Developer Reference for Intel® oneAPI Math Kernel Library for Fortran
?stein
Computes the eigenvectors corresponding to specified eigenvalues of a real symmetric tridiagonal matrix.
Syntax
call sstein ( n , d , e , m , w , iblock , isplit , z , ldz , work , iwork , ifailv , info )
call dstein ( n , d , e , m , w , iblock , isplit , z , ldz , work , iwork , ifailv , info )
call cstein ( n , d , e , m , w , iblock , isplit , z , ldz , work , iwork , ifailv , info )
call zstein ( n , d , e , m , w , iblock , isplit , z , ldz , work , iwork , ifailv , info )
call stein ( d , e , w , iblock , isplit , z [ , ifailv ] [ , info ] )
Include Files
mkl.fi , mkl_lapack.f90
Description
sstein dstein cstein zstein stein
The routine computes the eigenvectors of a real symmetric tridiagonal matrix T corresponding to specified eigenvalues, by inverse iteration. It is designed to be used in particular after the specified eigenvalues have been computed by ?stebz with order = 'B' , but may also be used when the eigenvalues have been computed by other routines.
If you use this routine after ?stebz , it can take advantage of the block structure by performing inverse iteration on each block Ti separately, which is more efficient than using the whole matrix T .
If T has been formed by reduction of a full symmetric or Hermitian matrix A to tridiagonal form, you can transform eigenvectors of T to eigenvectors of A by calling ?ormtr or ?opmtr (for real flavors) or by calling ?unmtr or ?upmtr (for complex flavors).
Input Parameters
n
INTEGER . The order of the matrix T ( n≥ 0 ).
m
INTEGER . The number of eigenvectors to be returned.
- d , e , w
-
REAL for single-precision flavors DOUBLE PRECISION for double-precision flavors. Arrays:
d (*) contains the diagonal elements of T .
The size of d must be at least max(1, n ).
e (*) contains the sub-diagonal elements of T stored in elements 1 to n -1
The size of e must be at least max(1, n -1).
w (*) contains the eigenvalues of T , stored in w(1) to w(m) (as returned by stebz ). Eigenvalues of T1 must be supplied first, in non-decreasing order; then those of T2 , again in non-decreasing order, and so on. Constraint:
if iblock(i) = iblock(i+1) , w(i) ≤w(i+1) . if iblock[i] = iblock[i+1] , w[i] ≤w[i+1] . The size of w must be at least max(1, n ).
iblock , isplit
INTEGER .
Arrays, size at least max(1, n ). The arrays iblock and isplit , as returned by ?stebz with order = 'B' .
If you did not call ?stebz with order = 'B' , set all elements of iblock to 1, and isplit(1)[0] to n .)
ldz
INTEGER . The leading dimension of the output array z ; ldz≥ max(1, n) .
- work
-
REAL for single-precision flavors DOUBLE PRECISION for double-precision flavors. Workspace array, size at least max(1, 5 n ).
iwork
INTEGER .
Workspace array, size at least max(1, n ).
Output Parameters
- z
-
REAL for sstein DOUBLE PRECISION for dstein COMPLEX for cstein DOUBLE COMPLEX for zstein .
Array, size ( ldz , *) .
If info = 0 , z contains an n -by- n matrix the columns of which are orthonormal eigenvectors. (The i -th column corresponds to the i th eigenvalue.)
ifailv
INTEGER .
Array, size at least max(1, m ). If info = i > 0, the first i elements of ifailv contain the indices of any eigenvectors that failed to converge.
info
INTEGER .
If info = 0 , the execution is successful.
If info = i , then i eigenvectors (as indicated by the parameter ifailv ) each failed to converge in 5 iterations. The current iterates are stored in the corresponding columns of the array z .
If info = -i , the i- th parameter had an illegal value.
Return Values
No return value, info is an Output Parameter.
LAPACK 95 Interface Notes
Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or restorable arguments, see LAPACK 95 Interface Conventions .
Specific details for the routine stein interface are the following:
d
Holds the vector of length n .
e
Holds the vector of length n .
w
Holds the vector of length n .
iblock
Holds the vector of length n .
isplit
Holds the vector of length n .
z
Holds the matrix Z of size ( n,m ).
ifailv
Holds the vector of length ( m ).
Application Notes
Each computed eigenvector zi is an exact eigenvector of a matrix \(T+E_{i}\) , where \(||E_{i}||_{2} = O(\varepsilon) ||T||_{2}\) . However, a set of eigenvectors computed by this routine may not be orthogonal to so high a degree of accuracy as those computed by ?steqr .