NAG CL Interface
f08jcc (dstevd)

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1 Purpose

f08jcc computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric tridiagonal matrix. If the eigenvectors are requested, then it uses a divide-and-conquer algorithm to compute eigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal–Walker–Kahan variant of the QL or QR algorithm.

2 Specification

#include <nag.h>
void  f08jcc (Nag_OrderType order, Nag_JobType job, Integer n, double d[], double e[], double z[], Integer pdz, NagError *fail)
The function may be called by the names: f08jcc, nag_lapackeig_dstevd or nag_dstevd.

3 Description

f08jcc computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric 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.  

4 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

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_DoNothing
Only eigenvalues are computed.
job=Nag_EigVecs
Eigenvalues and eigenvectors are computed.
Constraint: job=Nag_DoNothing or Nag_EigVecs.
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 n diagonal elements of the tridiagonal matrix T.
On exit: the eigenvalues of the matrix T in ascending order.
5: e[dim] double Input/Output
Note: the dimension, dim, of the array e must be at least max(1,n).
On entry: the n-1 off-diagonal elements of the tridiagonal matrix T. The nth element of this array is used as workspace.
On exit: e is overwritten with intermediate results.
6: z[dim] double Output
Note: the dimension, dim, of the array z must be at least
  • max(1,pdz×n) when job=Nag_EigVecs;
  • 1 when job=Nag_DoNothing.
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_EigVecs, z is overwritten by the orthogonal matrix Z which contains the eigenvectors of T.
If job=Nag_DoNothing, 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 job=Nag_EigVecs, pdz max(1,n) ;
  • if job=Nag_DoNothing, 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 failed to converge; value off-diagonal elements of e did not converge to zero.
NE_ENUM_INT_2
On entry, job=value, pdz=value and n=value.
Constraint: if job=Nag_EigVecs, pdz max(1,n) ;
if job=Nag_DoNothing, 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.

7 Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrix (T+E), where
E2 = O(ε) T2 ,  
and ε is the machine precision.
If λi is an exact eigenvalue and λ~i is the corresponding computed value, then
|λ~i-λi| c (n) ε T2 ,  
where c(n) is a modestly increasing function of n.
If zi is the corresponding exact eigenvector, 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 T2 min ij |λi-λj| .  
Thus the accuracy of a computed eigenvector depends on the gap between its eigenvalue and all the other eigenvalues.

8 Parallelism and Performance

f08jcc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08jcc 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

There is no complex analogue of this function.

10 Example

This example computes all the eigenvalues and eigenvectors of the symmetric tridiagonal matrix T, where
T = ( 1.0 1.0 0.0 0.0 1.0 4.0 2.0 0.0 0.0 2.0 9.0 3.0 0.0 0.0 3.0 16.0 ) .  

10.1 Program Text

Program Text (f08jcce.c)

10.2 Program Data

Program Data (f08jcce.d)

10.3 Program Results

Program Results (f08jcce.r)