NAG CL Interface
f08ubc (dsbgvx)

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1 Purpose

f08ubc computes selected eigenvalues and, optionally, eigenvectors of a real generalized symmetric-definite banded eigenproblem, of the form
Az=λBz ,  
where A and B are symmetric and banded, and B is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either all eigenvalues, a range of values or a range of indices for the desired eigenvalues.

2 Specification

#include <nag.h>
void  f08ubc (Nag_OrderType order, Nag_JobType job, Nag_RangeType range, Nag_UploType uplo, Integer n, Integer ka, Integer kb, double ab[], Integer pdab, double bb[], Integer pdbb, double q[], Integer pdq, double vl, double vu, Integer il, Integer iu, double abstol, Integer *m, double w[], double z[], Integer pdz, Integer jfail[], NagError *fail)
The function may be called by the names: f08ubc, nag_lapackeig_dsbgvx or nag_dsbgvx.

3 Description

The generalized symmetric-definite band problem
Az = λ Bz  
is first reduced to a standard band symmetric problem
Cx = λx ,  
where C is a symmetric band matrix, using Wilkinson's modification to Crawford's algorithm (see Crawford (1973) and Wilkinson (1977)). The symmetric eigenvalue problem is then solved for the required eigenvalues and eigenvectors, and the eigenvectors are then backtransformed to the eigenvectors of the original problem.
The eigenvectors are normalized so that
zT A z = λ   and   zT B z = 1 .  

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
Crawford C R (1973) Reduction of a band-symmetric generalized eigenvalue problem Comm. ACM 16 41–44
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
Wilkinson J H (1977) Some recent advances in numerical linear algebra The State of the Art in Numerical Analysis (ed D A H Jacobs) Academic Press

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: uplo Nag_UploType Input
On entry: if uplo=Nag_Upper, the upper triangles of A and B are stored.
If uplo=Nag_Lower, the lower triangles of A and B are stored.
Constraint: uplo=Nag_Upper or Nag_Lower.
5: n Integer Input
On entry: n, the order of the matrices A and B.
Constraint: n0.
6: ka Integer Input
On entry: if uplo=Nag_Upper, the number of superdiagonals, ka, of the matrix A.
If uplo=Nag_Lower, the number of subdiagonals, ka, of the matrix A.
Constraint: ka0.
7: kb Integer Input
On entry: if uplo=Nag_Upper, the number of superdiagonals, kb, of the matrix B.
If uplo=Nag_Lower, the number of subdiagonals, kb, of the matrix B.
Constraint: kakb0.
8: 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[ka+i-j+(j-1)×pdab], for j=1,,n and i=max(1,j-ka),,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+ka);
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+ka);
if order=Nag_RowMajor and uplo=Nag_Lower,
Aij is stored in ab[ka+j-i+(i-1)×pdab], for i=1,,n and j=max(1,i-ka),,i.
On exit: the contents of ab are overwritten.
9: 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: pdabka+1.
10: bb[dim] double Input/Output
Note: the dimension, dim, of the array bb must be at least max(1,pdbb×n).
On entry: the upper or lower triangle of the n×n symmetric positive definite band matrix B.
This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of Bij, depends on the order and uplo arguments as follows:
if order=Nag_ColMajor and uplo=Nag_Upper,
Bij is stored in bb[kb+i-j+(j-1)×pdbb], for j=1,,n and i=max(1,j-kb),,j;
if order=Nag_ColMajor and uplo=Nag_Lower,
Bij is stored in bb[i-j+(j-1)×pdbb], for j=1,,n and i=j,,min(n,j+kb);
if order=Nag_RowMajor and uplo=Nag_Upper,
Bij is stored in bb[j-i+(i-1)×pdbb], for i=1,,n and j=i,,min(n,i+kb);
if order=Nag_RowMajor and uplo=Nag_Lower,
Bij is stored in bb[kb+j-i+(i-1)×pdbb], for i=1,,n and j=max(1,i-kb),,i.
On exit: the factor S from the split Cholesky factorization B=STS, as returned by f08ufc.
11: pdbb Integer Input
On entry: the stride separating row or column elements (depending on the value of order) of the matrix B in the array bb.
Constraint: pdbbkb+1.
12: q[dim] double Output
Note: the dimension, dim, of the array q must be at least
  • max(1,pdq×n) when job=Nag_DoBoth;
  • 1 otherwise.
The (i,j)th 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×n matrix, Q used in the reduction of the standard form, i.e., Cx=λx, from symmetric banded to tridiagonal form.
If job=Nag_EigVals, q is not referenced.
13: pdq Integer Input
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 max(1,n) ;
  • otherwise pdq1.
14: vl double Input
15: 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.
16: il Integer Input
17: 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 .
18: 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, where T is the tridiagonal matrix obtained by reducing C 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).
19: 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.
20: w[n] double Output
On exit: the eigenvalues in ascending order.
21: 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, then
  • if fail.code= NE_NOERROR, the first m columns of Z contain the eigenvectors corresponding to the selected eigenvalues, with the ith column of Z holding the eigenvector associated with w[i-1]. The eigenvectors are normalized so that ZTBZ=I;
  • 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.
22: 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.
23: jfail[dim] Integer Output
Note: the dimension, dim, of the array jfail must be at least max(1,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, the first fail.errnum elements of jfail contains the indices of the eigenvectors that failed to converge.
If job=Nag_EigVals, jfail is not referenced.
24: 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 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 max(1,n) ;
otherwise pdq1.
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, ka=value.
Constraint: ka0.
On entry, n=value.
Constraint: n0.
On entry, pdab=value.
Constraint: pdab>0.
On entry, pdbb=value.
Constraint: pdbb>0.
On entry, pdq=value.
Constraint: pdq>0.
On entry, pdz=value.
Constraint: pdz>0.
NE_INT_2
On entry, ka=value and kb=value.
Constraint: kakb0.
On entry, pdab=value and ka=value.
Constraint: pdabka+1.
On entry, pdbb=value and kb=value.
Constraint: pdbbkb+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_MAT_NOT_POS_DEF
If fail.errnum=n+value, for 1valuen, f08ufc returned fail.errnum=value: B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed.
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

If B is ill-conditioned with respect to inversion, then the error bounds for the computed eigenvalues and vectors may be large, although when the diagonal elements of B differ widely in magnitude the eigenvalues and eigenvectors may be less sensitive than the condition of B would suggest. See Section 4.10 of Anderson et al. (1999) for details of the error bounds.

8 Parallelism and Performance

f08ubc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08ubc 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 n3 if job=Nag_DoBoth and range=Nag_AllValues, and assuming that nka , is approximately proportional to n2 ka if job=Nag_EigVals. Otherwise the number of floating-point operations depends upon the number of eigenvectors computed.
The complex analogue of this function is f08upc.

10 Example

This example finds the eigenvalues in the half-open interval (0.0,1.0] , and corresponding eigenvectors, of the generalized band symmetric eigenproblem Az = λ Bz , where
A = ( 0.24 0.39 0.42 0.00 0.39 -0.11 0.79 0.63 0.42 0.79 -0.25 0.48 0.00 0.63 0.48 -0.03 )   and   B = ( 2.07 0.95 0.00 0.00 0.95 1.69 -0.29 0.00 0.00 -0.29 0.65 -0.33 0.00 0.00 -0.33 1.17 ) .  

10.1 Program Text

Program Text (f08ubce.c)

10.2 Program Data

Program Data (f08ubce.d)

10.3 Program Results

Program Results (f08ubce.r)