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

Definition at line 227 of file clar1v.f.

  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

Definition at line 227 of file dlar1v.f.

  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

Definition at line 227 of file slar1v.f.

  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

Definition at line 227 of file zlar1v.f.

Author

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