NAG FL Interfacef08frf (zheevr)

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1Purpose

f08frf computes selected eigenvalues and, optionally, eigenvectors of a complex $n×n$ 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.

2Specification

Fortran Interface
 Subroutine f08frf ( jobz, uplo, n, a, lda, vl, vu, il, iu, m, w, z, ldz, work, info)
 Integer, Intent (In) :: n, lda, il, iu, ldz, lwork, lrwork, liwork Integer, Intent (Inout) :: isuppz(*) Integer, Intent (Out) :: m, iwork(max(1,liwork)), info Real (Kind=nag_wp), Intent (In) :: vl, vu, abstol Real (Kind=nag_wp), Intent (Inout) :: w(*) Real (Kind=nag_wp), Intent (Out) :: rwork(max(1,lrwork)) Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*), z(ldz,*) Complex (Kind=nag_wp), Intent (Out) :: work(max(1,lwork)) Character (1), Intent (In) :: jobz, range, uplo
#include <nag.h>
 void f08frf_ (const char *jobz, const char *range, const char *uplo, const Integer *n, Complex a[], const Integer *lda, const double *vl, const double *vu, const Integer *il, const Integer *iu, const double *abstol, Integer *m, double w[], Complex z[], const Integer *ldz, Integer isuppz[], Complex work[], const Integer *lwork, double rwork[], const Integer *lrwork, Integer iwork[], const Integer *liwork, Integer *info, const Charlen length_jobz, const Charlen length_range, const Charlen length_uplo)
The routine may be called by the names f08frf, nagf_lapackeig_zheevr or its LAPACK name zheevr.

3Description

The Hermitian matrix is first reduced to a real tridiagonal matrix $T$, using unitary similarity transformations. Then whenever possible, f08frf computes the eigenspectrum using Relatively Robust Representations. f08frf computes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are computed from various ‘good’ $LD{L}^{\mathrm{T}}$ 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 the $i$th unreduced block of $T$:
1. (a)compute $T-{\sigma }_{i}I={L}_{i}{D}_{i}{L}_{i}^{\mathrm{T}}$, such that ${L}_{i}{D}_{i}{L}_{i}^{\mathrm{T}}$ is a relatively robust representation,
2. (b)compute the eigenvalues, ${\lambda }_{j}$, of ${L}_{i}{D}_{i}{L}_{i}^{\mathrm{T}}$ to high relative accuracy by the dqds algorithm,
3. (c)if there is a cluster of close eigenvalues, ‘choose’ ${\sigma }_{i}$ close to the cluster, and go to (a),
4. (d)given the approximate eigenvalue ${\lambda }_{j}$ of ${L}_{i}{D}_{i}{L}_{i}^{\mathrm{T}}$, compute the corresponding eigenvector by forming a rank-revealing twisted factorization.
The desired accuracy of the output can be specified by the argument abstol. For more details, see Dhillon (1997) and Parlett and Dhillon (2000).

4References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia https://www.netlib.org/lapack/lug
Barlow J and Demmel J W (1990) Computing accurate eigensystems of scaled diagonally dominant matrices SIAM J. Numer. Anal. 27 762–791
Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices SIAM J. Sci. Statist. Comput. 11 873–912
Dhillon I (1997) A new $\mathit{O}\left({n}^{2}\right)$ algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem Computer Science Division Technical Report No. UCB//CSD-97-971 UC Berkeley
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Parlett B N and Dhillon I S (2000) Relatively robust representations of symmetric tridiagonals Linear Algebra Appl. 309 121–151

