NAG CL Interface
f08sac (dsygv)

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1 Purpose

f08sac computes all the eigenvalues and, optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form
Az=λBz ,   ABz=λz   or   BAz=λz ,  
where A and B are symmetric and B is also positive definite.

2 Specification

#include <nag.h>
void  f08sac (Nag_OrderType order, Integer itype, Nag_JobType job, Nag_UploType uplo, Integer n, double a[], Integer pda, double b[], Integer pdb, double w[], NagError *fail)
The function may be called by the names: f08sac, nag_lapackeig_dsygv or nag_dsygv.

3 Description

f08sac 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 eigenvalues and, optionally, the 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
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: 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: a[dim] double Input/Output
Note: the dimension, dim, of the array a must be at least max(1,pda×n).
On entry: the n×n symmetric matrix A.
If order=Nag_ColMajor, Aij is stored in a[(j-1)×pda+i-1].
If order=Nag_RowMajor, Aij is stored in a[(i-1)×pda+j-1].
If uplo=Nag_Upper, the upper triangular part of A must be stored and the elements of the array below the diagonal are not referenced.
If uplo=Nag_Lower, the lower triangular part of A must be stored and the elements of the array above the diagonal are not referenced.
On exit: if job=Nag_DoBoth, a contains the matrix Z of eigenvectors. The eigenvectors are normalized as follows:
  • if itype=1 or 2, ZTBZ=I;
  • if itype=3, ZTB-1Z=I.
If job=Nag_EigVals, the upper triangle (if uplo=Nag_Upper) or the lower triangle (if uplo=Nag_Lower) of a, including the diagonal, is overwritten.
7: pda Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: pdamax(1,n).
8: b[dim] double Input/Output
Note: the dimension, dim, of the array b must be at least max(1,pdb×n).
On entry: the n×n symmetric positive definite matrix B.
If order=Nag_ColMajor, Bij is stored in b[(j-1)×pdb+i-1].
If order=Nag_RowMajor, Bij is stored in b[(i-1)×pdb+j-1].
If uplo=Nag_Upper, the upper triangular part of B must be stored and the elements of the array below the diagonal are not referenced.
If uplo=Nag_Lower, the lower triangular part of B must be stored and the elements of the array above the diagonal are not referenced.
On exit: if fail.code= NE_NOERROR or NE_CONVERGENCE, the part of b containing the matrix is overwritten by the triangular factor U or L from the Cholesky factorization B=UTU or B=LLT.
9: pdb Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraint: pdbmax(1,n).
10: w[n] double Output
On exit: the eigenvalues in ascending order.
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
The algorithm failed to converge; value off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
NE_INT
On entry, itype=value.
Constraint: itype=1, 2 or 3.
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
On entry, pdb=value.
Constraint: pdb>0.
NE_INT_2
On entry, pda=value and n=value.
Constraint: pdamax(1,n).
On entry, pdb=value and n=value.
Constraint: pdbmax(1,n).
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.
The example program below illustrates the computation of approximate error bounds.

8 Parallelism and Performance

f08sac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08sac 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 f08snc.

10 Example

This example finds all the eigenvalues and eigenvectors of the generalized symmetric eigenproblem Az=λBz, 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 ) ,  
together with and estimate of the condition number of B, and approximate error bounds for the computed eigenvalues and eigenvectors.
The example program for f08scc illustrates solving a generalized symmetric eigenproblem of the form ABz=λz.

10.1 Program Text

Program Text (f08sace.c)

10.2 Program Data

Program Data (f08sace.d)

10.3 Program Results

Program Results (f08sace.r)