NAG CL Interface
f08hcc (dsbevd)

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1 Purpose

f08hcc computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric band 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  f08hcc (Nag_OrderType order, Nag_JobType job, Nag_UploType uplo, Integer n, Integer kd, double ab[], Integer pdab, double w[], double z[], Integer pdz, NagError *fail)
The function may be called by the names: f08hcc, nag_lapackeig_dsbevd or nag_dsbevd.

3 Description

f08hcc computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric band matrix A. In other words, it can compute the spectral factorization of A as
A=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
Azi=λizi,  i=1,2,,n.  

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
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: uplo Nag_UploType Input
On entry: indicates whether the upper or lower triangular part of A is stored.
uplo=Nag_Upper
The upper triangular part of A is stored.
uplo=Nag_Lower
The lower triangular part of A is stored.
Constraint: uplo=Nag_Upper or Nag_Lower.
4: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
5: kd Integer Input
On entry: if uplo=Nag_Upper, the number of superdiagonals, kd, of the matrix A.
If uplo=Nag_Lower, the number of subdiagonals, kd, of the matrix A.
Constraint: kd0.
6: ab[dim] double Input/Output
Note: the dimension, dim, of the array ab must be at least max(1,pdab×n).
On entry: the upper or lower triangle of the n×n symmetric band matrix A.
This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of Aij, depends on the order and uplo arguments as follows:
if order=Nag_ColMajor and uplo=Nag_Upper,
Aij is stored in ab[kd+i-j+(j-1)×pdab], for j=1,,n and i=max(1,j-kd),,j;
if order=Nag_ColMajor and uplo=Nag_Lower,
Aij is stored in ab[i-j+(j-1)×pdab], for j=1,,n and i=j,,min(n,j+kd);
if order=Nag_RowMajor and uplo=Nag_Upper,
Aij is stored in ab[j-i+(i-1)×pdab], for i=1,,n and j=i,,min(n,i+kd);
if order=Nag_RowMajor and uplo=Nag_Lower,
Aij is stored in ab[kd+j-i+(i-1)×pdab], for i=1,,n and j=max(1,i-kd),,i.
On exit: ab is overwritten by values generated during the reduction to tridiagonal form.
The first superdiagonal or subdiagonal and the diagonal of the tridiagonal matrix T are returned in ab using the same storage format as described above.
7: pdab Integer Input
On entry: the stride separating row or column elements (depending on the value of order) of the matrix A in the array ab.
Constraint: pdabkd+1.
8: w[dim] double Output
Note: the dimension, dim, of the array w must be at least max(1,n).
On exit: the eigenvalues of the matrix A in ascending order.
9: 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 A. The ith column of Z contains the eigenvector which corresponds to the eigenvalue w[i-1].
If job=Nag_DoNothing, z is not referenced.
10: 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.
11: 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
If fail.errnum=value and job=Nag_DoNothing, the algorithm failed to converge; value elements of an intermediate tridiagonal form did not converge to zero; if fail.errnum=value and job=Nag_EigVecs, then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and column value/(n+1) through value mod (n+1).
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, kd=value.
Constraint: kd0.
On entry, n=value.
Constraint: n0.
On entry, pdab=value.
Constraint: pdab>0.
On entry, pdz=value.
Constraint: pdz>0.
NE_INT_2
On entry, pdab=value and kd=value.
Constraint: pdabkd+1.
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 (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

f08hcc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08hcc 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 complex analogue of this function is f08hqc.

10 Example

This example computes all the eigenvalues and eigenvectors of the symmetric band matrix A, where
A = ( 1 2 3 0 0 2 2 3 4 0 3 3 3 4 5 0 4 4 4 5 0 0 5 5 5 ) .  

10.1 Program Text

Program Text (f08hcce.c)

10.2 Program Data

Program Data (f08hcce.d)

10.3 Program Results

Program Results (f08hcce.r)