NAG Library Function Document
nag_zheev (f08fnc) computes all the eigenvalues and, optionally, all the eigenvectors of a complex by Hermitian matrix .
||nag_zheev (Nag_OrderType order,
The Hermitian matrix is first reduced to real tridiagonal form, using unitary similarity transformations, and then the 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 http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
order – Nag_OrderTypeInput
: 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
. See Section 188.8.131.52
in the Essential Introduction for a more detailed explanation of the use of this argument.
job – Nag_JobTypeInput
: indicates whether eigenvectors are computed.
- Only eigenvalues are computed.
- Eigenvalues and eigenvectors are computed.
uplo – Nag_UploTypeInput
, the upper triangular part of
If , the lower triangular part of is stored.
n – IntegerInput
On entry: , the order of the matrix .
a – ComplexInput/Output
the dimension, dim
, of the array a
must be at least
If , is stored in .
If , is stored in .
If , the upper triangular part of must be stored and the elements of the array below the diagonal are not referenced.
If , the lower triangular part of must be stored and the elements of the array above the diagonal are not referenced.
, then a
contains the orthonormal eigenvectors of the matrix
, then on exit the lower triangle (if
) or the upper triangle (if
) of a
, including the diagonal, is overwritten.
pda – IntegerInput
: the stride separating row or column elements (depending on the value of order
) in the array a
w[n] – doubleOutput
On exit: the eigenvalues in ascending order.
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
Dynamic memory allocation failed.
On entry, argument had an illegal value.
The algorithm failed to converge; off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
On entry, .
On entry, .
On entry, and .
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
The computed eigenvalues and eigenvectors are exact for a nearby matrix
is the machine precision
. See Section 4.7 of Anderson et al. (1999)
for further details.
8 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 Users' Note
for your implementation for any additional implementation-specific information.
Each eigenvector is normalized so that the element of largest absolute value is real and positive.
The total number of floating-point operations is proportional to .
The real analogue of this function is nag_dsyev (f08fac)
This example finds all the eigenvalues and eigenvectors of the Hermitian matrix
together with approximate error bounds for the computed eigenvalues and eigenvectors.
10.1 Program Text
Program Text (f08fnce.c)
10.2 Program Data
Program Data (f08fnce.d)
10.3 Program Results
Program Results (f08fnce.r)