NAG Library Function Document

nag_zhbevx (f08hpc)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

nag_zhbevx (f08hpc) computes selected eigenvalues and, optionally, eigenvectors of a complex n by n Hermitian band matrix A of bandwidth 2kd+1 . 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>
#include <nagf08.h>
void  nag_zhbevx (Nag_OrderType order, Nag_JobType job, Nag_RangeType range, Nag_UploType uplo, Integer n, Integer kd, Complex ab[], Integer pdab, Complex q[], Integer pdq, double vl, double vu, Integer il, Integer iu, double abstol, Integer *m, double w[], Complex z[], Integer pdz, Integer jfail[], NagError *fail)

3
Description

The Hermitian band matrix A is first reduced to real tridiagonal form, using unitary similarity transformations. The required eigenvalues and eigenvectors are then computed from the tridiagonal matrix; the method used depends upon whether all, or selected, eigenvalues and eigenvectors are required.

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 http://www.netlib.org/lapack/lug
Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices SIAM J. Sci. Statist. Comput. 11 873–912
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5
Arguments

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 3.3.1.3 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.
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_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:     uplo Nag_UploTypeInput
On entry: if uplo=Nag_Upper, the upper triangular part of A is stored.
If uplo=Nag_Lower, the lower triangular part of A is stored.
Constraint: uplo=Nag_Upper or Nag_Lower.
5:     n IntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
6:     kd IntegerInput
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.
7:     ab[dim] ComplexInput/Output
Note: the dimension, dim, of the array ab must be at least max1,pdab×n.
On entry: the upper or lower triangle of the n by n Hermitian 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=max1,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,,minn,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,,minn,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=max1,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.
8:     pdab IntegerInput
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.
9:     q[dim] ComplexOutput
Note: the dimension, dim, of the array q must be at least
  • max1,pdq×n when job=Nag_DoBoth;
  • 1 otherwise.
The i,jth element of the matrix Q is stored in
  • q[j-1×pdq+i-1] when order=Nag_ColMajor;
  • q[i-1×pdq+j-1] when order=Nag_RowMajor.
On exit: if job=Nag_DoBoth, the n by n unitary matrix used in the reduction to tridiagonal form.
If job=Nag_EigVals, q is not referenced.
10:   pdq IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array q.
Constraints:
  • if job=Nag_DoBoth, pdq max1,n ;
  • otherwise pdq1.
11:   vl doubleInput
12:   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.
13:   il IntegerInput
14:   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.
Constraints:
  • if range=Nag_Indices and n=0, il=1 and iu=0;
  • if range=Nag_Indices and n>0, 1 il iu n .
15:   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, where T is the tridiagonal matrix obtained by reducing A to tridiagonal form. Eigenvalues will be computed most accurately when abstol is set to twice the underflow threshold 2 × nag_real_safe_small_number , not zero. If this function returns with fail.code= NE_CONVERGENCE, indicating that some eigenvectors did not converge, try setting abstol to 2 × nag_real_safe_small_number . See Demmel and Kahan (1990).
16:   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.
17:   w[n] doubleOutput
On exit: the first m elements contain the selected eigenvalues in ascending order.
18:   z[dim] ComplexOutput
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, then
  • if fail.code= NE_NOERROR, 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 an eigenvector fails to converge (fail.code= NE_CONVERGENCE), then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in jfail.
If job=Nag_EigVals, z is not referenced.
19:   pdz IntegerInput
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 max1,n ;
  • otherwise pdz1.
20:   jfail[dim] IntegerOutput
Note: the dimension, dim, of the array jfail must be at least max1,n.
On exit: if job=Nag_DoBoth, then
  • if fail.code= NE_NOERROR, the first m elements of jfail are zero;
  • if fail.code= NE_CONVERGENCE, jfail contains the indices of the eigenvectors that failed to converge.
If job=Nag_EigVals, jfail is not referenced.
21:   fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6
Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_CONVERGENCE
The algorithm failed to converge; value eigenvectors did not converge. Their indices are stored in array jfail.
NE_ENUM_INT_2
On entry, job=value, pdq=value and n=value.
Constraint: if job=Nag_DoBoth, pdq max1,n ;
otherwise pdq1.
On entry, job=value, pdz=value and n=value.
Constraint: if job=Nag_DoBoth, pdz max1,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, kd=value.
Constraint: kd0.
On entry, n=value.
Constraint: n0.
On entry, pdab=value.
Constraint: pdab>0.
On entry, pdq=value.
Constraint: pdq>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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
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.

7
Accuracy

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.

8
Parallelism and Performance

nag_zhbevx (f08hpc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zhbevx (f08hpc) 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 kd 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.
The real analogue of this function is nag_dsbevx (f08hbc).

10
Example

This example finds the eigenvalues in the half-open interval -2,2 , and the corresponding eigenvectors, of the Hermitian band matrix
A = 1 2-i 3-i 0 0 2+i 2 3-2i 4-2i 0 3+i 3+2i 3 4-3i 5-3i 0 4+2i 4+3i 4 5-4i 0 0 5+3i 5+4i 5 .  

10.1
Program Text

Program Text (f08hpce.c)

10.2
Program Data

Program Data (f08hpce.d)

10.3
Program Results

Program Results (f08hpce.r)

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