Visible to Intel only — GUID: GUID-24EE539E-D3CD-44A9-879A-0099F0EBC4DB
Visible to Intel only — GUID: GUID-24EE539E-D3CD-44A9-879A-0099F0EBC4DB
?lar1va
Computes scaled eigenvector corresponding to given eigenvalue.
Syntax
call slar1va(n, b1, bn, lambda, d, l, ld, lld, pivmin, gaptol, z, wantnc, negcnt, ztz, mingma, r, isuppz, nrminv, resid, rqcorr, work )
call dlar1va(n, b1, bn, lambda, d, l, ld, lld, pivmin, gaptol, z, wantnc, negcnt, ztz, mingma, r, isuppz, nrminv, resid, rqcorr, work )
Description
?slar1va computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI. 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, LDLT - λI = L+D+L+T,
Progressive qd transform, LDLT - λI = U-D-U-T,
Computation of the diagonal elements of the inverse of LDLT - λI 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 LDLT.
- b1
-
INTEGER
First index of the submatrix of LDLT.
- bn
-
INTEGER
Last index of the submatrix of LDLT.
- lambda
-
REAL for slar1va
DOUBLE PRECISION for dlar1va
The shift λ. In order to compute an accurate eigenvector, lambda should be a good approximation to an eigenvalue of LDLT.
- l
-
REAL for slar1va
DOUBLE PRECISION for dlar1va
Array of size n-1
The (n-1) subdiagonal elements of the unit bidiagonal matrix L, in elements 1 to n-1.
- d
-
REAL for slar1va
DOUBLE PRECISION for dlar1va
Array of size n
The n diagonal elements of the diagonal matrix D.
- ld
-
REAL for slar1va
DOUBLE PRECISION for dlar1va
Array of size n-1
The n-1 elements l(i)*d(i).
- lld
-
REAL for slar1va
DOUBLE PRECISION for dlar1va
Array of size n-1
The n-1 elements l(i)*l(i)*d(i).
- pivmin
-
REAL for slar1va
DOUBLE PRECISION for dlar1va
The minimum pivot in the Sturm sequence.
- gaptol
-
REAL for slar1va
DOUBLE PRECISION for dlar1va
Tolerance that indicates when eigenvector entries are negligible with respect to their contribution to the residual.
- z
-
REAL for slar1va
DOUBLE PRECISION for dlar1va
Array of size n
On input, 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 (LDLT - σI)-1 is largest in magnitude. If 1 ≤r≤n, r is unchanged.
Ideally, r designates the position of the maximum entry in the eigenvector.
- work
-
REAL for slar1va
DOUBLE PRECISION for dlar1va
(Workspace) array of size 4*n
OUTPUT Parameters
- z
-
On output, z contains the (scaled) r-th column of the inverse. The scaling is such that z(r) equals 1.
- negcnt
-
INTEGER
If wantncis .TRUE. then negcnt = the number of pivots < pivmin in the matrix factorization LDLT, and negcnt = -1 otherwise.
- ztz
-
REAL for slar1va
DOUBLE PRECISION for dlar1va
The square of the 2-norm of z.
- mingma
-
REAL for slar1va
DOUBLE PRECISION for dlar1va
The reciprocal of the largest (in magnitude) diagonal element of the inverse of LDLT - σI.
- r
-
On output, r contains the twist index used to compute z.
- isuppz
-
INTEGER array of size 2
The support of the vector in z, i.e., the vector z is non-zero only in elements isuppz(1) through isuppz(2).
- nrminv
-
REAL for slar1va
DOUBLE PRECISION for dlar1va
nrminv = 1/SQRT( ztz )
- resid
-
REAL for slar1va
DOUBLE PRECISION for dlar1va
The residual of the FP vector.
resid = ABS( mingma )/SQRT( ztz )
- rqcorr
-
REAL for slar1va
DOUBLE PRECISION for dlar1va
The Rayleigh Quotient correction to lambda.