Developer Reference for Intel® oneAPI Math Kernel Library for Fortran
?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)Include Files
mkl_scalapack.h
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 wantnc is .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 dlar1vanrminv = 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 .