table of contents
| stebz(3) | LAPACK | stebz(3) |
NAME¶
stebz - stebz: eig, Kahan
SYNOPSIS¶
Functions¶
subroutine dstebz (range, order, n, vl, vu, il, iu, abstol,
d, e, m, nsplit, w, iblock, isplit, work, iwork, info)
DSTEBZ subroutine sstebz (range, order, n, vl, vu, il, iu,
abstol, d, e, m, nsplit, w, iblock, isplit, work, iwork, info)
SSTEBZ
Detailed Description¶
Function Documentation¶
subroutine dstebz (character range, character order, integer n, double precision vl, double precision vu, integer il, integer iu, double precision abstol, double precision, dimension( * ) d, double precision, dimension( * ) e, integer m, integer nsplit, double precision, dimension( * ) w, integer, dimension( * ) iblock, integer, dimension( * ) isplit, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)¶
DSTEBZ
Purpose:
DSTEBZ computes the eigenvalues of a symmetric tridiagonal
matrix T. The user may ask for all eigenvalues, all eigenvalues
in the half-open interval (VL, VU], or the IL-th through IU-th
eigenvalues.
To avoid overflow, the matrix must be scaled so that its
largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest
accuracy, it should not be much smaller than that.
See W. Kahan 'Accurate Eigenvalues of a Symmetric Tridiagonal
Matrix', Report CS41, Computer Science Dept., Stanford
University, July 21, 1966.
Parameters
RANGE
RANGE is CHARACTER*1
= 'A': ('All') all eigenvalues will be found.
= 'V': ('Value') all eigenvalues in the half-open interval
(VL, VU] will be found.
= 'I': ('Index') the IL-th through IU-th eigenvalues (of the
entire matrix) will be found.
ORDER
ORDER is CHARACTER*1
= 'B': ('By Block') the eigenvalues will be grouped by
split-off block (see IBLOCK, ISPLIT) and
ordered from smallest to largest within
the block.
= 'E': ('Entire matrix')
the eigenvalues for the entire matrix
will be ordered from smallest to
largest.
N
N is INTEGER
The order of the tridiagonal matrix T. N >= 0.
VL
VL is DOUBLE PRECISION
If RANGE='V', the lower bound of the interval to
be searched for eigenvalues. Eigenvalues less than or equal
to VL, or greater than VU, will not be returned. 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. Eigenvalues less than or equal
to VL, or greater than VU, will not be returned. 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 tolerance for the eigenvalues. An eigenvalue
(or cluster) is considered to be located if it has been
determined to lie in an interval whose width is ABSTOL or
less. If ABSTOL is less than or equal to zero, then ULP*|T|
will be used, where |T| means the 1-norm of T.
Eigenvalues will be computed most accurately when ABSTOL is
set to twice the underflow threshold 2*DLAMCH('S'), not zero.
D
D is DOUBLE PRECISION array, dimension (N)
The n diagonal elements of the tridiagonal matrix T.
E
E is DOUBLE PRECISION array, dimension (N-1)
The (n-1) off-diagonal elements of the tridiagonal matrix T.
M
M is INTEGER
The actual number of eigenvalues found. 0 <= M <= N.
(See also the description of INFO=2,3.)
NSPLIT
NSPLIT is INTEGER
The number of diagonal blocks in the matrix T.
1 <= NSPLIT <= N.
W
W is DOUBLE PRECISION array, dimension (N)
On exit, the first M elements of W will contain the
eigenvalues. (DSTEBZ may use the remaining N-M elements as
workspace.)
IBLOCK
IBLOCK is INTEGER array, dimension (N)
At each row/column j where E(j) is zero or small, the
matrix T is considered to split into a block diagonal
matrix. On exit, if INFO = 0, IBLOCK(i) specifies to which
block (from 1 to the number of blocks) the eigenvalue W(i)
belongs. (DSTEBZ may use the remaining N-M elements as
workspace.)
ISPLIT
ISPLIT is INTEGER array, dimension (N)
The splitting points, at which T breaks up into submatrices.
The first submatrix consists of rows/columns 1 to ISPLIT(1),
the second of rows/columns ISPLIT(1)+1 through ISPLIT(2),
etc., and the NSPLIT-th consists of rows/columns
ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.
(Only the first NSPLIT elements will actually be used, but
since the user cannot know a priori what value NSPLIT will
have, N words must be reserved for ISPLIT.)
WORK
WORK is DOUBLE PRECISION array, dimension (4*N)
IWORK
IWORK is INTEGER array, dimension (3*N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: some or all of the eigenvalues failed to converge or
were not computed:
=1 or 3: Bisection failed to converge for some
eigenvalues; these eigenvalues are flagged by a
negative block number. The effect is that the
eigenvalues may not be as accurate as the
absolute and relative tolerances. This is
generally caused by unexpectedly inaccurate
arithmetic.
=2 or 3: RANGE='I' only: Not all of the eigenvalues
IL:IU were found.
Effect: M < IU+1-IL
Cause: non-monotonic arithmetic, causing the
Sturm sequence to be non-monotonic.
Cure: recalculate, using RANGE='A', and pick
out eigenvalues IL:IU. In some cases,
increasing the PARAMETER 'FUDGE' may
make things work.
= 4: RANGE='I', and the Gershgorin interval
initially used was too small. No eigenvalues
were computed.
Probable cause: your machine has sloppy
floating-point arithmetic.
Cure: Increase the PARAMETER 'FUDGE',
recompile, and try again.
Internal Parameters:
RELFAC DOUBLE PRECISION, default = 2.0e0
The relative tolerance. An interval (a,b] lies within
'relative tolerance' if b-a < RELFAC*ulp*max(|a|,|b|),
where 'ulp' is the machine precision (distance from 1 to
the next larger floating point number.)
