lar1v(3)

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, 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 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)^{-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 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 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)^{-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)^{-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 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.
    !> 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)^{-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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