5Arguments

1: $\mathbf{jobz}$Character(1) Input
On entry: indicates whether eigenvectors are computed.
${\mathbf{jobz}}=\text{'N'}$
Only eigenvalues are computed.
${\mathbf{jobz}}=\text{'V'}$
Eigenvalues and eigenvectors are computed.
Constraint: ${\mathbf{jobz}}=\text{'N'}$ or $\text{'V'}$.
2: $\mathbf{range}$Character(1) Input
On entry: if ${\mathbf{range}}=\text{'A'}$, all eigenvalues will be found.
If ${\mathbf{range}}=\text{'V'}$, all eigenvalues in the half-open interval $\left({\mathbf{vl}},{\mathbf{vu}}\right]$ will be found.
If ${\mathbf{range}}=\text{'I'}$, the ilth to iuth eigenvalues will be found.
For ${\mathbf{range}}=\text{'V'}$ or $\text{'I'}$ and ${\mathbf{iu}}-{\mathbf{il}}<{\mathbf{n}}-1$, f08jjf and f08jxf are called.
Constraint: ${\mathbf{range}}=\text{'A'}$, $\text{'V'}$ or $\text{'I'}$.
3: $\mathbf{uplo}$Character(1) Input
On entry: if ${\mathbf{uplo}}=\text{'U'}$, the upper triangular part of $A$ is stored.
If ${\mathbf{uplo}}=\text{'L'}$, the lower triangular part of $A$ is stored.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
4: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
5: $\mathbf{a}\left({\mathbf{lda}},*\right)$Complex (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $n×n$ Hermitian matrix $A$.
• If ${\mathbf{uplo}}=\text{'U'}$, the upper triangular part of $A$ must be stored and the elements of the array below the diagonal are not referenced.
• If ${\mathbf{uplo}}=\text{'L'}$, the lower triangular part of $A$ must be stored and the elements of the array above the diagonal are not referenced.
On exit: the lower triangle (if ${\mathbf{uplo}}=\text{'L'}$) or the upper triangle (if ${\mathbf{uplo}}=\text{'U'}$) of a, including the diagonal, is overwritten.
6: $\mathbf{lda}$Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f08frf is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
7: $\mathbf{vl}$Real (Kind=nag_wp) Input
8: $\mathbf{vu}$Real (Kind=nag_wp) Input
On entry: if ${\mathbf{range}}=\text{'V'}$, the lower and upper bounds of the interval to be searched for eigenvalues.
If ${\mathbf{range}}=\text{'A'}$ or $\text{'I'}$, vl and vu are not referenced.
Constraint: if ${\mathbf{range}}=\text{'V'}$, ${\mathbf{vl}}<{\mathbf{vu}}$.
9: $\mathbf{il}$Integer Input
10: $\mathbf{iu}$Integer Input
On entry: if ${\mathbf{range}}=\text{'I'}$, il and iu specify the indices (in ascending order) of the smallest and largest eigenvalues to be returned, respectively.
If ${\mathbf{range}}=\text{'A'}$ or $\text{'V'}$, il and iu are not referenced.
Constraints:
• if ${\mathbf{range}}=\text{'I'}$ and ${\mathbf{n}}=0$, ${\mathbf{il}}=1$ and ${\mathbf{iu}}=0$;
• if ${\mathbf{range}}=\text{'I'}$ and ${\mathbf{n}}>0$, $1\le {\mathbf{il}}\le {\mathbf{iu}}\le {\mathbf{n}}$.
11: $\mathbf{abstol}$Real (Kind=nag_wp) Input
On entry: the absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval $\left[a,b\right]$ of width less than or equal to
 $abstol+ε max(|a|,|b|) ,$
where $\epsilon$ is the machine precision. If abstol is less than or equal to zero, then $\epsilon {‖T‖}_{1}$ will be used in its place, where $T$ is the real tridiagonal matrix obtained by reducing $A$ to tridiagonal form. See Demmel and Kahan (1990).
If high relative accuracy is important, set abstol to , although doing so does not currently guarantee that eigenvalues are computed to high relative accuracy. See Barlow and Demmel (1990) for a discussion of which matrices can define their eigenvalues to high relative accuracy.
12: $\mathbf{m}$Integer Output
On exit: the total number of eigenvalues found. $0\le {\mathbf{m}}\le {\mathbf{n}}$.
If ${\mathbf{range}}=\text{'A'}$, ${\mathbf{m}}={\mathbf{n}}$.
If ${\mathbf{range}}=\text{'I'}$, ${\mathbf{m}}={\mathbf{iu}}-{\mathbf{il}}+1$.
13: $\mathbf{w}\left(*\right)$Real (Kind=nag_wp) array Output
Note: the dimension of the array w must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On exit: the first m elements contain the selected eigenvalues in ascending order.
14: $\mathbf{z}\left({\mathbf{ldz}},*\right)$Complex (Kind=nag_wp) array Output
Note: the second dimension of the array z must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$ if ${\mathbf{jobz}}=\text{'V'}$, and at least $1$ otherwise.
On exit: if ${\mathbf{jobz}}=\text{'V'}$, 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 ${\mathbf{w}}\left(i\right)$.
If ${\mathbf{jobz}}=\text{'N'}$, z is not referenced.
