NAG CL Interface
f08jdc (dstevr)

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1 Purpose

f08jdc computes selected eigenvalues and, optionally, eigenvectors of a real n×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.

2 Specification

#include <nag.h>
void  f08jdc (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)
The function may be called by the names: f08jdc, nag_lapackeig_dstevr or nag_dstevr.

3 Description

Whenever possible f08jdc computes the eigenspectrum using Relatively Robust Representations. 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:
  1. (a)compute T - σi I = Li Di LiT , such that Li Di LiT is a relatively robust representation,
  2. (b)compute the eigenvalues, λj, of Li Di LiT to high relative accuracy by the dqds algorithm,
  3. (c)if there is a cluster of close eigenvalues, ‘choose’ σi close to the cluster, and go to (a),
  4. (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).

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
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 O(n2) 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

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: job Nag_JobType Input
On entry: indicates whether eigenvectors are computed.
job=Nag_EigVals
Only eigenvalues are computed.
job=Nag_DoBoth
Eigenvalues and eigenvectors are computed.
Constraint: job=Nag_EigVals or Nag_DoBoth.
3: range Nag_RangeType Input
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 Integer Input
On entry: n, the order of the matrix.
Constraint: n0.
5: d[dim] double Input/Output
Note: the dimension, dim, of the array d must be at least max(1,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] double Input/Output
Note: the dimension, dim, of the array e must be at least max(1,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 double Input
8: vu double Input
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 Integer Input
10: iu Integer Input
On entry: if range=Nag_Indices, il and iu specify the indices (in ascending order) of the smallest and largest eigenvalues to be returned, respectively.
If range=Nag_AllValues or Nag_Interval, il and iu are not referenced.
Constraints:
  • 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 double Input
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 max(|a|,|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] double Output
Note: the dimension, dim, of the array w must be at least max(1,n).
On exit: the first m elements contain the selected eigenvalues in ascending order.
14: z[dim] double Output
Note: the dimension, dim, of the array z must be at least
  • max(1,pdz×n) when job=Nag_DoBoth;
  • 1 otherwise.
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 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 Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array z.
Constraints:
  • if job=Nag_DoBoth, pdz max(1,n) ;
  • otherwise pdz1.
16: isuppz[dim] Integer Output
Note: the dimension, dim, of the array isuppz must be at least max(1,2×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 range=Nag_Indices and iu-il=n-1.
17: 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_ENUM_INT_2
On entry, job=value, pdz=value and n=value.
Constraint: if job=Nag_DoBoth, pdz max(1,n) ;
otherwise pdz1.
NE_ENUM_INT_3
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 .
NE_ENUM_REAL_2
On entry, range=value, vl=value and vu=value.
Constraint: if range=Nag_Interval, vl<vu.
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.
An internal error has occurred in this function. Please refer to fail in f08jjc.
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 (A+E), where
E2 = O(ε) A2 ,  
and ε is the machine precision. See Section 4.7 of Anderson et al. (1999) for further details.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08jdc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
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.

9 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.

10 Example

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 ) .  

10.1 Program Text

Program Text (f08jdce.c)

10.2 Program Data

Program Data (f08jdce.d)

10.3 Program Results

Program Results (f08jdce.r)