nag_dsteqr (f08jec) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric tridiagonal matrix, or of a real symmetric matrix which has been reduced to tridiagonal form.

nag_dsteqr (f08jec) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric 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 may also be used to compute all the eigenvalues and eigenvectors of a real symmetric matrix  which has been reduced to tridiagonal form :

In this case, the matrix  must be formed explicitly and passed to nag_dsteqr (f08jec), which must be called with . The functions which must be called to perform the reduction to tridiagonal form and form  are:

nag_dsteqr (f08jec) uses the implicitly shifted  algorithm, switching between the  and  variants in order to handle graded matrices effectively (see Greenbaum and Dongarra (1980)). The eigenvectors are normalized so that , but are determined only to within a factor .

If only the eigenvalues of  are required, it is more efficient to call f08jfc (nag_dsterf) instead. If  is positive-definite, small eigenvalues can be computed more accurately by f08jgc (nag_dpteqr).

"NE_ALLOC_FAIL"

Dynamic memory allocation failed.

"NE_BAD_PARAM"

On entry, argument  had an illegal value.

"NE_CONVERGENCE"

 The algorithm has failed to find all the eigenvalues after a total of  iterations;  off-diagonal elements have not converged to zero. The arguments d and e contain the diagonal and off-diagonal elements, respectively, of a tridiagonal matrix orthogonally similar to .

"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 computed eigenvalues and eigenvectors are exact for a nearby matrix , where

and  is the machine precision.

If  is an exact eigenvalue and  is the corresponding computed value, then

where  is a modestly increasing function of .

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

Thus the accuracy of a computed eigenvector depends on the gap between its eigenvalue and all the other eigenvalues.

The total number of 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 complex analogue of this function is f08jsc (nag_zsteqr).