NAG CL Interface
f08usc (zhbgst)

1 Purpose

f08usc reduces a complex Hermitian-definite generalized eigenproblem Az=λBz to the standard form Cy=λy, where A and B are band matrices, A is a complex Hermitian matrix, and B has been factorized by f08utc.

2 Specification

#include <nag.h>
void  f08usc (Nag_OrderType order, Nag_VectType vect, Nag_UploType uplo, Integer n, Integer ka, Integer kb, Complex ab[], Integer pdab, const Complex bb[], Integer pdbb, Complex x[], Integer pdx, NagError *fail)
The function may be called by the names: f08usc, nag_lapackeig_zhbgst or nag_zhbgst.

3 Description

To reduce the complex Hermitian-definite generalized eigenproblem Az=λBz to the standard form Cy=λy, where A, B and C are banded, f08usc must be preceded by a call to f08utc which computes the split Cholesky factorization of the positive definite matrix B: B=SHS. The split Cholesky factorization, compared with the ordinary Cholesky factorization, allows the work to be approximately halved.
This function overwrites A with C=XHAX, where X=S-1Q and Q is a unitary matrix chosen (implicitly) to preserve the bandwidth of A. The function also has an option to allow the accumulation of X, and then, if z is an eigenvector of C, Xz is an eigenvector of the original system.

4 References

Crawford C R (1973) Reduction of a band-symmetric generalized eigenvalue problem Comm. ACM 16 41–44
Kaufman L (1984) Banded eigenvalue solvers on vector machines ACM Trans. Math. Software 10 73–86

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: vect Nag_VectType Input
On entry: indicates whether X is to be returned.
vect=Nag_DoNotForm
X is not returned.
vect=Nag_FormX
X is returned.
Constraint: vect=Nag_DoNotForm or Nag_FormX.
3: uplo Nag_UploType Input
On entry: indicates whether the upper or lower triangular part of A is stored.
uplo=Nag_Upper
The upper triangular part of A is stored.
uplo=Nag_Lower
The lower triangular part of A is stored.
Constraint: uplo=Nag_Upper or Nag_Lower.
4: n Integer Input
On entry: n, the order of the matrices A and B.
Constraint: n0.
5: 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.
6: 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.
7: ab[dim] Complex Input/Output
Note: the dimension, dim, of the array ab must be at least max1,pdab×n.
On entry: the upper or lower triangle of the n by n Hermitian 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=max1,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,,minn,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,,minn,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=max1,i-ka,,i.
On exit: the upper or lower triangle of ab is overwritten by the corresponding upper or lower triangle of C as specified by uplo.
8: 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.
9: bb[dim] const Complex Input
Note: the dimension, dim, of the array bb must be at least max1,pdbb×n.
On entry: the banded split Cholesky factor of B as specified by uplo, n and kb and returned by f08utc.
10: pdbb Integer Input
On entry: the stride separating row or column elements (depending on the value of order) of the matrix in the array bb.
Constraint: pdbbkb+1.
11: x[dim] Complex Output
Note: the dimension, dim, of the array x must be at least
  • max1,pdx×n when vect=Nag_FormX;
  • 1 when vect=Nag_DoNotForm.
The i,jth element of the matrix X is stored in
  • x[j-1×pdx+i-1] when order=Nag_ColMajor;
  • x[i-1×pdx+j-1] when order=Nag_RowMajor.
On exit: the n by n matrix X=S-1Q, if vect=Nag_FormX.
If vect=Nag_DoNotForm, x is not referenced.
12: pdx Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array x.
Constraints:
  • if vect=Nag_FormX, pdx max1,n ;
  • if vect=Nag_DoNotForm, pdx1.
13: 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_ENUM_INT_2
On entry, vect=value, pdx=value and n=value.
Constraint: if vect=Nag_FormX, pdx max1,n ;
if vect=Nag_DoNotForm, pdx1.
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, pdx=value.
Constraint: pdx>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_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

Forming the reduced matrix C is a stable procedure. However it involves implicit multiplication by B-1. When f08usc 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 B is ill-conditioned with respect to inversion.

8 Parallelism and Performance

f08usc 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 real floating-point operations is approximately 20n2kB, when vect=Nag_DoNotForm, assuming nkA,kB; there are an additional 5n3kB/kA operations when vect=Nag_FormX.
The real analogue of this function is f08uec.

10 Example

This example computes all the eigenvalues of Az=λBz, where
A = -1.13+0.00i 1.94-2.10i -1.40+0.25i 0.00+0.00i 1.94+2.10i -1.91+0.00i -0.82-0.89i -0.67+0.34i -1.40-0.25i -0.82+0.89i -1.87+0.00i -1.10-0.16i 0.00+0.00i -0.67-0.34i -1.10+0.16i 0.50+0.00i  
and
B = 9.89+0.00i 1.08-1.73i 0.00+0.00i 0.00+0.00i 1.08+1.73i 1.69+0.00i -0.04+0.29i 0.00+0.00i 0.00+0.00i -0.04-0.29i 2.65+0.00i -0.33+2.24i 0.00+0.00i 0.00+0.00i -0.33-2.24i 2.17+0.00i .  
Here A is Hermitian, B is Hermitian positive definite, and A and B are treated as band matrices. B must first be factorized by f08utc. The program calls f08usc to reduce the problem to the standard form Cy=λy, then f08hsc to reduce C to tridiagonal form, and f08jfc to compute the eigenvalues.

10.1 Program Text

Program Text (f08usce.c)

10.2 Program Data

Program Data (f08usce.d)

10.3 Program Results

Program Results (f08usce.r)