NAG Library Function Document

nag_zheev (f08fnc)


    1  Purpose
    7  Accuracy


nag_zheev (f08fnc) computes all the eigenvalues and, optionally, all the eigenvectors of a complex n by n Hermitian matrix A.


#include <nag.h>
#include <nagf08.h>
void  nag_zheev (Nag_OrderType order, Nag_JobType job, Nag_UploType uplo, Integer n, Complex a[], Integer pda, double w[], NagError *fail)


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.


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


1:     order Nag_OrderTypeInput
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 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     job Nag_JobTypeInput
On entry: indicates whether eigenvectors are computed.
Only eigenvalues are computed.
Eigenvalues and eigenvectors are computed.
Constraint: job=Nag_EigVals or Nag_DoBoth.
3:     uplo Nag_UploTypeInput
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 IntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
5:     a[dim] ComplexInput/Output
Note: the dimension, dim, of the array a must be at least max1,pda×n.
On entry: the n by n Hermitian matrix A.
If order=Nag_ColMajor, Aij is stored in a[j-1×pda+i-1].
If order=Nag_RowMajor, Aij is stored in a[i-1×pda+j-1].
If uplo=Nag_Upper, the upper triangular part of A must be stored and the elements of the array below the diagonal are not referenced.
If uplo=Nag_Lower, the lower triangular part of A must be stored and the elements of the array above the diagonal are not referenced.
On exit: if job=Nag_DoBoth, a contains the orthonormal eigenvectors of the matrix A.
If job=Nag_EigVals then on exit the lower triangle (if uplo=Nag_Lower) or the upper triangle (if uplo=Nag_Upper) of a, including the diagonal, is overwritten.
6:     pda IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: pdamax1,n.
7:     w[n] doubleOutput
On exit: the eigenvalues in ascending order.
8:     fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

Error Indicators and Warnings

Dynamic memory allocation failed.
See Section in How to Use the NAG Library and its Documentation for further information.
On entry, argument value had an illegal value.
The algorithm failed to converge; value off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
On entry, pda=value and n=value.
Constraint: pdamax1,n.
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.


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

Parallelism and Performance

nag_zheev (f08fnc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zheev (f08fnc) 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.

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 nag_dsyev (f08fac).


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

Program Text

Program Text (f08fnce.c)

Program Data

Program Data (f08fnce.d)

Program Results

Program Results (f08fnce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017