f08uef reduces a real symmetric-definite generalized eigenproblem to the standard form , where and are band matrices, is a real symmetric matrix, and has been factorized by f08uff.
The routine may be called by the names f08uef, nagf_lapackeig_dsbgst or its LAPACK name dsbgst.
3Description
To reduce the real symmetric-definite generalized eigenproblem to the standard form , where , and are banded, f08uef must be preceded by a call to f08uff which computes the split Cholesky factorization of the positive definite matrix : . The split Cholesky factorization, compared with the ordinary Cholesky factorization, allows the work to be approximately halved.
This routine overwrites with , where and is a orthogonal matrix chosen (implicitly) to preserve the bandwidth of . The routine also has an option to allow the accumulation of , and then, if is an eigenvector of , is an eigenvector of the original system.
4References
Crawford C R (1973) Reduction of a band-symmetric generalized eigenvalue problem Comm. ACM16 41–44
Kaufman L (1984) Banded eigenvalue solvers on vector machines ACM Trans. Math. Software10 73–86
5Arguments
1: – Character(1)Input
On entry: indicates whether is to be returned.
is not returned.
is returned.
Constraint:
or .
2: – Character(1)Input
On entry: indicates whether the upper or lower triangular part of is stored.
The upper triangular part of is stored.
The lower triangular part of is stored.
Constraint:
or .
3: – IntegerInput
On entry: , the order of the matrices and .
Constraint:
.
4: – IntegerInput
On entry: if , the number of superdiagonals, , of the matrix .
If , the number of subdiagonals, , of the matrix .
Constraint:
.
5: – IntegerInput
On entry: if , the number of superdiagonals, , of the matrix .
If , the number of subdiagonals, , of the matrix .
Constraint:
.
6: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array ab
must be at least
.
On entry: the upper or lower triangle of the symmetric band matrix .
The matrix is stored in rows to , more precisely,
if , the elements of the upper triangle of within the band must be stored with element in ;
if , the elements of the lower triangle of within the band must be stored with element in
On exit: the upper or lower triangle of ab is overwritten by the corresponding upper or lower triangle of as specified by uplo.
7: – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f08uef is called.
Constraint:
.
8: – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array bb
must be at least
.
On entry: the banded split Cholesky factor of as specified by uplo, n and kb and returned by f08uff.
9: – IntegerInput
On entry: the first dimension of the array bb as declared in the (sub)program from which f08uef is called.
Constraint:
.
10: – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array x
must be at least
if and at least if .
On entry: the first dimension of the array x as declared in the (sub)program from which f08uef is called.
Constraints:
if , ;
if , .
12: – Real (Kind=nag_wp) arrayWorkspace
13: – 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.
7Accuracy
Forming the reduced matrix is a stable procedure. However it involves implicit multiplication by . When f08uef is used as a step in the computation of eigenvalues and eigenvectors of the original problem, there may be a significant loss of accuracy if is ill-conditioned with respect to inversion.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f08uef 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.
9Further Comments
The total number of floating-point operations is approximately , when , assuming ; there are an additional operations when .
This example computes all the eigenvalues of , where
Here is symmetric, is symmetric positive definite, and and are treated as band matrices. must first be factorized by f08uff. The program calls f08uef to reduce the problem to the standard form , then f08hef to reduce to tridiagonal form, and f08jff to compute the eigenvalues.