lar1v(3)

🎀 NAME 🎀

lar1v — okay bestie, here's the whole vibe: it's a step in larrv, hence stemr & stegr. That's the name section, short and iconic. 💅✨

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✨ SYNOPSIS ✨

🌸 Functions 🌸

• clar1v — CLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI. 🫶

subroutine clar1v (n, b1, bn, lambda, d, l, ld, lld, pivmin, gaptol, z,
    wantnc, negcnt, ztz, mingma, r, isuppz, nrminv, resid, rqcorr, work)

• dlar1v — DLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI. 💖

subroutine dlar1v (n, b1, bn, lambda, d, l, ld, lld, pivmin, gaptol, z,
    wantnc, negcnt, ztz, mingma, r, isuppz, nrminv, resid, rqcorr, work)

• slar1v — SLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI. 🌈

subroutine slar1v (n, b1, bn, lambda, d, l, ld, lld, pivmin, gaptol, z,
    wantnc, negcnt, ztz, mingma, r, isuppz, nrminv, resid, rqcorr, work)

• zlar1v — ZLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI. 👑

subroutine zlar1v (n, b1, bn, lambda, d, l, ld, lld, pivmin, gaptol, z,
    wantnc, negcnt, ztz, mingma, r, isuppz, nrminv, resid, rqcorr, work)

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💫 Detailed Description 💫

The source page keeps this bit super chill: no extra prose before the function docs, just the section heading and then we're moving. No cap. ✨

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📖 Function Documentation 📖

🌸 subroutine clar1v 🌸

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 💫

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 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 227 of file clar1v.f. 📌

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💖 subroutine dlar1v 💖

subroutine dlar1v (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, double
precision, 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)

DLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI. 💜

Purpose 💫

DLAR1V 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 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 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 — L is DOUBLE PRECISION array, dimension (N-1). The (n-1) subdiagonal elements of the unit bidiagonal matrix L, in elements 1 to N-1.
• D — D is DOUBLE PRECISION array, dimension (N). The n diagonal elements of the diagonal matrix D.
• LD — LD is DOUBLE PRECISION array, dimension (N-1). The n-1 elements L(i)*D(i).
• LLD — LLD is DOUBLE PRECISION array, dimension (N-1). The n-1 elements L(i)*L(i)*D(i).
• PIVMIN — PIVMIN is DOUBLE PRECISION. The minimum pivot in the Sturm sequence.
• GAPTOL — GAPTOL is DOUBLE PRECISION. Tolerance that indicates when eigenvector entries are negligible w.r.t. their contribution to the residual.
• Z — Z is DOUBLE PRECISION 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 — the part 1 source cuts off right after this label, bestie; part 2 picks up the rest. No invented deets. 🫶
• WANTNC — LOGICAL. Tells us whether NEGCNT actually needs to be computed. Like, do we care, or nah 💅
• 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. that's the whole vibe 🫶
• 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. Lowkey the main character of this whole routine, fr:
  • 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 ). outside that range it's giving absolutely nothing 🔥
• 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). scratch space, it's a whole thing ✨

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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 dlar1v.f.

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💖 subroutine slar1v 💖

okay so same routine, single precision era. iconic of her honestly. here's the signature, verbatim, no cap:

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 🎀

!> !> 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. 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.

so bestie, the tea ☕: when sigma gets close to an eigenvalue, the vector it spits out is a genuinely accurate eigenvector. and usually r lines up with wherever the eigenvector is biggest in magnitude. slay.

!> 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.

like literally four steps, all of them ate:

1. stationary qd transform
2. progressive qd transform
3. combine them to get the diagonal of the inverse, then pick r where that diagonal is (one of the) biggest in magnitude
4. twisted factorization time — top half from the stationary transform, bottom half from the progressive one — and that gives you the (scaled) r-th column of the inverse 🦋

✨ 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 — REAL. The shift. In order to compute an accurate eigenvector, LAMBDA should be a good approximation to an eigenvalue of L D L**T. like, be a good approximation, bestie, that's the whole assignment 🙈
• L — REAL array, dimension (N-1). The (n-1) subdiagonal elements of the unit bidiagonal matrix L, in elements 1 to N-1.
• D — REAL array, dimension (N). The n diagonal elements of the diagonal matrix D.
• LD — REAL array, dimension (N-1). The n-1 elements L(i)*D(i).
• LLD — REAL array, dimension (N-1). The n-1 elements L(i)*L(i)*D(i).
• PIVMIN — REAL. The minimum pivot in the Sturm sequence.
• GAPTOL — REAL. Tolerance that indicates when eigenvector entries are negligible w.r.t. their contribution to the residual. i.e. at what point an entry is contributing nothing and can be politely ignored 💀
• Z — 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.

