Computes selected eigenvalues and, optionally, eigenvectors of a complex generalized Hermitian positivedefinite eigenproblem with banded matrices.
Syntax

lapack_int LAPACKE_chbgvx( int matrix_layout, char jobz, char range, char uplo, lapack_int n, lapack_int ka, lapack_int kb, lapack_complex_float* ab, lapack_int ldab, lapack_complex_float* bb, lapack_int ldbb, lapack_complex_float* q, lapack_int ldq, float vl, float vu, lapack_int il, lapack_int iu, float abstol, lapack_int* m, float* w, lapack_complex_float* z, lapack_int ldz, lapack_int* ifail );
lapack_int LAPACKE_zhbgvx( int matrix_layout, char jobz, char range, char uplo, lapack_int n, lapack_int ka, lapack_int kb, lapack_complex_double* ab, lapack_int ldab, lapack_complex_double* bb, lapack_int ldbb, lapack_complex_double* q, lapack_int ldq, double vl, double vu, lapack_int il, lapack_int iu, double abstol, lapack_int* m, double* w, lapack_complex_double* z, lapack_int ldz, lapack_int* ifail );
Include Files
 mkl.h
Description
The routine computes selected eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian positivedefinite banded eigenproblem, of the form A*x = λ*B*x. Here A and B are assumed to be Hermitian and banded, and B is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either all eigenvalues, a range of values or a range of indices for the desired eigenvalues.
Input Parameters
 matrix_layout

Specifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).
 jobz

Must be 'N' or 'V'.
If jobz = 'N', then compute eigenvalues only.
If jobz = 'V', then compute eigenvalues and eigenvectors.
 range

Must be 'A' or 'V' or 'I'.
If range = 'A', the routine computes all eigenvalues.
If range = 'V', the routine computes eigenvalues w[i] in the halfopen interval:
vl< w[i]≤vu.
If range = 'I', the routine computes eigenvalues with indices il to iu.
 uplo

Must be 'U' or 'L'.
If uplo = 'U', arrays ab and bb store the upper triangles of A and B;
If uplo = 'L', arrays ab and bb store the lower triangles of A and B.
 n

The order of the matrices A and B (n≥ 0).
 ka

The number of super or subdiagonals in A
(ka≥ 0).
 kb

The number of super or subdiagonals in B (kb≥ 0).
 ab, bb

Arrays:
ab(size at least max(1, ldab*n) for column major layout and max(1, ldab*(ka + 1)) for row major layout) is an array containing either upper or lower triangular part of the Hermitian matrix A (as specified by uplo) in band storage format.
bb(size at least max(1, ldbb*n) for column major layout and max(1, ldbb*(kb + 1)) for row major layout) is an array containing either upper or lower triangular part of the Hermitian matrix B (as specified by uplo) in band storage format.
 ldab

The leading dimension of the array ab; must be at least ka+1 for column major layout and at least max(1, n) for row major layout.
 ldbb

The leading dimension of the array bb; must be at least kb+1 for column major layout and at least max(1, n) for row major layout.
 vl, vu

If range = 'V', the lower and upper bounds of the interval to be searched for eigenvalues.
Constraint: vl< vu.
If range = 'A' or 'I', vl and vu are not referenced.
 il, iu

If range = 'I', the indices in ascending order of the smallest and largest eigenvalues to be returned.
Constraint: 1 ≤il≤iu≤n, if n > 0; il=1 and iu=0
if n = 0.
If range = 'A' or 'V', il and iu are not referenced.
 abstol

The absolute error tolerance for the eigenvalues. See Application Notes for more information.
 ldz

The leading dimension of the output array z; ldz≥ 1. If jobz = 'V', ldz≥ max(1, n) for column major layout and at least max(1, m) for row major layout.
 ldq

The leading dimension of the output array q; ldq≥ 1. If jobz = 'V', ldq≥ max(1, n).
Output Parameters
 ab

On exit, the contents of ab are overwritten.
 bb

On exit, contains the factor S from the split Cholesky factorization B = S^{H}*S, as returned by pbstf/pbstf.
 m

The total number of eigenvalues found,
0 ≤m≤n. If range = 'A', m = n, and if range = 'I',
m = iuil+1.
 w

Array w, size at least max(1, n).
If info = 0, contains the eigenvalues in ascending order.
 z, q

Arrays:
z(size max(1, ldz*m) for column major layout and max(1, ldz*n) for row major layout).
If jobz = 'V', then if info = 0, z contains the matrix Z of eigenvectors, with the ith column of z holding the eigenvector associated with w[i  1]. The eigenvectors are normalized so that Z^{H}*B*Z = I.
If jobz = 'N', then z is not referenced.
q (size max(1, ldq*n)).
If jobz = 'V', then q contains the nbyn matrix used in the reduction of Ax = λBx to standard form, that is, Cx = λx and consequently C to tridiagonal form.
If jobz = 'N', then q is not referenced.
 ifail

Array, size at least max(1, n).
If jobz = 'V', then if info = 0, the first m elements of ifail are zero; if info > 0, the ifail contains the indices of the eigenvectors that failed to converge.
If jobz = 'N', then ifail is not referenced.
Return Values
This function returns a value info.
If info=0, the execution is successful.
If info = i, the ith parameter had an illegal value.
If info > 0, and
if i≤n, the algorithm failed to converge, and i offdiagonal elements of an intermediate tridiagonal did not converge to zero;
if info = n + i, for 1 ≤i≤n, then pbstf/pbstf returned info = i and B is not positivedefinite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed.
Application Notes
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*T_{1} will be used in its place, where T is the tridiagonal matrix obtained by reducing A to tridiagonal form. Eigenvalues will be computed most accurately when abstol is set to twice the underflow threshold 2*?lamch('S'), not zero.
If this routine returns with info > 0, indicating that some eigenvectors did not converge, try setting abstol to 2*?lamch('S').