?stevd

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

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

call sstevd(jobz, n, d, e, z, ldz, work, lwork, iwork, liwork, info)

call dstevd(jobz, n, d, e, z, ldz, work, lwork, iwork, liwork, info)

call stevd(d, e [,z] [,info])

Include Files

mkl.fi, lapack.f90

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

jobz

CHARACTER*1. Must be 'N' or 'V'.

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

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

n

INTEGER. The order of the matrix T (n ≥ 0).

d, e, work

REAL for sstevd

DOUBLE PRECISION for dstevd.

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.

work(*) is a workspace array.

The dimension of work must be at least lwork.

ldz

INTEGER. The leading dimension of the output array z. Constraints:

ldz≥ 1 if job = 'N';

ldz≥ max(1, n) if job = 'V'.

lwork

INTEGER.

The dimension of the array work.

Constraints:

if jobz = 'N' or n ≤ 1, then lwork ≥ 1;

if jobz = 'V' and n > 1, then lwork ≥n²+ 4*n + 1.

If lwork = -1, then a workspace query is assumed; the routine only calculates the required sizes of the work and iwork arrays, returns these values as the first entries of the work and iwork arrays, and no error message related to lwork or liwork is issued by xerbla. See Application Notes for details.

iwork

INTEGER. Workspace array, its dimension max(1, liwork).

liwork

INTEGER.

The dimension of the array iwork.

Constraints:

if jobz = 'N' or n ≤ 1, then liwork ≥ 1;

if jobz = 'V' and n > 1, then liwork ≥ 5*n+3.

If liwork = -1, then a workspace query is assumed; the routine only calculates the required sizes of the work and iwork arrays, returns these values as the first entries of the work and iwork arrays, and no error message related to lwork or liwork is issued by xerbla. See Application Notes for details.

Output Parameters

d

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

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 stevd interface are the following:

d

Holds the vector of length n.

e

Holds the vector of length n.

z

Holds the matrix Z of size (n, n).

jobz

Restored based on the presence of the argument z as follows:

jobz = 'V', if z is present,

jobz = 'N', if z is omitted.

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.

If it is not clear how much workspace to supply, use a generous value of lwork (or liwork) for the first run, or set lwork = -1 (liwork = -1).

If lwork (or liwork) has any of admissible sizes, which is no less than the minimal value described, then the routine completes the task, though probably not so fast as with a recommended workspace, and provides the recommended workspace in the first element of the corresponding array (work, iwork) on exit. Use this value (work(1), iwork(1)) for subsequent runs.

If lwork = -1 (liwork = -1), then the routine returns immediately and provides the recommended workspace in the first element of the corresponding array (work, iwork). This operation is called a workspace query.

Note that if lwork (liwork) is less than the minimal required value and is not equal to -1, then the routine returns immediately with an error exit and does not provide any information on the recommended workspace.

Parent topic: Symmetric Eigenvalue Problems: LAPACK Driver Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?stevd

Syntax

Include Files

Description

Input Parameters

Output Parameters

LAPACK 95 Interface Notes

Application Notes