, work)

ZLAR1V 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 💅

Here’s the full signature, character-for-character, because we do not play about exact types:

subroutine ZLAR1V (integer n, integer b1, integer bn, double precision
    lambda, double precision, dimension( * ) d, double precision, dimension(
    * ) l, double precision, dimension( * ) ld, double precision, dimension(
    * ) lld, double precision pivmin, double precision gaptol, complex*16,
    dimension( * ) z, logical wantnc, integer negcnt, double precision ztz,
    double precision mingma, integer r, integer, dimension( * ) isuppz,
    double precision nrminv, double precision resid, double precision
    rqcorr, double precision, dimension( * ) work)

ZLAR1V 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: ZLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows B1 through BN of the tridiagonal matrix L D L**T - sigma I. 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 — INTEGER. The order of the matrix L D L**T.
• B1 — INTEGER. First index of the submatrix of L D L**T.
• BN — INTEGER. Last index of the submatrix of L D L**T.
• LAMBDA — DOUBLE PRECISION. The shift. In order to compute an accurate eigenvector, LAMBDA should be a good approximation to an eigenvalue of L D L**T.
• L — DOUBLE PRECISION array, dimension (N-1). The (n-1) subdiagonal elements of the unit bidiagonal matrix L, in elements 1 to N-1.
• D — DOUBLE PRECISION array, dimension (N). The n diagonal elements of the diagonal matrix D.
• LD — DOUBLE PRECISION array, dimension (N-1). The n-1 elements L(i)*D(i).
• LLD — DOUBLE PRECISION array, dimension (N-1). The n-1 elements L(i)*L(i)*D(i).
• PIVMIN — DOUBLE PRECISION. The minimum pivot in the Sturm sequence.
• GAPTOL — DOUBLE PRECISION. Tolerance that indicates when eigenvector entries are negligible w.r.t. their contribution to the residual.
• Z — COMPLEX*16 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 — LOGICAL. Specifies whether NEGCNT has to be computed.
• NEGCNT — 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 — DOUBLE PRECISION. The square of the 2-norm of Z.
• MINGMA — DOUBLE PRECISION. The reciprocal of the largest (in magnitude) diagonal element of the inverse of L D L**T - sigma I.
• R — 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 — 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 — DOUBLE PRECISION. NRMINV = 1/SQRT( ZTZ )
• RESID — DOUBLE PRECISION. The residual of the FP vector. RESID = ABS( MINGMA )/SQRT( ZTZ )
• RQCORR — DOUBLE PRECISION. The Rayleigh Quotient correction to LAMBDA. RQCORR = MINGMA*TMP
• WORK — DOUBLE PRECISION 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 at line 227 of file zlar1v.f.

Author

Generated automatically by Doxygen for LAPACK from the source code.

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💖 cute little sign-off, bestie: LAPACK Version 3.12.0, from /home/abui...RC/zlar1v.f(3) — that’s the footer, and yes, it ate. 🫶✨