NAG CL Interface
f08gnc (zhpev)

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1 Purpose

f08gnc computes all the eigenvalues and, optionally, all the eigenvectors of a complex n×n Hermitian matrix A in packed storage.

2 Specification

#include <nag.h>
void  f08gnc (Nag_OrderType order, Nag_JobType job, Nag_UploType uplo, Integer n, Complex ap[], double w[], Complex z[], Integer pdz, NagError *fail)
The function may be called by the names: f08gnc, nag_lapackeig_zhpev or nag_zhpev.

3 Description

The Hermitian matrix A is first reduced to real tridiagonal form, using unitary similarity transformations, and then the QR algorithm is applied to the tridiagonal matrix to compute the eigenvalues and (optionally) the eigenvectors.

4 References

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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Arguments

1: order Nag_OrderType Input
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2: job Nag_JobType Input
On entry: indicates whether eigenvectors are computed.
job=Nag_EigVals
Only eigenvalues are computed.
job=Nag_DoBoth
Eigenvalues and eigenvectors are computed.
Constraint: job=Nag_EigVals or Nag_DoBoth.
3: uplo Nag_UploType Input
On entry: if uplo=Nag_Upper, the upper triangular part of A is stored.
If uplo=Nag_Lower, the lower triangular part of A is stored.
Constraint: uplo=Nag_Upper or Nag_Lower.
4: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
5: ap[dim] Complex Input/Output
Note: the dimension, dim, of the array ap must be at least max(1,n×(n+1)/2).
On entry: the upper or lower triangle of the n×n Hermitian matrix A, packed by rows or columns.
The storage of elements Aij depends on the order and uplo arguments as follows:
if order=Nag_ColMajor and uplo=Nag_Upper,
Aij is stored in ap[(j-1)×j/2+i-1], for ij;
if order=Nag_ColMajor and uplo=Nag_Lower,
Aij is stored in ap[(2n-j)×(j-1)/2+i-1], for ij;
if order=Nag_RowMajor and uplo=Nag_Upper,
Aij is stored in ap[(2n-i)×(i-1)/2+j-1], for ij;
if order=Nag_RowMajor and uplo=Nag_Lower,
Aij is stored in ap[(i-1)×i/2+j-1], for ij.
On exit: ap is overwritten by the values generated during the reduction to tridiagonal form. The elements of the diagonal and the off-diagonal of the tridiagonal matrix overwrite the corresponding elements of A.
6: w[n] double Output
On exit: the eigenvalues in ascending order.
7: z[dim] Complex Output
Note: the dimension, dim, of the array z must be at least
  • max(1,pdz×n) when job=Nag_DoBoth;
  • 1 otherwise.
The (i,j)th element of the matrix Z is stored in
  • z[(j-1)×pdz+i-1] when order=Nag_ColMajor;
  • z[(i-1)×pdz+j-1] when order=Nag_RowMajor.
On exit: if job=Nag_DoBoth, z contains the orthonormal eigenvectors of the matrix A, with the ith column of Z holding the eigenvector associated with w[i-1].
If job=Nag_EigVals, z is not referenced.
8: pdz Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array z.
Constraints:
  • if job=Nag_DoBoth, pdz max(1,n) ;
  • otherwise pdz1.
9: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_CONVERGENCE
The algorithm failed to converge; value off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
NE_ENUM_INT_2
On entry, job=value, pdz=value and n=value.
Constraint: if job=Nag_DoBoth, pdz max(1,n) ;
otherwise pdz1.
NE_INT
On entry, n=value.
Constraint: n0.
On entry, pdz=value.
Constraint: pdz>0.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

7 Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrix (A+E), where
E2 = O(ε) A2 ,  
and ε is the machine precision. See Section 4.7 of Anderson et al. (1999) for further details.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08gnc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08gnc 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

Each eigenvector is normalized so that the element of largest absolute value is real.
The total number of floating-point operations is proportional to n3.
The real analogue of this function is f08gac.

10 Example

This example finds all the eigenvalues 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 ) ,  
together with approximate error bounds for the computed eigenvalues.

10.1 Program Text

Program Text (f08gnce.c)

10.2 Program Data

Program Data (f08gnce.d)

10.3 Program Results

Program Results (f08gnce.r)