NAG Library Function Document

nag_dstevr (f08jdc)


    1  Purpose
    7  Accuracy


nag_dstevr (f08jdc) computes selected eigenvalues and, optionally, eigenvectors of a real n by n symmetric tridiagonal matrix T. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.


#include <nag.h>
#include <nagf08.h>
void  nag_dstevr (Nag_OrderType order, Nag_JobType job, Nag_RangeType range, Integer n, double d[], double e[], double vl, double vu, Integer il, Integer iu, double abstol, Integer *m, double w[], double z[], Integer pdz, Integer isuppz[], NagError *fail)


Whenever possible nag_dstevr (f08jdc) computes the eigenspectrum using Relatively Robust Representations. nag_dstevr (f08jdc) computes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are computed from various ‘good’ LDLT representations (also known as Relatively Robust Representations). Gram–Schmidt orthogonalization is avoided as far as possible. More specifically, the various steps of the algorithm are as follows. For the ith unreduced block of T:
(a) compute T - σi I = Li Di LiT , such that Li Di LiT  is a relatively robust representation,
(b) compute the eigenvalues, λj, of Li Di LiT  to high relative accuracy by the dqds algorithm,
(c) if there is a cluster of close eigenvalues, ‘choose’ σi close to the cluster, and go to (a),
(d) given the approximate eigenvalue λj of Li Di LiT , compute the corresponding eigenvector by forming a rank-revealing twisted factorization.
The desired accuracy of the output can be specified by the argument abstol. For more details, see Dhillon (1997) and Parlett and Dhillon (2000).


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
Barlow J and Demmel J W (1990) Computing accurate eigensystems of scaled diagonally dominant matrices SIAM J. Numer. Anal. 27 762–791
Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices SIAM J. Sci. Statist. Comput. 11 873–912
Dhillon I (1997) A new On2 algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem Computer Science Division Technical Report No. UCB//CSD-97-971 UC Berkeley
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Parlett B N and Dhillon I S (2000) Relatively robust representations of symmetric tridiagonals Linear Algebra Appl. 309 121–151


1:     order Nag_OrderTypeInput
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 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     job Nag_JobTypeInput
On entry: indicates whether eigenvectors are computed.
Only eigenvalues are computed.
Eigenvalues and eigenvectors are computed.
Constraint: job=Nag_EigVals or Nag_DoBoth.
3:     range Nag_RangeTypeInput
On entry: if range=Nag_AllValues, all eigenvalues will be found.
If range=Nag_Interval, all eigenvalues in the half-open interval vl,vu will be found.
If range=Nag_Indices, the ilth to iuth eigenvalues will be found.
Constraint: range=Nag_AllValues, Nag_Interval or Nag_Indices.
4:     n IntegerInput
On entry: n, the order of the matrix.
Constraint: n0.
5:     d[dim] doubleInput/Output
Note: the dimension, dim, of the array d must be at least max1,n.
On entry: the n diagonal elements of the tridiagonal matrix T.
On exit: may be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues.
6:     e[dim] doubleInput/Output
Note: the dimension, dim, of the array e must be at least max1,n-1.
On entry: the n-1 subdiagonal elements of the tridiagonal matrix T.
On exit: may be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues.
7:     vl doubleInput
8:     vu doubleInput
On entry: if range=Nag_Interval, the lower and upper bounds of the interval to be searched for eigenvalues.
If range=Nag_AllValues or Nag_Indices, vl and vu are not referenced.
Constraint: if range=Nag_Interval, vl<vu.
9:     il IntegerInput
10:   iu IntegerInput
On entry: if range=Nag_Indices, the indices (in ascending order) of the smallest and largest eigenvalues to be returned.
If range=Nag_AllValues or Nag_Interval, il and iu are not referenced.
  • if range=Nag_Indices and n=0, il=1 and iu=0;
  • if range=Nag_Indices and n>0, 1 il iu n .
11:   abstol doubleInput
On entry: the absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval a,b  of width less than or equal to
abstol+ε maxa,b ,  
where ε  is the machine precision. If abstol is less than or equal to zero, then ε T1  will be used in its place. See Demmel and Kahan (1990).
If high relative accuracy is important, set abstol to nag_real_safe_small_number  , although doing so does not currently guarantee that eigenvalues are computed to high relative accuracy. See Barlow and Demmel (1990) for a discussion of which matrices can define their eigenvalues to high relative accuracy.
12:   m Integer *Output
On exit: the total number of eigenvalues found. 0mn.
If range=Nag_AllValues, m=n.
If range=Nag_Indices, m=iu-il+1.
13:   w[dim] doubleOutput
Note: the dimension, dim, of the array w must be at least max1,n.
On exit: the first m elements contain the selected eigenvalues in ascending order.
14:   z[dim] doubleOutput
Note: the dimension, dim, of the array z must be at least
  • max1,pdz×n when job=Nag_DoBoth;
  • 1 otherwise.
The i,jth 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 exit: if job=Nag_DoBoth, the first m columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the ith column of Z holding the eigenvector associated with w[i-1].
If job=Nag_EigVals, z is not referenced.
15:   pdz IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array z.
  • if job=Nag_DoBoth, pdz max1,n ;
  • otherwise pdz1.
16:   isuppz[dim] IntegerOutput
Note: the dimension, dim, of the array isuppz must be at least max1,2×m.
On exit: the support of the eigenvectors in z, i.e., the indices indicating the nonzero elements in z. The ith eigenvector is nonzero only in elements isuppz[2×i-2] through isuppz[2×i-1]. Implemented only for range=Nag_AllValues or Nag_Indices and iu-il=n-1.
17:   fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

Error Indicators and Warnings

Dynamic memory allocation failed.
See Section in How to Use the NAG Library and its Documentation for further information.
On entry, argument value had an illegal value.
On entry, job=value, pdz=value and n=value.
Constraint: if job=Nag_DoBoth, pdz max1,n ;
otherwise pdz1.
On entry, range=value, il=value, iu=value and n=value.
Constraint: if range=Nag_Indices and n=0, il=1 and iu=0;
if range=Nag_Indices and n>0, 1 il iu n .
On entry, range=value, vl=value and vu=value.
Constraint: if range=Nag_Interval, vl<vu.
On entry, n=value.
Constraint: n0.
On entry, pdz=value.
Constraint: pdz>0.
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
An internal error has occurred in this function. Please refer to fail in nag_dstebz (f08jjc).
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.


The computed eigenvalues and eigenvectors are exact for a nearby matrix A+E, where
E2 = Oε A2 ,  
and ε is the machine precision. See Section 4.7 of Anderson et al. (1999) for further details.

Parallelism and Performance

nag_dstevr (f08jdc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dstevr (f08jdc) 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.

Further Comments

The total number of floating-point operations is proportional to n2 if job=Nag_EigVals and is proportional to n3 if job=Nag_DoBoth and range=Nag_AllValues, otherwise the number of floating-point operations will depend upon the number of computed eigenvectors.


This example finds the eigenvalues with indices in the range 2,3 , and the corresponding eigenvectors, of the symmetric tridiagonal matrix
T = 1 1 0 0 1 4 2 0 0 2 9 3 0 0 3 16 .  

Program Text

Program Text (f08jdce.c)

Program Data

Program Data (f08jdce.d)

Program Results

Program Results (f08jdce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017