Computes selected eigenvalues and, optionally, eigenvectors of a Hermitian matrix using the Relatively Robust Representations.

## Syntax

lapack_int LAPACKE_cheevr( int matrix_layout, char jobz, char range, char uplo, lapack_int n, lapack_complex_float* a, lapack_int lda, float vl, float vu, lapack_int il, lapack_int iu, float abstol, lapack_int* m, float* w, lapack_complex_float* z, lapack_int ldz, lapack_int* isuppz );

lapack_int LAPACKE_zheevr( int matrix_layout, char jobz, char range, char uplo, lapack_int n, lapack_complex_double* a, lapack_int lda, double vl, double vu, lapack_int il, lapack_int iu, double abstol, lapack_int* m, double* w, lapack_complex_double* z, lapack_int ldz, lapack_int* isuppz );

• mkl.h

## Description

The routine computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

The routine first reduces the matrix A to tridiagonal form T with a call to hetrd. Then, whenever possible, ?heevr calls stegr to compute the eigenspectrum using Relatively Robust Representations. ?stegr computes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are computed from various "good" L*D*LT representations (also known as Relatively Robust Representations). Gram-Schmidt orthogonalization is avoided as far as possible. More specifically, the various steps of the algorithm are as follows. For each unreduced block (submatrix) of T:

1. Compute T - σ*I = L*D*LT, so that L and D define all the wanted eigenvalues to high relative accuracy. This means that small relative changes in the entries of D and L cause only small relative changes in the eigenvalues and eigenvectors. The standard (unfactored) representation of the tridiagonal matrix T does not have this property in general.

2. Compute the eigenvalues to suitable accuracy. If the eigenvectors are desired, the algorithm attains full accuracy of the computed eigenvalues only right before the corresponding vectors have to be computed, see Steps c) and d).

3. For each cluster of close eigenvalues, select a new shift close to the cluster, find a new factorization, and refine the shifted eigenvalues to suitable accuracy.

4. For each eigenvalue with a large enough relative separation, compute the corresponding eigenvector by forming a rank revealing twisted factorization. Go back to Step c) for any clusters that remain.

The desired accuracy of the output can be specified by the input parameter abstol.

The routine ?heevr calls stemr when the full spectrum is requested on machines which conform to the IEEE-754 floating point standard, or stebz and stein on non-IEEE machines and when partial spectrum requests are made.

Note that the routine ?heevr is preferable for most cases of complex Hermitian eigenvalue problems as its underlying algorithm is fast and uses less workspace.

## 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 job = 'N', then only eigenvalues are computed.

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

range

Must be 'A' or 'V' or 'I'.

If range = 'A', the routine computes all eigenvalues.

If range = 'V', the routine computes eigenvalues lambda(i) in the half-open interval: vl< lambda(i)vu.

If range = 'I', the routine computes eigenvalues with indices il to iu.

For range = 'V'or 'I', sstebz/dstebz and cstein/zstein are called.

uplo

Must be 'U' or 'L'.

If uplo = 'U', a stores the upper triangular part of A.

If uplo = 'L', a stores the lower triangular part of A.

n

The order of the matrix A (n 0).

a

a (size max(1, lda*n)) is an array containing either upper or lower triangular part of the Hermitian matrix A, as specified by uplo.

lda

The leading dimension of the array a.

Must be at least max(1, n).

vl, vu

If range = 'V', the lower and upper bounds of the interval to be searched for eigenvalues.

Constraint: vl< vu.

If range = 'A' or 'I', vl and vu are not referenced.

il, iu

If range = 'I', the indices in ascending order of the smallest and largest eigenvalues to be returned.

Constraint: 1 iliun, if n > 0; il=1 and iu=0 if n = 0.

If range = 'A' or 'V', il and iu are not referenced.

abstol

The absolute error tolerance to which each eigenvalue/eigenvector is required.

If jobz = 'V', the eigenvalues and eigenvectors output have residual norms bounded by abstol, and the dot products between different eigenvectors are bounded by abstol.

If abstol < n *eps*||T||, then n *eps*||T|| is used instead, where eps is the machine precision, and ||T|| is the 1-norm of the matrix T. The eigenvalues are computed to an accuracy of eps*||T|| irrespective of abstol.

If high relative accuracy is important, set abstol to ?lamch('S').

ldz

The leading dimension of the output array z. Constraints:

ldz 1 if jobz = 'N';

ldz max(1, n) for column major layout and ldz max(1, m) for row major layout if jobz = 'V'.

## Output Parameters

a

On exit, the lower triangle (if uplo = 'L') or the upper triangle (if uplo = 'U') of A, including the diagonal, is overwritten.

m

The total number of eigenvalues found,

0 mn.

If range = 'A', m = n, if range = 'I', m = iu-il+1, and if range = 'V' the exact value of m is not known in advance.

w

Array, size at least max(1, n), contains the selected eigenvalues in ascending order, stored in w[0] to w[m - 1].

z

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

If jobz = 'V', then if info = 0, the first m columns of z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of z holding the eigenvector associated with w[i - 1].

If jobz = 'N', then z is not referenced.

isuppz

Array, size at least 2 *max(1, m).

The support of the eigenvectors in z, i.e., the indices indicating the nonzero elements in z. The i-th eigenvector is nonzero only in elements isuppz[2i - 2] through isuppz[2i - 1]. Referenced only if eigenvectors are needed (jobz = 'V') and all eigenvalues are needed, that is, range = 'A' or range = 'I' and il = 1 and iu = n.

## Return Values

This function returns a value info.

If info=0, the execution is successful.

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

If info = i, an internal error has occurred.

## Application Notes

Normal execution of ?stemr may create NaNs and infinities and hence may abort due to a floating point exception in environments which do not handle NaNs and infinities in the IEEE standard default manner.

For more details, see ?stemr and these references:

• Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations to compute orthogonal eigenvectors of symmetric tridiagonal matrices," Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.

• Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25, 2004. Also LAPACK Working Note 154.

• Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem", Computer Science Division Technical Report No. UCB/CSD-97-971, UC Berkeley, May 1997.

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