?stevd

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 3/22/2024

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?stevd

Computes all eigenvalues and, optionally, all eigenvectors of a real symmetric tridiagonal matrix using divide and conquer algorithm.

Syntax

lapack_int LAPACKE_sstevd (int matrix_layout, char jobz, lapack_int n, float* d, float* e, float* z, lapack_int ldz);

lapack_int LAPACKE_dstevd (int matrix_layout, char jobz, lapack_int n, double* d, double* e, double* z, lapack_int ldz);

Include Files

mkl.h

Description

The routine computes all the eigenvalues, and optionally all the eigenvectors, of a real symmetric tridiagonal matrix T. In other words, the routine can compute the spectral factorization of T as: T = Z*Λ*Z^T.

Here Λ is a diagonal matrix whose diagonal elements are the eigenvalues λ_i, and Z is the orthogonal matrix whose columns are the eigenvectors z_i. Thus,

T*z_i = λ_i*z_i for i = 1, 2, ..., n.

If the eigenvectors are requested, then this routine uses a divide and conquer algorithm to compute eigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal-Walker-Kahan variant of the QL or QR algorithm.

There is no complex analogue of this routine.

Input Parameters

matrix_layout

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

jobz

Must be 'N' or 'V'.

If jobz = 'N', then only eigenvalues are computed.

If jobz = 'V', then eigenvalues and eigenvectors are computed.

n

The order of the matrix T (n ≥ 0).

d, e

Arrays:

d contains the n diagonal elements of the tridiagonal matrix T.

The dimension of d must be at least max(1, n).

e contains the n-1 off-diagonal elements of T.

The dimension of e must be at least max(1, n). The n-th element of this array is used as workspace.

ldz

The leading dimension of the output array z. Constraints:

ldz≥ 1 if job = 'N';

ldz≥ max(1, n) if job = 'V'.

Output Parameters

d

On exit, if info = 0, contains the eigenvalues of the matrix T in ascending order.

Return Values

This function returns a value info.

If info=0, the execution is successful.

If info = i, then the algorithm failed to converge; i indicates the number of elements of an intermediate tridiagonal form which did not converge to zero.

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

Application Notes

The computed eigenvalues and eigenvectors are exact for a matrix T+E such that ||E||₂ = O(ε)*||T||₂, where ε is the machine precision.

If λ_i is an exact eigenvalue, and μ_i is the corresponding computed value, then

|μ_i - λ_i| ≤ c(n)*ε*||T||₂

where c(n) is a modestly increasing function of n.

If z_i is the corresponding exact eigenvector, and w_i is the corresponding computed vector, then the angle θ(z_i, w_i) between them is bounded as follows:

θ(z_i, w_i) ≤ c(n)*ε*||T||₂ / min _i≠j|λ_i - λ_j|.

Thus the accuracy of a computed eigenvector depends on the gap between its eigenvalue and all the other eigenvalues.

Parent topic: Symmetric Eigenvalue Problems: LAPACK Driver Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for C

?stevd

Syntax

Include Files

Description

Input Parameters

Output Parameters

Return Values

Application Notes