NAG CL Interface
f08tbc (dspgvx)

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1 Purpose

f08tbc computes selected eigenvalues and, optionally, eigenvectors of a real generalized symmetric-definite eigenproblem, of the form
Az=λBz ,   ABz=λz   or   BAz=λz ,  
where A and B are symmetric, stored in packed storage, and B is also positive definite. 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  f08tbc (Nag_OrderType order, Integer itype, Nag_JobType job, Nag_RangeType range, Nag_UploType uplo, Integer n, double ap[], double bp[], 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: f08tbc, nag_lapackeig_dspgvx or nag_dspgvx.

3 Description

f08tbc first performs a Cholesky factorization of the matrix B as B=UTU , when uplo=Nag_Upper or B=LLT , when uplo=Nag_Lower. The generalized problem is then reduced to a standard symmetric eigenvalue problem
Cx=λx ,  
which is solved for the desired eigenvalues and eigenvectors; the eigenvectors are then backtransformed to give the eigenvectors of the original problem.
For the problem Az=λBz , the eigenvectors are normalized so that the matrix of eigenvectors, Z, satisfies
ZT A Z = Λ   and   ZT B Z = I ,  
where Λ is the diagonal matrix whose diagonal elements are the eigenvalues. For the problem A B z = λ z we correspondingly have
Z-1 A Z-T = Λ   and   ZT B Z = I ,  
and for B A z = λ z we have
ZT A Z = Λ   and   ZT B-1 Z = I .  

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
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_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: itype Integer Input
On entry: specifies the problem type to be solved.
itype=1
Az=λBz.
itype=2
ABz=λz.
itype=3
BAz=λz.
Constraint: itype=1, 2 or 3.
3: 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.
4: 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.
5: 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.
6: n Integer Input
On entry: n, the order of the matrices A and B.
Constraint: n0.
7: ap[dim] double Input/Output
Note: the dimension, dim, of the array ap must be at least max(1,n×(n+1)/2).
On entry: the upper or lower triangle of the n×n symmetric matrix A, packed by rows or columns.
The storage of elements Aij depends on the order and uplo arguments as follows:
if order=Nag_ColMajor and uplo=Nag_Upper,
Aij is stored in ap[(j-1)×j/2+i-1], for ij;
if order=Nag_ColMajor and uplo=Nag_Lower,
Aij is stored in ap[(2n-j)×(j-1)/2+i-1], for ij;
if order=Nag_RowMajor and uplo=Nag_Upper,
Aij is stored in ap[(2n-i)×(i-1)/2+j-1], for ij;
if order=Nag_RowMajor and uplo=Nag_Lower,
Aij is stored in ap[(i-1)×i/2+j-1], for ij.
On exit: the contents of ap are destroyed.
8: bp[dim] double Input/Output
Note: the dimension, dim, of the array bp must be at least max(1,n×(n+1)/2).
On entry: the upper or lower triangle of the n×n symmetric matrix B, packed by rows or columns.
The storage of elements Bij depends on the order and uplo arguments as follows:
if order=Nag_ColMajor and uplo=Nag_Upper,
Bij is stored in bp[(j-1)×j/2+i-1], for ij;
if order=Nag_ColMajor and uplo=Nag_Lower,
Bij is stored in bp[(2n-j)×(j-1)/2+i-1], for ij;
if order=Nag_RowMajor and uplo=Nag_Upper,
Bij is stored in bp[(2n-i)×(i-1)/2+j-1], for ij;
if order=Nag_RowMajor and uplo=Nag_Lower,
Bij is stored in bp[(i-1)×i/2+j-1], for ij.
On exit: the triangular factor U or L from the Cholesky factorization B=UTU or B=LLT, in the same storage format as B.
9: vl double Input
10: 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.
11: il Integer Input
12: 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 .
13: 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).
14: 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.
15: w[n] double Output
On exit: the first m elements contain the selected eigenvalues in ascending order.
16: 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 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]. The eigenvectors are normalized as follows:
    • if itype=1 or 2, ZTBZ=I;
    • if itype=3, ZTB-1Z=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.
17: 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.
18: 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.
19: 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, 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, itype=value.
Constraint: itype=1, 2 or 3.
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_MAT_NOT_POS_DEF
If fail.errnum=n+value, for 1valuen, then the leading minor of order value of 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

Background information to multithreading can be found in the Multithreading documentation.
f08tbc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08tbc 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 .
The complex analogue of this function is f08tpc.

10 Example

This example finds the eigenvalues in the half-open interval (-1.0,1.0] , and corresponding eigenvectors, of the generalized symmetric eigenproblem Az = λ Bz , where
A = ( 0.24 0.39 0.42 -0.16 0.39 -0.11 0.79 0.63 0.42 0.79 -0.25 0.48 -0.16 0.63 0.48 -0.03 )   and   B = ( 4.16 -3.12 0.56 -0.10 -3.12 5.03 -0.83 1.09 0.56 -0.83 0.76 0.34 -0.10 1.09 0.34 1.18 ) .  
The example program for f08tcc illustrates solving a generalized symmetric eigenproblem of the form ABz=λz .

10.1 Program Text

Program Text (f08tbce.c)

10.2 Program Data

Program Data (f08tbce.d)

10.3 Program Results

Program Results (f08tbce.r)