Purpose ======= LA_SBGV, LA_SBGVD, LA_HBGV and LA_HBGVD compute all eigenvalues and, optionally, all eigenvectors of the generalized eigenvalue problem A*z = lambda*B*z, where A and B are real symmetric in the cases of LA_SBGV and LA_SBGVD and complex Hermitian in the cases of LA_HBGV and LA_HBGVD. Matrix B is positive definite. Matrices A and B are stored in a band format. LA_SBGVD and LA_HBGVD use a divide and conquer algorithm. If eigenvectors are desired, they can be much faster than LA_SBGV and LA_HBGV for large matrices but use more workspace. ========= SUBROUTINE LA_SBGV / LA_SBGVD / LA_HBGV / LA_HBGVD( AB, BB, & W, UPLO=uplo, Z=z, INFO=info ) (), INTENT(INOUT) :: AB(:,:), BB(:,:) REAL(), INTENT(OUT) :: W(:) CHARACTER(LEN=1), INTENT(IN), OPTIONAL :: UPLO (), INTENT(OUT), OPTIONAL :: Z(:,:) INTEGER, INTENT(OUT), OPTIONAL :: INFO where ::= REAL | COMPLEX ::= KIND(1.0) | KIND(1.0D0) Arguments ========= AB (input/output) REAL or COMPLEX array, shape (:,:) with size(AB,1) = ka + 1 and size(AB,2) = n, where ka is the number of subdiagonals or superdiagonals in the band and n is the order of A and B. On entry, the upper (if UPLO = 'U') or lower (if UPLO = 'L') triangle of matrix A in band storage. The ka + 1 diagonals of A are stored in the rows of AB so that the j-th column of A is stored in the j-th column of AB as follows: if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j, 1<=j<=n if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka), 1<=j<=n. On exit, the contents of AB are destroyed. BB (input/output) REAL or COMPLEX array, shape (:,:) with size(BB,1) = kb + 1 and size(BB,2) = n, where kb is the number of subdiagonals or superdiagonals in the band of B. On entry, the upper (if UPLO = 'U') or lower (if UPLO = 'L') triangle of matrix B in band storage. The kb + 1 diagonals of B are stored in the rows of BB so that the j-th column of B is stored in the j-th column of BB as follows: if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j, 1<=j<=n if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb), 1<=j<=n. On exit, the factor S from the split Cholesky factorization B = S^H*S. W (output) REAL array, shape (:) with size(W) = n. The eigenvalues in ascending order. UPLO Optional (input) CHARACTER(LEN=1). = 'U': Upper triangles of A and B are stored; = 'L': Lower triangles of A and B are stored. Default value: 'U'. Z Optional (output) REAL or COMPLEX square array, shape (:,:) with size(Z,1) = n. The matrix Z of eigenvectors, normalized so that Z^H*B*Z = I. INFO Optional (output) INTEGER. = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: the algorithm failed to converge or matrix B is not positive definite: <= n: if INFO = i, i off-diagonal elements of an intermediate tridiagonal form did not converge to zero. > n: if INFO = n+i, for 1<=i<=n, then the leading minor of order i of B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed. If INFO is not present and an error occurs, then the program is terminated with an error message.