Provided by: liblapack-doc_3.12.0-3build1.1_all
NAME
hbgvx - {hb,sb}gvx: eig, bisection
SYNOPSIS
Functions subroutine chbgvx (jobz, range, uplo, n, ka, kb, ab, ldab, bb, ldbb, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, rwork, iwork, ifail, info) CHBGVX subroutine dsbgvx (jobz, range, uplo, n, ka, kb, ab, ldab, bb, ldbb, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info) DSBGVX subroutine ssbgvx (jobz, range, uplo, n, ka, kb, ab, ldab, bb, ldbb, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info) SSBGVX subroutine zhbgvx (jobz, range, uplo, n, ka, kb, ab, ldab, bb, ldbb, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, rwork, iwork, ifail, info) ZHBGVX
Detailed Description
Function Documentation
subroutine chbgvx (character jobz, character range, character uplo, integer n, integer ka, integer kb, complex, dimension( ldab, * ) ab, integer ldab, complex, dimension( ldbb, * ) bb, integer ldbb, complex, dimension( ldq, * ) q, integer ldq, real vl, real vu, integer il, integer iu, real abstol, integer m, real, dimension( * ) w, complex, dimension( ldz, * ) z, integer ldz, complex, dimension( * ) work, real, dimension( * ) rwork, integer, dimension( * ) iwork, integer, dimension( * ) ifail, integer info) CHBGVX Purpose: CHBGVX computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite banded eigenproblem, of the form A*x=(lambda)*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. Parameters JOBZ JOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors. RANGE RANGE is CHARACTER*1 = 'A': all eigenvalues will be found; = 'V': all eigenvalues in the half-open interval (VL,VU] will be found; = 'I': the IL-th through IU-th eigenvalues will be found. UPLO UPLO is CHARACTER*1 = 'U': Upper triangles of A and B are stored; = 'L': Lower triangles of A and B are stored. N N is INTEGER The order of the matrices A and B. N >= 0. KA KA is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KA >= 0. KB KB is INTEGER The number of superdiagonals of the matrix B if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KB >= 0. AB AB is COMPLEX array, dimension (LDAB, N) On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first ka+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka). On exit, the contents of AB are destroyed. LDAB LDAB is INTEGER The leading dimension of the array AB. LDAB >= KA+1. BB BB is COMPLEX array, dimension (LDBB, N) On entry, the upper or lower triangle of the Hermitian band matrix B, stored in the first kb+1 rows of the array. The j-th column of B is stored in the j-th column of the array BB as follows: if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j; if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb). On exit, the factor S from the split Cholesky factorization B = S**H*S, as returned by CPBSTF. LDBB LDBB is INTEGER The leading dimension of the array BB. LDBB >= KB+1. Q Q is COMPLEX array, dimension (LDQ, N) If JOBZ = 'V', the n-by-n matrix used in the reduction of A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x, and consequently C to tridiagonal form. If JOBZ = 'N', the array Q is not referenced. LDQ LDQ is INTEGER The leading dimension of the array Q. If JOBZ = 'N', LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N). VL VL is REAL If RANGE='V', the lower bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'. VU VU is REAL If RANGE='V', the upper bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'. IL IL is INTEGER If RANGE='I', the index of the smallest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'. IU IU is INTEGER If RANGE='I', the index of the largest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'. ABSTOL ABSTOL is 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 + EPS * max( |a|,|b| ) , where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-norm of the tridiagonal matrix obtained by reducing AP to tridiagonal form. Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*SLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*SLAMCH('S'). M M is INTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. W W is REAL array, dimension (N) If INFO = 0, the eigenvalues in ascending order. Z Z is COMPLEX array, dimension (LDZ, N) If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of eigenvectors, with the i-th column of Z holding the eigenvector associated with W(i). The eigenvectors are normalized so that Z**H*B*Z = I. If JOBZ = 'N', then Z is not referenced. LDZ LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= N. WORK WORK is COMPLEX array, dimension (N) RWORK RWORK is REAL array, dimension (7*N) IWORK IWORK is INTEGER array, dimension (5*N) IFAIL IFAIL is INTEGER array, dimension (N) If JOBZ = 'V', then 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. If JOBZ = 'N', then IFAIL is not referenced. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, and i is: <= N: then i eigenvectors failed to converge. Their indices are stored in array IFAIL. > N: if INFO = N + i, for 1 <= i <= N, then CPBSTF returned INFO = i: B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed. Author Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Contributors: Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA subroutine dsbgvx (character jobz, character range, character uplo, integer n, integer ka, integer kb, double precision, dimension( ldab, * ) ab, integer ldab, double precision, dimension( ldbb, * ) bb, integer ldbb, double precision, dimension( ldq, * ) q, integer ldq, double precision vl, double precision vu, integer il, integer iu, double precision abstol, integer m, double precision, dimension( * ) w, double precision, dimension( ldz, * ) z, integer ldz, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer, dimension( * ) ifail, integer info) DSBGVX Purpose: DSBGVX computes selected eigenvalues, and optionally, eigenvectors of a real generalized symmetric-definite banded eigenproblem, of the form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric 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. Parameters JOBZ JOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors. RANGE RANGE is CHARACTER*1 = 'A': all eigenvalues will be found. = 'V': all eigenvalues in the half-open interval (VL,VU] will be found. = 'I': the IL-th through IU-th eigenvalues will be found. UPLO UPLO is CHARACTER*1 = 'U': Upper triangles of A and B are stored; = 'L': Lower triangles of A and B are stored. N N is INTEGER The order of the matrices A and B. N >= 0. KA KA is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KA >= 0. KB KB is INTEGER The number of superdiagonals of the matrix B if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KB >= 0. AB AB is DOUBLE PRECISION array, dimension (LDAB, N) On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first ka+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka). On exit, the contents of AB are destroyed. LDAB LDAB is INTEGER The leading dimension of the array AB. LDAB >= KA+1. BB BB is DOUBLE PRECISION array, dimension (LDBB, N) On entry, the upper or lower triangle of the symmetric band matrix B, stored in the first kb+1 rows of the array. The j-th column of B is stored in the j-th column of the array BB as follows: if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j; if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb). On exit, the factor S from the split Cholesky factorization B = S**T*S, as returned by DPBSTF. LDBB LDBB is INTEGER The leading dimension of the array BB. LDBB >= KB+1. Q Q is DOUBLE PRECISION array, dimension (LDQ, N) If JOBZ = 'V', the n-by-n matrix used in the reduction of A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x, and consequently C to tridiagonal form. If JOBZ = 'N', the array Q is not referenced. LDQ LDQ is INTEGER The leading dimension of the array Q. If JOBZ = 'N', LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N). VL VL is DOUBLE PRECISION If RANGE='V', the lower bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'. VU VU is DOUBLE PRECISION If RANGE='V', the upper bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'. IL IL is INTEGER If RANGE='I', the index of the smallest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'. IU IU is INTEGER If RANGE='I', the index of the largest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'. ABSTOL ABSTOL is DOUBLE PRECISION 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 + EPS * max( |a|,|b| ) , where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-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*DLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*DLAMCH('S'). M M is INTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. W W is DOUBLE PRECISION array, dimension (N) If INFO = 0, the eigenvalues in ascending order. Z Z is DOUBLE PRECISION array, dimension (LDZ, N) If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of eigenvectors, with the i-th column of Z holding the eigenvector associated with W(i). The eigenvectors are normalized so Z**T*B*Z = I. If JOBZ = 'N', then Z is not referenced. LDZ LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N). WORK WORK is DOUBLE PRECISION array, dimension (7*N) IWORK IWORK is INTEGER array, dimension (5*N) IFAIL IFAIL is INTEGER array, dimension (M) If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvalues that failed to converge. If JOBZ = 'N', then IFAIL is not referenced. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value <= N: if INFO = i, then i eigenvectors failed to converge. Their indices are stored in IFAIL. > N: DPBSTF returned an error code; i.e., if INFO = N + i, for 1 <= i <= N, then the leading principal minor of order i of B is not positive. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed. Author Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Contributors: Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA subroutine ssbgvx (character jobz, character range, character uplo, integer n, integer ka, integer kb, real, dimension( ldab, * ) ab, integer ldab, real, dimension( ldbb, * ) bb, integer ldbb, real, dimension( ldq, * ) q, integer ldq, real vl, real vu, integer il, integer iu, real abstol, integer m, real, dimension( * ) w, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer, dimension( * ) iwork, integer, dimension( * ) ifail, integer info) SSBGVX Purpose: SSBGVX computes selected eigenvalues, and optionally, eigenvectors of a real generalized symmetric-definite banded eigenproblem, of the form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric 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. Parameters JOBZ JOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors. RANGE RANGE is CHARACTER*1 = 'A': all eigenvalues will be found. = 'V': all eigenvalues in the half-open interval (VL,VU] will be found. = 'I': the IL-th through IU-th eigenvalues will be found. UPLO UPLO is CHARACTER*1 = 'U': Upper triangles of A and B are stored; = 'L': Lower triangles of A and B are stored. N N is INTEGER The order of the matrices A and B. N >= 0. KA KA is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KA >= 0. KB KB is INTEGER The number of superdiagonals of the matrix B if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KB >= 0. AB AB is REAL array, dimension (LDAB, N) On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first ka+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka). On exit, the contents of AB are destroyed. LDAB LDAB is INTEGER The leading dimension of the array AB. LDAB >= KA+1. BB BB is REAL array, dimension (LDBB, N) On entry, the upper or lower triangle of the symmetric band matrix B, stored in the first kb+1 rows of the array. The j-th column of B is stored in the j-th column of the array BB as follows: if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j; if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb). On exit, the factor S from the split Cholesky factorization B = S**T*S, as returned by SPBSTF. LDBB LDBB is INTEGER The leading dimension of the array BB. LDBB >= KB+1. Q Q is REAL array, dimension (LDQ, N) If JOBZ = 'V', the n-by-n matrix used in the reduction of A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x, and consequently C to tridiagonal form. If JOBZ = 'N', the array Q is not referenced. LDQ LDQ is INTEGER The leading dimension of the array Q. If JOBZ = 'N', LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N). VL VL is REAL If RANGE='V', the lower bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'. VU VU is REAL If RANGE='V', the upper bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'. IL IL is INTEGER If RANGE='I', the index of the