CLAR1V(3)

🎀 NAME 🎀

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/clar1v.f

Okay bestie, this is the CLAR1V(3) page from the Library Functions Manual — yes, the manual-name vibe is folded right in here instead of copied as a boring header line. The file is the path right above, and we're about to get into some very specific LAPACK math 💅✨

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✨ SYNOPSIS ✨

🧩 Functions/Subroutines

       subroutine CLAR1V (n, b1, bn, lambda, d, l, ld, lld, pivmin, gaptol, z,
           wantnc, negcnt, ztz, mingma, r, isuppz, nrminv, resid, rqcorr, work)

CLAR1V 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 CLAR1V (integer n, integer b1, integer bn, real lambda, real,
       dimension( * ) d, real, dimension( * ) l, real, dimension( * ) ld, real,
       dimension( * ) lld, real pivmin, real gaptol, complex, dimension( * ) z,
       logical wantnc, integer negcnt, real ztz, real mingma, integer r,
       integer, dimension( * ) isuppz, real nrminv, real resid, real rqcorr,
       real, dimension( * ) work)

CLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI.

🧚 Purpose

So, CLAR1V 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:

1. Stationary qd transform: L D L**T - sigma I = L(+) D(+) L(+)**T
2. Progressive qd transform: L D L**T - sigma I = U(-) D(-) U(-)**T
3. 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.
4. Computation of the (scaled) r-th column of the inverse using the twisted factorization obtained by combining the top part of the stationary transform 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 COMPLEX 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 at line

Definition at line 227 of file clar1v.f.

🤖 Author (the generated note)

Generated automatically by Doxygen for LAPACK from the source code.

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okay that's the whole CLAR1V(3) ride, bestie — LAPACK Version 3.12.0, footer source /home/abui...RC/clar1v.f(3), and all the math tea included 💅💖