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

nag_zpteqr (f08juc) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric positive-definite tridiagonal matrix . In other words, it can compute the spectral factorization of  as

where  is a diagonal matrix whose diagonal elements are the eigenvalues , and  is the orthogonal matrix whose columns are the eigenvectors . Thus

The function stores the real orthogonal matrix  in a complex array, so that it may be used to compute all the eigenvalues and eigenvectors of a complex Hermitian positive-definite matrix  which has been reduced to tridiagonal form :

In this case, the matrix  must be formed explicitly and passed to nag_zpteqr (f08juc), which must be called with . The functions which must be called to perform the reduction to tridiagonal form and form  are:

nag_zpteqr (f08juc) first factorizes  as  where  is unit lower bidiagonal and  is diagonal. It forms the bidiagonal matrix , and then calls f08msc (nag_zbdsqr) to compute the singular values of  which are the same as the eigenvalues of . The method used by the function allows high relative accuracy to be achieved in the small eigenvalues of . The eigenvectors are normalized so that , but are determined only to within a complex factor of absolute value 1.

"NE_ALLOC_FAIL"

Dynamic memory allocation failed.

"NE_BAD_PARAM"

On entry, argument  had an illegal value.

"NE_CONVERGENCE"

 The algorithm to compute the singular values of the Cholesky factor failed to converge;  off-diagonalelements did not converge to zero.

 The leading minor of order  is not positive-definiteand the Cholesky factorization of  could not becompleted. Hence  itself is not positive-definite.

"NE_INT"

On entry, . Constraint: .

"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 consult NAG for assistance.

The eigenvalues and eigenvectors of  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  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  be the tridiagonal matrix defined by , where  is diagonal with , and  for all . If  is an exact eigenvalue of  and  is the corresponding computed value, then

where  is a modestly increasing function of ,  is the machine precision, and  is the condition number of  with respect to inversion defined by: .

If  is the corresponding exact eigenvector of , and  is the corresponding computed eigenvector, then the angle  between them is bounded as follows:

where  is the relative gap between  and the other eigenvalues, defined by

The total number of real floating-point operations is typically about  if  and about  if   "Nag_UpdateZ" or "Nag_InitZ", but depends on how rapidly the algorithm converges. When , the operations are all performed in scalar mode; the additional operations to compute the eigenvectors when   "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 (nag_dpteqr).