Man page - lar1v(3)

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Manual

lar1v

NAME
SYNOPSIS
Functions
Detailed Description
Function 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, integerr, integer, dimension( * ) isuppz, real nrminv, real resid, realrqcorr, real, dimension( * ) work)
subroutine dlar1v (integer n, integer b1, integer bn, double precisionlambda, double precision, dimension( * ) d, double precision,dimension( * ) l, double precision, dimension( * ) ld, doubleprecision, dimension( * ) lld, double precision pivmin, doubleprecision gaptol, double precision, dimension( * ) z, logical wantnc,integer negcnt, double precision ztz, double precision mingma, integerr, integer, dimension( * ) isuppz, double precision nrminv, doubleprecision resid, double precision rqcorr, double precision, dimension(* ) work)
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)
subroutine zlar1v (integer n, integer b1, integer bn, double precisionlambda, double precision, dimension( * ) d, double precision,dimension( * ) l, double precision, dimension( * ) ld, doubleprecision, dimension( * ) lld, double precision pivmin, doubleprecision gaptol, complex*16, dimension( * ) z, logical wantnc, integernegcnt, double precision ztz, double precision mingma, integer r,integer, dimension( * ) isuppz, double precision nrminv, doubleprecision resid, double precision rqcorr, double precision, dimension(* ) work)
Author

NAME

lar1v - lar1v: step in larrv, hence stemr & stegr

SYNOPSIS

Functions

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

Detailed Description

Function 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, integerr, integer, dimension( * ) isuppz, real nrminv, real resid, realrqcorr, 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 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.
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.

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)ˆ{-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

subroutine dlar1v (integer n, integer b1, integer bn, double precisionlambda, double precision, dimension( * ) d, double precision,dimension( * ) l, double precision, dimension( * ) ld, doubleprecision, dimension( * ) lld, double precision pivmin, doubleprecision gaptol, double precision, dimension( * ) z, logical wantnc,integer negcnt, double precision ztz, double precision mingma, integerr, integer, dimension( * ) isuppz, double precision nrminv, doubleprecision 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 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.
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.

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

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 DOUBLE PRECISION
The square of the 2-norm of Z.

MINGMA

MINGMA is DOUBLE PRECISION
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)ˆ{-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 DOUBLE PRECISION
NRMINV = 1/SQRT( ZTZ )

RESID

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

RQCORR

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

WORK

WORK is 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

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.
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.

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)ˆ{-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

subroutine zlar1v (integer n, integer b1, integer bn, double precisionlambda, double precision, dimension( * ) d, double precision,dimension( * ) l, double precision, dimension( * ) ld, doubleprecision, dimension( * ) lld, double precision pivmin, doubleprecision gaptol, complex*16, dimension( * ) z, logical wantnc, integernegcnt, double precision ztz, double precision mingma, integer r,integer, dimension( * ) isuppz, double precision nrminv, doubleprecision 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.
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.

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 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

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 DOUBLE PRECISION
The square of the 2-norm of Z.

MINGMA

MINGMA is DOUBLE PRECISION
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)ˆ{-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 DOUBLE PRECISION
NRMINV = 1/SQRT( ZTZ )

RESID

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

RQCORR

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

WORK

WORK is 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

Author

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