Computes all eigenvalues and, optionally, all eigenvectors of a real symmetric tridiagonal matrix using divide and conquer algorithm.
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);
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*ZT.
Here Λ is a diagonal matrix whose diagonal elements are the eigenvalues λi, and Z is the orthogonal matrix whose columns are the eigenvectors zi. Thus,
T*zi = λi*zi 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.
Specifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).
Must be 'N' or 'V'.
If jobz = 'N', then only eigenvalues are computed.
If jobz = 'V', then eigenvalues and eigenvectors are computed.
The order of the matrix T (n ≥ 0).
- d, e
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.
The leading dimension of the output array z. Constraints:
ldz≥ 1 if job = 'N';
ldz≥ max(1, n) if job = 'V'.
On exit, if info = 0, contains the eigenvalues of the matrix T in ascending order.
See also info.
Array, size max(1, ldz*n) if jobz = 'V' and 1 if jobz = 'N' .
If jobz = 'V', then this array is overwritten by the orthogonal matrix Z which contains the eigenvectors of T.
If jobz = 'N', then z is not referenced.
On exit, this array is overwritten with intermediate results.
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.
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 λi is an exact eigenvalue, and μi is the corresponding computed value, then
|μi - λi| ≤ c(n)*ε*||T||2
where c(n) is a modestly increasing function of n.
If zi is the corresponding exact eigenvector, and wi is the corresponding computed vector, then the angle θ(zi, wi) between them is bounded as follows:
θ(zi, wi) ≤ c(n)*ε*||T||2 / min i≠j|λi - λj|.
Thus the accuracy of a computed eigenvector depends on the gap between its eigenvalue and all the other eigenvalues.