Developer Reference for Intel® oneAPI Math Kernel Library for Fortran
?larrv
Computes the eigenvectors of the tridiagonal matrix T = L*D*LT given L , D and the eigenvalues of L*D*LT .
Syntax
call slarrv(n, vl, vu, d, l, pivmin, isplit, m, dol, dou, minrgp, rtol1, rtol2, w, werr,
wgap, iblock, indexw, gers, z, ldz, isuppz, work, iwork, info)
call dlarrv(n, vl, vu, d, l, pivmin, isplit, m, dol, dou, minrgp, rtol1, rtol2, w, werr,
wgap, iblock, indexw, gers, z, ldz, isuppz, work, iwork, info)
call clarrv(n, vl, vu, d, l, pivmin, isplit, m, dol, dou, minrgp, rtol1, rtol2, w, werr,
wgap, iblock, indexw, gers, z, ldz, isuppz, work, iwork, info)
call zlarrv(n, vl, vu, d, l, pivmin, isplit, m, dol, dou, minrgp, rtol1, rtol2, w, werr,
wgap, iblock, indexw, gers, z, ldz, isuppz, work, iwork, info)Include Files
mkl.fi
Description
The routine ?larrv computes the eigenvectors of the tridiagonal matrix T = L*D*LT given L , D and approximations to the eigenvalues of L*D*LT .
The input eigenvalues should have been computed by slarre for real flavors ( slarrv/clarrv ) and by dlarre for double precision flavors ( dlarre/zlarre ).
Input Parameters
- n
-
INTEGER . The order of the matrix. n≥ 0 .
- vl , vu
-
REAL for slarrv / clarrv DOUBLE PRECISION for dlarrv / zlarrv Lower and upper bounds respectively of the interval that contains the desired eigenvalues. vl < vu . Needed to compute gaps on the left or right end of the extremal eigenvalues in the desired range.
- d
-
REAL for slarrv / clarrv DOUBLE PRECISION for dlarrv / zlarrv Array, DIMENSION ( n ). On entry, the n diagonal elements of the diagonal matrix D .
- l
-
REAL for slarrv / clarrv DOUBLE PRECISION for dlarrv / zlarrv Array, DIMENSION ( n ). On entry, the ( n -1) subdiagonal elements of the unit bidiagonal matrix L are contained in elements 1 to n -1 of L if the matrix is not splitted. At the end of each block the corresponding shift is stored as given by slarre for real flavors and by dlarre for double precision flavors.
- pivmin
-
REAL for slarrv / clarrv DOUBLE PRECISION for dlarrv / zlarrv The minimum pivot allowed in the Sturm sequence.
- isplit
-
INTEGER . Array, DIMENSION ( n ). The splitting points, at which T breaks up into blocks. The first block consists of rows/columns 1 to isplit (1), the second of rows/columns isplit (1)+1 through isplit (2), etc.
- m
-
INTEGER . The total number of eigenvalues found. 0 ≤ m ≤ n . If range = 'A' , m = n , and if range = 'I' , m = iu - il +1 .
- dol , dou
-
INTEGER . If you want to compute only selected eigenvectors from all the eigenvalues supplied, specify an index range dol : dou . Or else apply the setting dol =1, dou = m . Note that dol and dou refer to the order in which the eigenvalues are stored in w . If you want to compute only selected eigenpairs, then the columns dol -1 to dou +1 of the eigenvector space Z contain the computed eigenvectors. All other columns of Z are set to zero.
- minrgp , rtol1 , rtol2
-
REAL for slarrv / clarrv DOUBLE PRECISION for dlarrv / zlarrv Parameters for bisection. An interval [LEFT,RIGHT] has converged if RIGHT-LEFT.LT.MAX( rtol1*gap, rtol2*max(|LEFT|,|RIGHT|) ) .
- w
-
REAL for slarrv / clarrv DOUBLE PRECISION for dlarrv / zlarrv Array, DIMENSION ( n ). The first m elements of w contain the approximate eigenvalues for which eigenvectors are to be computed. The eigenvalues should be grouped by split-off block and ordered from smallest to largest within the block (the output array w from ?larre is expected here). These eigenvalues are set with respect to the shift of the corresponding root representation for their block.
- werr
-
REAL for slarrv / clarrv DOUBLE PRECISION for dlarrv / zlarrv Array, DIMENSION ( n ). The first m elements contain the semiwidth of the uncertainty interval of the corresponding eigenvalue in w .
- wgap
-
REAL for slarrv / clarrv DOUBLE PRECISION for dlarrv / zlarrv Array, DIMENSION ( n ). The separation from the right neighbor eigenvalue in w .
- iblock
-
INTEGER . Array, DIMENSION ( n ). The indices of the blocks (submatrices) associated with the corresponding eigenvalues in w ; iblock(i)=1 if eigenvalue w(i) belongs to the first block from the top, =2 if w(i) belongs to the second block, etc.
- indexw
-
INTEGER . Array, DIMENSION ( n ). The indices of the eigenvalues within each block (submatrix); for example, indexw(i)`= 10 and :code:`iblock(i)`=2 imply that the :code:`i -th eigenvalue w(i) is the 10-th eigenvalue in the second block.
- gers
-
REAL for slarrv / clarrv DOUBLE PRECISION for dlarrv / zlarrv Array, DIMENSION (2* n ). The n Gerschgorin intervals (the i -th Gerschgorin interval is (gers(2*i-1), gers(2*i)) . The Gerschgorin intervals should be computed from the original unshifted matrix.
- ldz
-
INTEGER . The leading dimension of the output array Z . ldz≥ 1 , and if jobz = 'V' , ldz≥ max(1,n) .
- work
-
REAL for slarrv / clarrv DOUBLE PRECISION for dlarrv / zlarrv Workspace array, DIMENSION (12* n ).
- iwork
-
INTEGER . Workspace array, DIMENSION (7* n ).
Output Parameters
- d
-
On exit, d may be overwritten.
- l
-
On exit, l is overwritten.
- w
-
On exit, w holds the eigenvalues of the unshifted matrix.
- werr
-
On exit, werr contains refined values of its input approximations.
- wgap
-
On exit, wgap contains refined values of its input approximations. Very small gaps are changed.
- z
-
REAL for slarrv DOUBLE PRECISION for dlarrv COMPLEX for clarrv DOUBLE COMPLEX for zlarrv Array, DIMENSION ( ldz , max(1, m ) ). If info = 0, the first m columns of z contain the orthonormal eigenvectors of the matrix T corresponding to the input eigenvalues, with the i -th column of z holding the eigenvector associated with w(i).
- isuppz
-
INTEGER . Array, DIMENSION (2*max(1,m)) . The support of the eigenvectors in z , that is, the indices indicating the nonzero elements in z . The i -th eigenvector is nonzero only in elements isuppz (2 i -1) through isuppz (2 i ).
info
INTEGER .
If info = 0 : successful exit
If info > 0 : A problem occured in ?larrv . If info = 5 , the Rayleigh Quotient Iteration failed to converge to full accuracy.
If info < 0 : One of the called subroutines signaled an internal problem. Inspection of the corresponding parameter info for further information is required.