DSBEVX(3F) DSBEVX(3F)
DSBEVX - compute selected eigenvalues and, optionally, eigenvectors of a
real symmetric band matrix A
SUBROUTINE DSBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, VU,
IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO
)
CHARACTER JOBZ, RANGE, UPLO
INTEGER IL, INFO, IU, KD, LDAB, LDQ, LDZ, M, N
DOUBLE PRECISION ABSTOL, VL, VU
INTEGER IFAIL( * ), IWORK( * )
DOUBLE PRECISION AB( LDAB, * ), Q( LDQ, * ), W( * ), WORK( *
), Z( LDZ, * )
DSBEVX computes selected eigenvalues and, optionally, eigenvectors of a
real symmetric 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.
JOBZ (input) CHARACTER*1
= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.
RANGE (input) 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 (input) CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N (input) INTEGER
The order of the matrix A. N >= 0.
KD (input) INTEGER
The number of superdiagonals of the matrix A if UPLO = 'U', or
the number of subdiagonals if UPLO = 'L'. KD >= 0.
AB (input/output) DOUBLE PRECISION array, dimension (LDAB, N)
On entry, the upper or lower triangle of the symmetric band
matrix A, stored in the first KD+1 rows of the array. The j-th
column of A is stored in the j-th column of the array AB as
Page 1
DSBEVX(3F) DSBEVX(3F)
follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,jkd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j) = A(i,j) for
j<=i<=min(n,j+kd).
On exit, AB is overwritten by values generated during the
reduction to tridiagonal form. If UPLO = 'U', the first
superdiagonal and the diagonal of the tridiagonal matrix T are
returned in rows KD and KD+1 of AB, and if UPLO = 'L', the
diagonal and first subdiagonal of T are returned in the first two
rows of AB.
LDAB (input) INTEGER
The leading dimension of the array AB. LDAB >= KD + 1.
Q (output) DOUBLE PRECISION array, dimension (LDQ, N)
If JOBZ = 'V', the N-by-N orthogonal matrix used in the reduction
to tridiagonal form. If JOBZ = 'N', the array Q is not
referenced.
LDQ (input) INTEGER
The leading dimension of the array Q. If JOBZ = 'V', then LDQ >=
max(1,N).
VL (input) DOUBLE PRECISION
VU (input) DOUBLE PRECISION If RANGE='V', the lower and
upper bounds of the interval to be searched for eigenvalues. VL <
VU. Not referenced if RANGE = 'A' or 'I'.
IL (input) INTEGER
IU (input) INTEGER If RANGE='I', the indices (in ascending
order) of the smallest and largest eigenvalues 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 (input) 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
AB 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').
Page 2
DSBEVX(3F) DSBEVX(3F)
See "Computing Small Singular Values of Bidiagonal Matrices with
Guaranteed High Relative Accuracy," by Demmel and Kahan, LAPACK
Working Note #3.
M (output) INTEGER
The total number of eigenvalues found. 0 <= M <= N. If RANGE =
'A', M = N, and if RANGE = 'I', M = IU-IL+1.
W (output) DOUBLE PRECISION array, dimension (N)
The first M elements contain the selected eigenvalues in
ascending order.
Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M))
If JOBZ = 'V', then if INFO = 0, 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 holding the
eigenvector associated with 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. If JOBZ = 'N', then Z is not referenced. Note: the
user must ensure that at least max(1,M) columns are supplied in
the array Z; if RANGE = 'V', the exact value of M is not known in
advance and an upper bound must be used.
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= 1, and if JOBZ =
'V', LDZ >= max(1,N).
WORK (workspace) DOUBLE PRECISION array, dimension (7*N)
IWORK (workspace) INTEGER array, dimension (5*N)
IFAIL (output) 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 (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.
DSBEVX(3F) DSBEVX(3F)
DSBEVX - compute selected eigenvalues and, optionally, eigenvectors of a
real symmetric band matrix A
SUBROUTINE DSBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, VU,
IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO
)
CHARACTER JOBZ, RANGE, UPLO
INTEGER IL, INFO, IU, KD, LDAB, LDQ, LDZ, M, N
DOUBLE PRECISION ABSTOL, VL, VU
INTEGER IFAIL( * ), IWORK( * )
DOUBLE PRECISION AB( LDAB, * ), Q( LDQ, * ), W( * ), WORK( *
), Z( LDZ, * )
DSBEVX computes selected eigenvalues and, optionally, eigenvectors of a
real symmetric 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.
JOBZ (input) CHARACTER*1
= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.
RANGE (input) 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 (input) CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N (input) INTEGER
The order of the matrix A. N >= 0.
KD (input) INTEGER
The number of superdiagonals of the matrix A if UPLO = 'U', or
the number of subdiagonals if UPLO = 'L'. KD >= 0.
AB (input/output) DOUBLE PRECISION array, dimension (LDAB, N)
On entry, the upper or lower triangle of the symmetric band
matrix A, stored in the first KD+1 rows of the array. The j-th
column of A is stored in the j-th column of the array AB as
Page 1
DSBEVX(3F) DSBEVX(3F)
follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,jkd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j) = A(i,j) for
j<=i<=min(n,j+kd).
On exit, AB is overwritten by values generated during the
reduction to tridiagonal form. If UPLO = 'U', the first
superdiagonal and the diagonal of the tridiagonal matrix T are
returned in rows KD and KD+1 of AB, and if UPLO = 'L', the
diagonal and first subdiagonal of T are returned in the first two
rows of AB.
LDAB (input) INTEGER
The leading dimension of the array AB. LDAB >= KD + 1.
Q (output) DOUBLE PRECISION array, dimension (LDQ, N)
If JOBZ = 'V', the N-by-N orthogonal matrix used in the reduction
to tridiagonal form. If JOBZ = 'N', the array Q is not
referenced.
LDQ (input) INTEGER
The leading dimension of the array Q. If JOBZ = 'V', then LDQ >=
max(1,N).
VL (input) DOUBLE PRECISION
VU (input) DOUBLE PRECISION If RANGE='V', the lower and
upper bounds of the interval to be searched for eigenvalues. VL <
VU. Not referenced if RANGE = 'A' or 'I'.
IL (input) INTEGER
IU (input) INTEGER If RANGE='I', the indices (in ascending
order) of the smallest and largest eigenvalues 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 (input) 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
AB 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').
Page 2
DSBEVX(3F) DSBEVX(3F)
See "Computing Small Singular Values of Bidiagonal Matrices with
Guaranteed High Relative Accuracy," by Demmel and Kahan, LAPACK
Working Note #3.
M (output) INTEGER
The total number of eigenvalues found. 0 <= M <= N. If RANGE =
'A', M = N, and if RANGE = 'I', M = IU-IL+1.
W (output) DOUBLE PRECISION array, dimension (N)
The first M elements contain the selected eigenvalues in
ascending order.
Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M))
If JOBZ = 'V', then if INFO = 0, 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 holding the
eigenvector associated with 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. If JOBZ = 'N', then Z is not referenced. Note: the
user must ensure that at least max(1,M) columns are supplied in
the array Z; if RANGE = 'V', the exact value of M is not known in
advance and an upper bound must be used.
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= 1, and if JOBZ =
'V', LDZ >= max(1,N).
WORK (workspace) DOUBLE PRECISION array, dimension (7*N)
IWORK (workspace) INTEGER array, dimension (5*N)
IFAIL (output) 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 (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.
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