table of contents
| lar1v(3) | LAPACK | 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
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
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
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
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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| Wed Feb 7 2024 11:30:40 | Version 3.12.0 |