Note:  you must ensure that at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$ columns are supplied in the array z; if ${\mathbf{range}}=\text{'V'}$, the exact value of m is not known in advance and an upper bound of at least n must be used.
15: $\mathbf{ldz}$Integer Input
On entry: the first dimension of the array z as declared in the (sub)program from which f08frf is called.
Constraints:
• if ${\mathbf{jobz}}=\text{'V'}$, ${\mathbf{ldz}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{ldz}}\ge 1$.
16: $\mathbf{isuppz}\left(*\right)$Integer array Output
Note: the dimension of the array isuppz must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2×{\mathbf{m}}\right)$.
On exit: 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 ${\mathbf{isuppz}}\left(2×i-1\right)$ through ${\mathbf{isuppz}}\left(2×i\right)$. Implemented only for ${\mathbf{range}}=\text{'A'}$ or ${\mathbf{range}}=\text{'I'}$ and ${\mathbf{iu}}-{\mathbf{il}}={\mathbf{n}}-1$.
17: $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$Complex (Kind=nag_wp) array Workspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, the real part of ${\mathbf{work}}\left(1\right)$ contains the minimum value of lwork required for optimal performance.
18: $\mathbf{lwork}$Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f08frf is called.
If ${\mathbf{lwork}}=-1$, a workspace query is assumed; the routine only calculates the optimal sizes of the work, rwork and iwork arrays, returns these values as the first entries of the work, rwork and iwork arrays, and no error message related to lwork, lrwork or liwork is issued.
Suggested value: for optimal performance, ${\mathbf{lwork}}\ge \left(\mathit{nb}+1\right)×{\mathbf{n}}$, where $\mathit{nb}$ is the largest optimal block size for f08fsf and for f08fuf.
Constraint: ${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2×{\mathbf{n}}\right)$ or ${\mathbf{lwork}}=-1$.
19: $\mathbf{rwork}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lrwork}}\right)\right)$Real (Kind=nag_wp) array Workspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, ${\mathbf{rwork}}\left(1\right)$ returns the optimal (and minimal) lrwork.
20: $\mathbf{lrwork}$Integer Input
On entry: the dimension of the array rwork as declared in the (sub)program from which f08frf is called.
If ${\mathbf{lrwork}}=-1$, a workspace query is assumed; the routine only calculates the optimal sizes of the work, rwork and iwork arrays, returns these values as the first entries of the work, rwork and iwork arrays, and no error message related to lwork, lrwork or liwork is issued.
Constraint: ${\mathbf{lrwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,24×{\mathbf{n}}\right)$ or ${\mathbf{lrwork}}=-1$.
21: $\mathbf{iwork}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{liwork}}\right)\right)$Integer array Workspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, ${\mathbf{iwork}}\left(1\right)$ returns the optimal (and minimal) liwork.
22: $\mathbf{liwork}$Integer Input
On entry: the dimension of the array iwork as declared in the (sub)program from which f08frf is called.
If ${\mathbf{liwork}}=-1$, a workspace query is assumed; the routine only calculates the optimal sizes of the work, rwork and iwork arrays, returns these values as the first entries of the work, rwork and iwork arrays, and no error message related to lwork, lrwork or liwork is issued.
Constraint: ${\mathbf{liwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,10×{\mathbf{n}}\right)$ or ${\mathbf{liwork}}=-1$.
23: $\mathbf{info}$Integer Output
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{info}}>0$
f08frf failed to converge.

7Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrix $\left(A+E\right)$, where
 $‖E‖2 = O(ε) ‖A‖2 ,$
and $\epsilon$ is the machine precision. See Section 4.7 of Anderson et al. (1999) for further details.

8Parallelism and Performance

f08frf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08frf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The total number of floating-point operations is proportional to ${n}^{3}$.
The real analogue of this routine is f08fdf.

10Example

This example finds the eigenvalues with indices in the range $\left[2,3\right]$, and the corresponding eigenvectors, of the Hermitian matrix
 $A = ( 1 2-i 3-i 4-i 2+i 2 3-2i 4-2i 3+i 3+2i 3 4-3i 4+i 4+2i 4+3i 4 ) .$
Information on required and provided workspace is also output.

10.1Program Text

Program Text (f08frfe.f90)

10.2Program Data

Program Data (f08frfe.d)

10.3Program Results

Program Results (f08frfe.r)