Purpose ======= LA_SBEVX / LA_HBEVX compute selected eigenvalues and, optionally, the corresponding eigenvectors of a real symmetric/complex Hermitian band matrix A. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. ========= SUBROUTINE LA_SBEVX / LA_HBEVX( AB, W, UPLO=uplo, Z=z, & VL=vl, VU=vu, IL=il, IU=iu, M=m, IFAIL=ifail, & Q=q, ABSTOL=abstol, INFO=info ) (), INTENT(INOUT) :: AB(:,:) REAL(), INTENT(OUT) :: W(:) CHARACTER(LEN=1), INTENT(IN), OPTIONAL :: UPLO (), INTENT(OUT), OPTIONAL :: Z(:,:) REAL(), INTENT(IN), OPTIONAL :: VL, VU INTEGER, INTENT(IN), OPTIONAL :: IL, IU INTEGER, INTENT(OUT), OPTIONAL :: M INTEGER, INTENT(OUT), OPTIONAL :: IFAIL(:) (), INTENT(OUT), OPTIONAL :: Q(:,:) REAL(), INTENT(IN), OPTIONAL :: ABSTOL INTEGER, INTENT(OUT), OPTIONAL :: INFO where ::= REAL j COMPLEX ::= KIND(1.0) j KIND(1.0D0) Arguments ========= AB (input/output) REAL or COMPLEX array, shape (:,:) with size(AB,1) = kd + 1 and size(AB,2) = n, where kd is the number of subdiagonals or superdiagonals in the band and n is the order of A. On entry, the upper (if UPLO = 'U') or lower (if UPLO = 'L') triangle of matrix A in band storage. The kd + 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(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j 1<=j<=n if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd) 1<=j<=n. On exit, AB is overwritten by values generated during the reduction of A to a tridiagonal matrix T . If UPLO = 'U' the first superdiagonal and the diagonal of T are returned in rows kd and kd + 1 of AB. If UPLO = 'L', the diagonal and first subdiagonal of T are returned in the first two rows of AB. W (output) REAL array, shape (:) with size(W) = n. The first M elements contain the selected eigenvalues in ascending order. UPLO Optional (input) CHARACTER(LEN=1). = 'U' : Upper triangle of A is stored; = 'L' : Lower triangle of A is stored. Default value: 'U'. Z Optional (output) REAL or COMPLEX array, shape (:,:) with size(Z,1) = n and size(Z,2) = M. The first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z containing the eigenvector associated with the eigenvalue in W(i). If an eigenvector fails to converge, then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in IFAIL. Note: The user must ensure that at least M columns are supplied in the array Z. When the exact value of M is not known in advance, an upper bound must be used. In all cases M<=n. VL,VU Optional (input) REAL. The lower and upper bounds of the interval to be searched for eigenvalues. VL < VU. Default values: VL = -HUGE() and VU = HUGE(), where ::= KIND(1.0) | KIND(1.0D0). Note: Neither VL nor VU may be present if IL and/or IU is present. IL,IU Optional (input) INTEGER. The indices of the smallest and largest eigenvalues to be returned. The IL-th through IU-th eigenvalues will be found. 1<=IL<=IU<=size(A,1). Default values: IL = 1 and IU = size(A,1). Note: Neither IL nor IU may be present if VL and/or VU is present. Note: All eigenvalues are calculated if none of the arguments VL, VU, IL and IU are present. M Optional (output) INTEGER. The total number of eigenvalues found. 0<=M<=size(A,1). Note: If IL and IU are present then M = IU - IL + 1. IFAIL Optional (output) INTEGER array, shape (:) with size(IFAIL) = n. If INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvectors that failed to converge. Note: If Z is present then IFAIL should also be present. Q Optional (output) REAL or COMPLEX square array, shape(:,:) with size(Q,1) = n. The n by n unitary matrix used in the reduction to tridiagonal form. This is computed only if Z is present. ABSTOL Optional (input) REAL. The absolute error tolerance for the eigenvalues. 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 + EPSILON(1.0_) * max(|a|, |b|), where is the working precision. If ABSTOL<=0, then EPSILON(1.0_)* ||T||1 will be used in its place, where ||T||1 is the l1 norm of 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 * LA_LAMCH(1.0_, 'Save minimum'), not zero. Default value: 0.0_. Note: If this routine returns with INFO > 0, then some eigenvectors did not converge. Try setting ABSTOL to 2 * LA_LAMCH(1.0_, 'Save minimum'). INFO Optional (output) INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = i, then i eigenvectors failed to converge. Their indices are stored in array IFAIL. If INFO is not present and an error occurs, then the program is terminated with an error message.