NAG CL Interface
f08jvc (zstedc)

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1 Purpose

f08jvc computes all the eigenvalues and, optionally, all the eigenvectors of a real n×n symmetric tridiagonal matrix, or of a complex full or banded Hermitian matrix which has been reduced to tridiagonal form.

2 Specification

#include <nag.h>
void  f08jvc (Nag_OrderType order, Nag_ComputeEigVecsType compz, Integer n, double d[], double e[], Complex z[], Integer pdz, NagError *fail)
The function may be called by the names: f08jvc, nag_lapackeig_zstedc or nag_zstedc.

3 Description

f08jvc computes all the eigenvalues and, optionally, the eigenvectors of a real symmetric tridiagonal matrix T. That is, the function computes the spectral factorization of T given by
T = Z Λ ZT ,  
where Λ is a diagonal matrix whose diagonal elements are the eigenvalues, λi, of T and Z is an orthogonal matrix whose columns are the eigenvectors, zi, of T. Thus
Tzi = λi zi ,   i = 1,2,,n .  
The function may also be used to compute all the eigenvalues and eigenvectors of a complex full, or banded, Hermitian matrix A which has been reduced to real tridiagonal form T as
A = QTQH ,  
where Q is unitary. The spectral factorization of A is then given by
A = (QZ) Λ (QZ)H .  
In this case Q must be formed explicitly and passed to f08jvc in the array z, and the function called with compz=Nag_OrigEigVecs. Functions which may be called to form T and Q are
full matrix f08fsc and f08ftc
full matrix, packed storage f08gsc and f08gtc
band matrix f08hsc, with vect=Nag_FormQ
When only eigenvalues are required then this function calls f08jfc to compute the eigenvalues of the tridiagonal matrix T, but when eigenvectors of T are also required and the matrix is not too small, then a divide and conquer method is used, which can be much faster than f08jsc, although more storage is required.

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: compz Nag_ComputeEigVecsType Input
On entry: indicates whether the eigenvectors are to be computed.
compz=Nag_NotEigVecs
Only the eigenvalues are computed (and the array z is not referenced).
compz=Nag_OrigEigVecs
The eigenvalues and eigenvectors of A are computed (and the array z must contain the matrix Q on entry).
compz=Nag_TridiagEigVecs
The eigenvalues and eigenvectors of T are computed (and the array z is initialized by the function).
Constraint: compz=Nag_NotEigVecs, Nag_OrigEigVecs or Nag_TridiagEigVecs.
3: n Integer Input
On entry: n, the order of the symmetric tridiagonal matrix T.
Constraint: n0.
4: d[dim] double Input/Output
Note: the dimension, dim, of the array d must be at least max(1,n).
On entry: the diagonal elements of the tridiagonal matrix.
On exit: if fail.code= NE_NOERROR, the eigenvalues in ascending order.
5: e[dim] double Input/Output
Note: the dimension, dim, of the array e must be at least max(1,n-1).
On entry: the subdiagonal elements of the tridiagonal matrix.
On exit: e is overwritten.
6: z[dim] Complex Input/Output
Note: the dimension, dim, of the array z must be at least
  • max(1,pdz×n) when compz=Nag_OrigEigVecs or Nag_TridiagEigVecs;
  • 1 otherwise.
if compz=Nag_OrigEigVecs then the(i,j)th element of the matrix Q is stored in
  • z[(j-1)×pdz+i-1] when order=Nag_ColMajor;
  • z[(i-1)×pdz+j-1] when order=Nag_RowMajor.
On entry: if compz=Nag_OrigEigVecs, z must contain the unitary matrix Q used in the reduction to tridiagonal form.
On exit: if compz=Nag_OrigEigVecs, z contains the orthonormal eigenvectors of the original Hermitian matrix A, and if compz=Nag_TridiagEigVecs, z contains the orthonormal eigenvectors of the symmetric tridiagonal matrix T.
If compz=Nag_NotEigVecs, z is not referenced.
7: pdz Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array z.
Constraints:
  • if compz=Nag_OrigEigVecs or Nag_TridiagEigVecs, pdz max(1,n) ;
  • otherwise pdz1.
8: 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 compute an eigenvalue while working on the submatrix lying in rows and columns value/(n+1) through value mod (n+1).
NE_ENUM_INT_2
On entry, compz=value, pdz=value and n=value.
Constraint: if compz=Nag_OrigEigVecs or Nag_TridiagEigVecs, 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 (T+E), where
E2 = O(ε) T2 ,  
and ε is the machine precision.
If λi is an exact eigenvalue and λ~i is the corresponding computed value, then
|λ~i-λi| c (n) ε T2 ,  
where c(n) is a modestly increasing function of n.
If zi is the corresponding exact eigenvector, and z~i is the corresponding computed eigenvector, then the angle θ(z~i,zi) between them is bounded as follows:
θ (z~i,zi) c(n)εT2 minij|λi-λj| .  
Thus the accuracy of a computed eigenvector depends on the gap between its eigenvalue and all the other eigenvalues.
See Section 4.7 of Anderson et al. (1999) for further details. See also f08flc.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08jvc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08jvc 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

If only eigenvalues are required, the total number of floating-point operations is approximately proportional to n2. When eigenvectors are required the number of operations is bounded above by approximately the same number of operations as f08jsc, but for large matrices f08jvc is usually much faster.
The real analogue of this function is f08jhc.

10 Example

This example finds all the eigenvalues and eigenvectors of the Hermitian band matrix
A = ( -3.13i+0.00 1.94-2.10i -3.40+0.25i 0.00i+0.00 1.94+2.10i -1.91i+0.00 -0.82-0.89i -0.67+0.34i -3.40-0.25i -0.82+0.89i -2.87i+0.00 -2.10-0.16i 0.00i+0.00 -0.67-0.34i -2.10+0.16i 0.50i+0.00 ) .  
A is first reduced to tridiagonal form by a call to f08hsc.

10.1 Program Text

Program Text (f08jvce.c)

10.2 Program Data

Program Data (f08jvce.d)

10.3 Program Results

Program Results (f08jvce.r)