Visible to Intel only — GUID: GUID-5274024C-916C-466D-925E-399A1B5762FE
Visible to Intel only — GUID: GUID-5274024C-916C-466D-925E-399A1B5762FE
?lar1v
Computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of tridiagonal matrix.
Syntax
call slar1v( n, b1, bn, lambda, d, l, ld, lld, pivmin, gaptol, z, wantnc, negcnt, ztz, mingma, r, isuppz, nrminv, resid, rqcorr, work )
call dlar1v( n, b1, bn, lambda, d, l, ld, lld, pivmin, gaptol, z, wantnc, negcnt, ztz, mingma, r, isuppz, nrminv, resid, rqcorr, work )
call clar1v( n, b1, bn, lambda, d, l, ld, lld, pivmin, gaptol, z, wantnc, negcnt, ztz, mingma, r, isuppz, nrminv, resid, rqcorr, work )
call zlar1v( n, b1, bn, lambda, d, l, ld, lld, pivmin, gaptol, z, wantnc, negcnt, ztz, mingma, r, isuppz, nrminv, resid, rqcorr, work )
Include Files
- mkl.fi
Description
The routine ?lar1v computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix L*D*LT - λ*I. When λ 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 :
Stationary qd transform, L*D*LT - λ*I = L(+)*D(+)*L(+)T
Progressive qd transform, L*D*LT - λ*I = U(-)*D(-)*U(-)T,
Computation of the diagonal elements of the inverse of L*D*LT - λ*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.
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.
Input Parameters
- n
-
INTEGER. The order of the matrix L*D*LT.
- b1
-
INTEGER. First index of the submatrix of L*D*LT.
- bn
-
INTEGER. Last index of the submatrix of L*D*LT.
- lambda
-
REAL for slar1v/clar1v
DOUBLE PRECISION for dlar1v/zlar1v
The shift. To compute an accurate eigenvector, lambda should be a good approximation to an eigenvalue of L*D*LT.
- l
-
REAL for slar1v/clar1v
DOUBLE PRECISION for dlar1v/zlar1v
Array, DIMENSION (n-1).
The (n-1) subdiagonal elements of the unit bidiagonal matrix L, in elements 1 to n-1.
- d
-
REAL for slar1v/clar1v
DOUBLE PRECISION for dlar1v/zlar1v
Array, DIMENSION (n).
The n diagonal elements of the diagonal matrix D.
- ld
-
REAL for slar1v/clar1v
DOUBLE PRECISION for dlar1v/zlar1v
Array, DIMENSION (n-1).
The n-1 elements Li*Di.
- lld
-
REAL for slar1v/clar1v
DOUBLE PRECISION for dlar1v/zlar1v
Array, DIMENSION (n-1).
The n-1 elements Li*Li*Di.
- pivmin
-
REAL for slar1v/clar1v
DOUBLE PRECISION for dlar1v/zlar1v
The minimum pivot in the Sturm sequence.
- gaptol
-
REAL for slar1v/clar1v
DOUBLE PRECISION for dlar1v/zlar1v
Tolerance that indicates when eigenvector entries are negligible with respect to their contribution to the residual.
- z
-
REAL for slar1v
DOUBLE PRECISION for dlar1v
COMPLEX for clar1v
DOUBLE COMPLEX for zlar1v
Array, DIMENSION (n). All entries of z must be set to 0.
- wantnc
-
LOGICAL.
Specifies whether negcnt has to be computed.
- 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*LT - lambda*I)-1 is largest in magnitude. If 1 ≤ r ≤ n, r is unchanged.
- work
-
REAL for slar1v/clar1v
DOUBLE PRECISION for dlar1v/zlar1v
Workspace array, DIMENSION (4*n).
Output Parameters
- z
-
REAL for slar1v
DOUBLE PRECISION for dlar1v
COMPLEX for clar1v
DOUBLE COMPLEX for zlar1v
Array, DIMENSION (n). The (scaled) r-th column of the inverse. z(r) is returned to be 1.
- negcnt
-
INTEGER. If wantnc is .TRUE. then negcnt = the number of pivots < pivmin in the matrix factorization L*D*LT, and negcnt = -1 otherwise.
- ztz
-
REAL for slar1v/clar1v
DOUBLE PRECISION for dlar1v/zlar1v
The square of the 2-norm of z.
- mingma
-
REAL for slar1v/clar1v
DOUBLE PRECISION for dlar1v/zlar1v
The reciprocal of the largest (in magnitude) diagonal element of the inverse of L*D*LT - lambda*I.
- r
-
On output, r is 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, that is, the vector z is nonzero only in elements isuppz(1) through isuppz(2).
- nrminv
-
REAL for slar1v/clar1v
DOUBLE PRECISION for dlar1v/zlar1v
Equals 1/sqrt( ztz ).
- resid
-
REAL for slar1v/clar1v
DOUBLE PRECISION for dlar1v/zlar1v
The residual of the FP vector.
resid = ABS( mingma )/sqrt( ztz ).
- rqcorr
-
REAL for slar1v/clar1v
DOUBLE PRECISION for dlar1v/zlar1v
The Rayleigh Quotient correction to lambda.
rqcorr = mingma/ztz.