FUDGE DOUBLE PRECISION, default = 2
A 'fudge factor' to widen the Gershgorin intervals. Ideally,
a value of 1 should work, but on machines with sloppy
arithmetic, this needs to be larger. The default for
publicly released versions should be large enough to handle
the worst machine around. Note that this has no effect
on accuracy of the solution.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sstebz (character range, character order, integer n, real vl, real vu, integer il, integer iu, real abstol, real, dimension( * ) d, real, dimension( * ) e, integer m, integer nsplit, real, dimension( * ) w, integer, dimension( * ) iblock, integer, dimension( * ) isplit, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)¶
SSTEBZ
Purpose:
SSTEBZ computes the eigenvalues of a symmetric tridiagonal
matrix T. The user may ask for all eigenvalues, all eigenvalues
in the half-open interval (VL, VU], or the IL-th through IU-th
eigenvalues.
To avoid overflow, the matrix must be scaled so that its
largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest
accuracy, it should not be much smaller than that.
See W. Kahan 'Accurate Eigenvalues of a Symmetric Tridiagonal
Matrix', Report CS41, Computer Science Dept., Stanford
University, July 21, 1966.
Parameters
RANGE
RANGE is CHARACTER*1
= 'A': ('All') all eigenvalues will be found.
= 'V': ('Value') all eigenvalues in the half-open interval
(VL, VU] will be found.
= 'I': ('Index') the IL-th through IU-th eigenvalues (of the
entire matrix) will be found.
ORDER
ORDER is CHARACTER*1
= 'B': ('By Block') the eigenvalues will be grouped by
split-off block (see IBLOCK, ISPLIT) and
ordered from smallest to largest within
the block.
= 'E': ('Entire matrix')
the eigenvalues for the entire matrix
will be ordered from smallest to
largest.
N
N is INTEGER
The order of the tridiagonal matrix T. N >= 0.
VL
VL is REAL
If RANGE='V', the lower bound of the interval to
be searched for eigenvalues. Eigenvalues less than or equal
to VL, or greater than VU, will not be returned. 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. Eigenvalues less than or equal
to VL, or greater than VU, will not be returned. 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 tolerance for the eigenvalues. An eigenvalue
(or cluster) is considered to be located if it has been
determined to lie in an interval whose width is ABSTOL or
less. If ABSTOL is less than or equal to zero, then ULP*|T|
will be used, where |T| means the 1-norm of T.
Eigenvalues will be computed most accurately when ABSTOL is
set to twice the underflow threshold 2*SLAMCH('S'), not zero.
D
D is REAL array, dimension (N)
The n diagonal elements of the tridiagonal matrix T.
E
E is REAL array, dimension (N-1)
The (n-1) off-diagonal elements of the tridiagonal matrix T.
M
M is INTEGER
The actual number of eigenvalues found. 0 <= M <= N.
(See also the description of INFO=2,3.)
NSPLIT
NSPLIT is INTEGER
The number of diagonal blocks in the matrix T.
1 <= NSPLIT <= N.
W
W is REAL array, dimension (N)
On exit, the first M elements of W will contain the
eigenvalues. (SSTEBZ may use the remaining N-M elements as
workspace.)
IBLOCK
IBLOCK is INTEGER array, dimension (N)
At each row/column j where E(j) is zero or small, the
matrix T is considered to split into a block diagonal
matrix. On exit, if INFO = 0, IBLOCK(i) specifies to which
block (from 1 to the number of blocks) the eigenvalue W(i)
belongs. (SSTEBZ may use the remaining N-M elements as
workspace.)
ISPLIT
ISPLIT is INTEGER array, dimension (N)
The splitting points, at which T breaks up into submatrices.
The first submatrix consists of rows/columns 1 to ISPLIT(1),
the second of rows/columns ISPLIT(1)+1 through ISPLIT(2),
etc., and the NSPLIT-th consists of rows/columns
ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.
(Only the first NSPLIT elements will actually be used, but
since the user cannot know a priori what value NSPLIT will
have, N words must be reserved for ISPLIT.)
WORK
WORK is REAL array, dimension (4*N)
IWORK
IWORK is INTEGER array, dimension (3*N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: some or all of the eigenvalues failed to converge or
were not computed:
=1 or 3: Bisection failed to converge for some
eigenvalues; these eigenvalues are flagged by a
negative block number. The effect is that the
eigenvalues may not be as accurate as the
absolute and relative tolerances. This is
generally caused by unexpectedly inaccurate
arithmetic.
=2 or 3: RANGE='I' only: Not all of the eigenvalues
IL:IU were found.
Effect: M < IU+1-IL
Cause: non-monotonic arithmetic, causing the
Sturm sequence to be non-monotonic.
Cure: recalculate, using RANGE='A', and pick
out eigenvalues IL:IU. In some cases,
increasing the PARAMETER 'FUDGE' may
make things work.
= 4: RANGE='I', and the Gershgorin interval
initially used was too small. No eigenvalues
were computed.
Probable cause: your machine has sloppy
floating-point arithmetic.
Cure: Increase the PARAMETER 'FUDGE',
recompile, and try again.
Internal Parameters:
RELFAC REAL, default = 2.0e0
The relative tolerance. An interval (a,b] lies within
'relative tolerance' if b-a < RELFAC*ulp*max(|a|,|b|),
where 'ulp' is the machine precision (distance from 1 to
the next larger floating point number.)
FUDGE REAL, default = 2
A 'fudge factor' to widen the Gershgorin intervals. Ideally,
a value of 1 should work, but on machines with sloppy
arithmetic, this needs to be larger. The default for
publicly released versions should be large enough to handle
the worst machine around. Note that this has no effect
on accuracy of the solution.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Author¶
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