smallest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'. IU IU is INTEGER If RANGE='I', the index of the largest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'. ABSTOL ABSTOL is 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 + EPS * max( |a|,|b| ) , where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-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*SLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*SLAMCH('S'). M M is INTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. W W is REAL array, dimension (N) If INFO = 0, the eigenvalues in ascending order. Z Z is REAL array, dimension (LDZ, N) If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of eigenvectors, with the i-th column of Z holding the eigenvector associated with W(i). The eigenvectors are normalized so Z**T*B*Z = I. If JOBZ = 'N', then Z is not referenced. LDZ LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N). WORK WORK is REAL array, dimension (7*N) IWORK IWORK is INTEGER array, dimension (5*N) IFAIL IFAIL is INTEGER array, dimension (M) If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvalues that failed to converge. If JOBZ = 'N', then IFAIL is not referenced. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value <= N: if INFO = i, then i eigenvectors failed to converge. Their indices are stored in IFAIL. > N: SPBSTF returned an error code; i.e., if INFO = N + i, for 1 <= i <= N, then the leading principal minor of order i of B is not positive. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed. Author Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Contributors: Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA subroutine zhbgvx (character jobz, character range, character uplo, integer n, integer ka, integer kb, complex*16, dimension( ldab, * ) ab, integer ldab, complex*16, dimension( ldbb, * ) bb, integer ldbb, complex*16, dimension( ldq, * ) q, integer ldq, double precision vl, double precision vu, integer il, integer iu, double precision abstol, integer m, double precision, dimension( * ) w, complex*16, dimension( ldz, * ) z, integer ldz, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer, dimension( * ) iwork, integer, dimension( * ) ifail, integer info) ZHBGVX Purpose: ZHBGVX computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite banded eigenproblem, of the form A*x=(lambda)*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. Parameters JOBZ JOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors. RANGE RANGE is CHARACTER*1 = 'A': all eigenvalues will be found; = 'V': all eigenvalues in the half-open interval (VL,VU] will be found; = 'I': the IL-th through IU-th eigenvalues will be found. UPLO UPLO is CHARACTER*1 = 'U': Upper triangles of A and B are stored; = 'L': Lower triangles of A and B are stored. N N is INTEGER The order of the matrices A and B. N >= 0. KA KA is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KA >= 0. KB KB is INTEGER The number of superdiagonals of the matrix B if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KB >= 0. AB AB is COMPLEX*16 array, dimension (LDAB, N) On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first ka+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka). On exit, the contents of AB are destroyed. LDAB LDAB is INTEGER The leading dimension of the array AB. LDAB >= KA+1. BB BB is COMPLEX*16 array, dimension (LDBB, N) On entry, the upper or lower triangle of the Hermitian band matrix B, stored in the first kb+1 rows of the array. The j-th column of B is stored in the j-th column of the array BB as follows: if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j; if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb). On exit, the factor S from the split Cholesky factorization B = S**H*S, as returned by ZPBSTF. LDBB LDBB is INTEGER The leading dimension of the array BB. LDBB >= KB+1. Q Q is COMPLEX*16 array, dimension (LDQ, N) If JOBZ = 'V', the n-by-n matrix used in the reduction of A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x, and consequently C to tridiagonal form. If JOBZ = 'N', the array Q is not referenced. LDQ LDQ is INTEGER The leading dimension of the array Q. If JOBZ = 'N', LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N). VL VL is DOUBLE PRECISION If RANGE='V', the lower bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'. VU VU is DOUBLE PRECISION If RANGE='V', the upper bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'. IL IL is INTEGER If RANGE='I', the index of the smallest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'. IU IU is INTEGER If RANGE='I', the index of the largest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'. ABSTOL ABSTOL is DOUBLE PRECISION 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 + EPS * max( |a|,|b| ) , where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-norm of the tridiagonal matrix obtained by reducing AP to tridiagonal form. Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*DLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*DLAMCH('S'). M M is INTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. W W is DOUBLE PRECISION array, dimension (N) If INFO = 0, the eigenvalues in ascending order. Z Z is COMPLEX*16 array, dimension (LDZ, N) If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of eigenvectors, with the i-th column of Z holding the eigenvector associated with W(i). The eigenvectors are normalized so that Z**H*B*Z = I. If JOBZ = 'N', then Z is not referenced. LDZ LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= N. WORK WORK is COMPLEX*16 array, dimension (N) RWORK RWORK is DOUBLE PRECISION array, dimension (7*N) IWORK IWORK is INTEGER array, dimension (5*N) IFAIL IFAIL is INTEGER array, dimension (N) If JOBZ = 'V', then 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. If JOBZ = 'N', then IFAIL is not referenced. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, and i is: <= N: then i eigenvectors failed to converge. Their indices are stored in array IFAIL. > N: if INFO = N + i, for 1 <= i <= N, then ZPBSTF returned INFO = i: B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed. Author Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Contributors: Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
Author
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