f08jgf computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric positive definite tridiagonal matrix, or of a real symmetric positive definite matrix which has been reduced to tridiagonal form.
f08jgf first factorizes as where is unit lower bidiagonal and is diagonal. It forms the bidiagonal matrix , and then calls f08mef to compute the singular values of which are the same as the eigenvalues of . The method used by the routine allows high relative accuracy to be achieved in the small eigenvalues of . The eigenvectors are normalized so that , but are determined only to within a factor .
Barlow J and Demmel J W (1990) Computing accurate eigensystems of scaled diagonally dominant matrices SIAM J. Numer. Anal.27 762–791
1: – Character(1)Input
On entry: indicates whether the eigenvectors are to be computed.
Only the eigenvalues are computed (and the array z is not referenced).
The eigenvalues and eigenvectors of are computed (and the array z must contain the matrix on entry).
The eigenvalues and eigenvectors of are computed (and the array z is initialized by the routine).
, or .
2: – IntegerInput
On entry: , the order of the matrix .
3: – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array d
must be at least
On entry: the diagonal elements of the tridiagonal matrix .
On exit: the eigenvalues in descending order, unless , in which case d is overwritten.
4: – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array e
must be at least
On entry: the off-diagonal elements of the tridiagonal matrix .
On entry: the first dimension of the array z as declared in the (sub)program from which f08jgf is called.
if or , ;
if , .
7: – Real (Kind=nag_wp) arrayWorkspace
8: – IntegerOutput
On exit: unless the routine detects an error (see Section 6).
6Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
The leading minor of order is not positive definite and the Cholesky factorization of could not be completed. Hence itself is not positive definite.
The algorithm to compute the singular values of the Cholesky factor failed to converge; off-diagonal elements did not converge to zero.
The eigenvalues and eigenvectors of are computed to high relative accuracy which means that if they vary widely in magnitude, then any small eigenvalues (and corresponding eigenvectors) will be computed more accurately than, for example, with the standard method. However, the reduction to tridiagonal form (prior to calling the routine) may exclude the possibility of obtaining high relative accuracy in the small eigenvalues of the original matrix if its eigenvalues vary widely in magnitude.
To be more precise, let be the tridiagonal matrix defined by , where is diagonal with , and for all . If is an exact eigenvalue of and is the corresponding computed value, then
where is a modestly increasing function of , is the machine precision, and is the condition number of with respect to inversion defined by: .
If is the corresponding exact eigenvector of , and is the corresponding computed eigenvector, then the angle between them is bounded as follows:
where is the relative gap between and the other eigenvalues, defined by
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f08jgf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08jgf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
The total number of floating-point operations is typically about if and about if or , but depends on how rapidly the algorithm converges. When , the operations are all performed in scalar mode; the additional operations to compute the eigenvectors when or can be vectorized and on some machines may be performed much faster.