?stein

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 11/07/2023

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?stein

Computes the eigenvectors corresponding to specified eigenvalues of a real symmetric tridiagonal matrix.

Syntax

lapack_int LAPACKE_sstein( int matrix_layout, lapack_int n, const float* d, const float* e, lapack_int m, const float* w, const lapack_int* iblock, const lapack_int* isplit, float* z, lapack_int ldz, lapack_int* ifailv );

lapack_int LAPACKE_dstein( int matrix_layout, lapack_int n, const double* d, const double* e, lapack_int m, const double* w, const lapack_int* iblock, const lapack_int* isplit, double* z, lapack_int ldz, lapack_int* ifailv );

lapack_int LAPACKE_cstein( int matrix_layout, lapack_int n, const float* d, const float* e, lapack_int m, const float* w, const lapack_int* iblock, const lapack_int* isplit, lapack_complex_float* z, lapack_int ldz, lapack_int* ifailv );

lapack_int LAPACKE_zstein( int matrix_layout, lapack_int n, const double* d, const double* e, lapack_int m, const double* w, const lapack_int* iblock, const lapack_int* isplit, lapack_complex_double* z, lapack_int ldz, lapack_int* ifailv );

Include Files

mkl.h

Description

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 T_i 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

matrix_layout

Specifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).

n

The order of the matrix T (n≥ 0).

m

The number of eigenvectors to be returned.

d, e, w

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[0] to w[m - 1] (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].

The size of w must be at least max(1, n).

iblock, isplit

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[0] to n.)

ldz

The leading dimension of the output array z; ldz≥ max(1, n) for column major layout and ldz>=max(1,m) for row major layout.

Output Parameters

z

Array, size at least max(1,ldz*m) for column major layout and max(1,ldz*n) for row major layout.

If info = 0, z contains an n-by-n matrix the columns of which are orthonormal eigenvectors. (The i-th column corresponds to the ith eigenvalue.)

ifailv

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.

Return Values

This function returns a value info.

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/rows of the array z.

If info = -i, the i-th parameter had an illegal value.

Application Notes

Each computed eigenvector z_i is an exact eigenvector of a matrix T+E_i, where ||E_i||₂ = O(ε)*||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.

Parent topic: Symmetric Eigenvalue Problems: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for C

?stein

Syntax

Include Files

Description

Input Parameters

Output Parameters

Return Values

Application Notes