NAG CL Interface
f08juc (zpteqr)

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1 Purpose

f08juc computes all the eigenvalues and, optionally, all the eigenvectors of a complex Hermitian positive definite matrix which has been reduced to tridiagonal form.

2 Specification

#include <nag.h>
void  f08juc (Nag_OrderType order, Nag_ComputeZType compz, Integer n, double d[], double e[], Complex z[], Integer pdz, NagError *fail)
The function may be called by the names: f08juc, nag_lapackeig_zpteqr or nag_zpteqr.

3 Description

f08juc computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric positive definite tridiagonal matrix T. In other words, it can compute the spectral factorization of T as
T=ZΛZT,  
where Λ is a diagonal matrix whose diagonal elements are the eigenvalues λi, and Z is the orthogonal matrix whose columns are the eigenvectors zi. Thus
Tzi=λizi,  i=1,2,,n.  
The function stores the real orthogonal matrix Z in a complex array, so that it may be used to compute all the eigenvalues and eigenvectors of a complex Hermitian positive definite matrix A which has been reduced to tridiagonal form T:
A =QTQH, where ​Q​ is unitary =(QZ)Λ(QZ)H.  
In this case, the matrix Q must be formed explicitly and passed to f08juc, which must be called with compz=Nag_UpdateZ. The functions which must be called to perform the reduction to tridiagonal form and form Q are:
full matrix f08fsc and f08ftc
full matrix, packed storage f08gsc and f08gtc
band matrix f08hsc with vect=Nag_FormQ.
f08juc first factorizes T as LDLH where L is unit lower bidiagonal and D is diagonal. It forms the bidiagonal matrix B=LD12, and then calls f08msc to compute the singular values of B which are the same as the eigenvalues of T. The method used by the function allows high relative accuracy to be achieved in the small eigenvalues of T. The eigenvectors are normalized so that zi2=1, but are determined only to within a complex factor of absolute value 1.

4 References

Barlow J and Demmel J W (1990) Computing accurate eigensystems of scaled diagonally dominant matrices SIAM J. Numer. Anal. 27 762–791

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_ComputeZType Input
On entry: indicates whether the eigenvectors are to be computed.
compz=Nag_NotZ
Only the eigenvalues are computed (and the array z is not referenced).
compz=Nag_UpdateZ
The eigenvalues and eigenvectors of A are computed (and the array z must contain the matrix Q on entry).
compz=Nag_InitZ
The eigenvalues and eigenvectors of T are computed (and the array z is initialized by the function).
Constraint: compz=Nag_NotZ, Nag_UpdateZ or Nag_InitZ.
3: n Integer Input
On entry: n, the order of the 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 T.
On exit: the n eigenvalues in descending order, unless fail.code= NE_CONVERGENCE or NE_POS_DEF, in which case d is overwritten.
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 off-diagonal elements of the tridiagonal matrix T.
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_UpdateZ or Nag_InitZ;
  • 1 when compz=Nag_NotZ.
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 entry: if compz=Nag_UpdateZ, z must contain the unitary matrix Q from the reduction to tridiagonal form.
If compz=Nag_InitZ, z need not be set.
On exit: if compz=Nag_UpdateZ or Nag_InitZ, the n required orthonormal eigenvectors stored as columns of Z; the ith column corresponds to the ith eigenvalue, where i=1,2,,n, unless fail.code= NE_CONVERGENCE or NE_POS_DEF.
If compz=Nag_NotZ, 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_UpdateZ or Nag_InitZ, pdz max(1,n) ;
  • if compz=Nag_NotZ, 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 to compute the singular values of the Cholesky factor B failed to converge; value off-diagonal elements did not converge to zero.
NE_ENUM_INT_2
On entry, compz=value, pdz=value and n=value.
Constraint: if compz=Nag_UpdateZ or Nag_InitZ, pdz max(1,n) ;
if compz=Nag_NotZ, 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.
NE_POS_DEF
The leading minor of order value is not positive definite and the Cholesky factorization of T could not be completed. Hence T itself is not positive definite.

7 Accuracy

The eigenvalues and eigenvectors of T are computed to high relative accuracy which means that if they vary widely in magnitude, then any small eigenvalues (and corresponding eigenvectors) will be computed more accurately than, for example, with the standard QR method. However, the reduction to tridiagonal form (prior to calling the function) may exclude the possibility of obtaining high relative accuracy in the small eigenvalues of the original matrix if its eigenvalues vary widely in magnitude.
To be more precise, let H be the tridiagonal matrix defined by H=DTD, where D is diagonal with dii = t ii -12 , and hii = 1 for all i. If λi is an exact eigenvalue of T and λ~i is the corresponding computed value, then
|λ~i-λi| c (n) ε κ2 (H) λi  
where c(n) is a modestly increasing function of n, ε is the machine precision, and κ2(H) is the condition number of H with respect to inversion defined by: κ2(H)=H·H-1.
If zi is the corresponding exact eigenvector of T, 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) ε κ2 (H) relgapi  
where relgapi is the relative gap between λi and the other eigenvalues, defined by
relgapi = min ij |λi-λj| (λi+λj) .  

8 Parallelism and Performance

f08juc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08juc 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

The total number of real floating-point operations is typically about 30n2 if compz=Nag_NotZ and about 12n3 if compz=Nag_UpdateZ or Nag_InitZ, but depends on how rapidly the algorithm converges. When compz=Nag_NotZ, the operations are all performed in scalar mode; the additional operations to compute the eigenvectors when compz=Nag_UpdateZ or Nag_InitZ can be vectorized and on some machines may be performed much faster.
The real analogue of this function is f08jgc.

10 Example

This example computes all the eigenvalues and eigenvectors of the complex Hermitian positive definite matrix A, where
A = ( 6.02+0.00i -0.45+0.25i -1.30+1.74i 1.45-0.66i -0.45-0.25i 2.91+0.00i 0.05+1.56i -1.04+1.27i -1.30-1.74i 0.05-1.56i 3.29+0.00i 0.14+1.70i 1.45+0.66i -1.04-1.27i 0.14-1.70i 4.18+0.00i ) .  

10.1 Program Text

Program Text (f08juce.c)

10.2 Program Data

Program Data (f08juce.d)

10.3 Program Results

Program Results (f08juce.r)