heads up bestie: Z has to come in as all zeros. that's not a suggestion, that's the contract 🫶

• 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 — REAL. The square of the 2-norm of Z.
• MINGMA — REAL. 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 — REAL. NRMINV = 1/SQRT( ZTZ )
• RESID — REAL. The residual of the FP vector. RESID = ABS( MINGMA )/SQRT( ZTZ )
• RQCORR — REAL. The Rayleigh Quotient correction to LAMBDA. RQCORR = MINGMA*TMP
• WORK — 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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👑 subroutine zlar1v 👑

and now for the complex*16 girlypop of the family. signature, again verbatim, do not touch:

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 🎀

!> !> ZLAR1V 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. 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.

same deal as her sisters, no cap — sigma close to an eigenvalue means the computed vector is an accurate eigenvector, and r usually lands on the index where the eigenvector is largest in magnitude. obsessed 🌸

!> 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.

same four steps, same energy:

1. stationary qd transform
2. progressive qd transform
3. smash them together for the diagonal of the inverse, pick r where it's (one of the) biggest in magnitude
4. twisted factorization — top from stationary, bottom from progressive — and you get the (scaled) r-th column of the inverse ✨

🌟 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.

okay bestie we're still in the parameter list and these next ones are literally the rest of the arguments — stay with me, we're almost at the glow-up finish line 💖

• L — DOUBLE PRECISION array, dimension (N-1). the (n-1) subdiagonal elements of the unit bidiagonal matrix L, in elements 1 to N-1. she's giving "the lower lane" energy ✨
• D — DOUBLE PRECISION array, dimension (N). the n diagonal elements of the diagonal matrix D. main character diagonal, no cap 💅
• LD — DOUBLE PRECISION array, dimension (N-1). the n-1 elements L(i)*D(i). yes bestie, multiplication is real 🔥
• LLD — DOUBLE PRECISION array, dimension (N-1). the n-1 elements L(i)*L(i)*D(i). that's L squared times D, periodt 💀
• PIVMIN — DOUBLE PRECISION. the minimum pivot in the Sturm sequence. tiny but iconic 🎀
• GAPTOL — DOUBLE PRECISION. tolerance that indicates when eigenvector entries are negligible w.r.t. their contribution to the residual. basically the "girl, that's close enough to zero" threshold 🙈
• Z — COMPLEX*16 array, dimension (N). on input, all entries of Z must be set to 0 (not me stressing this, but fr fr do it). on output, Z contains the (scaled) r-th column of the inverse. the scaling is such that Z(R) equals 1. she's serving scaled inverse column realness 👑
• WANTNC — LOGICAL. specifies whether NEGCNT has to be computed. a little yes/no vibe check 🦋
• 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. math said "count the tiny pivots" and we said slay 💅
• ZTZ — DOUBLE PRECISION. the square of the 2-norm of Z. norm squared, cute ✨
• MINGMA — DOUBLE PRECISION. the reciprocal of the largest (in magnitude) diagonal element of the inverse of L D L**T - sigma I. yes that's a whole sentence, but it's literally the definition, so 🫶
• 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. she's the twist index, the main character of the factorization 💃
• 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 ). support system but make it math 🌸
• NRMINV — DOUBLE PRECISION. NRMINV = 1/SQRT( ZTZ ). that's a whole formula, keep it exact bestie 🧚
• RESID — DOUBLE PRECISION. the residual of the FP vector. RESID = ABS( MINGMA )/SQRT( ZTZ ). residual said "I'm the error term" and we said iconic 😭
• RQCORR — DOUBLE PRECISION. the Rayleigh Quotient correction to LAMBDA. RQCORR = MINGMA*TMP. correction era 💜
• WORK — DOUBLE PRECISION array, dimension (4*N). workspace, but make it spacious 🎀

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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 zlar1v.f.

👑 Author (again, bestie) 👑

Generated automatically by Doxygen for LAPACK from the source code.

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later bestie! this was LAPACK version 3.12.0, page lar1v(3), and you ate that whole parameter list up 💅✨