SUBROUTINE SLAEBZ( IJOB, NITMAX, N, MMAX, MINP, NBMIN, ABSTOL,
$ RELTOL, PIVMIN, D, E, E2, NVAL, AB, C, MOUT,
$ NAB, WORK, IWORK, INFO )
*
* -- LAPACK auxiliary routine (version 3.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2006
*
* .. Scalar Arguments ..
INTEGER IJOB, INFO, MINP, MMAX, MOUT, N, NBMIN, NITMAX
REAL ABSTOL, PIVMIN, RELTOL
* ..
* .. Array Arguments ..
INTEGER IWORK( * ), NAB( MMAX, * ), NVAL( * )
REAL AB( MMAX, * ), C( * ), D( * ), E( * ), E2( * ),
$ WORK( * )
* ..
*
* Purpose
* =======
*
* SLAEBZ contains the iteration loops which compute and use the
* function N(w), which is the count of eigenvalues of a symmetric
* tridiagonal matrix T less than or equal to its argument w. It
* performs a choice of two types of loops:
*
* IJOB=1, followed by
* IJOB=2: It takes as input a list of intervals and returns a list of
* sufficiently small intervals whose union contains the same
* eigenvalues as the union of the original intervals.
* The input intervals are (AB(j,1),AB(j,2)], j=1,...,MINP.
* The output interval (AB(j,1),AB(j,2)] will contain
* eigenvalues NAB(j,1)+1,...,NAB(j,2), where 1 <= j <= MOUT.
*
* IJOB=3: It performs a binary search in each input interval
* (AB(j,1),AB(j,2)] for a point w(j) such that
* N(w(j))=NVAL(j), and uses C(j) as the starting point of
* the search. If such a w(j) is found, then on output
* AB(j,1)=AB(j,2)=w. If no such w(j) is found, then on output
* (AB(j,1),AB(j,2)] will be a small interval containing the
* point where N(w) jumps through NVAL(j), unless that point
* lies outside the initial interval.
*
* Note that the intervals are in all cases half-open intervals,
* i.e., of the form (a,b] , which includes b but not a .
*
* To avoid underflow, the matrix should be scaled so that its largest
* element is no greater than overflow**(1/2) * underflow**(1/4)
* in absolute value. To assure the most accurate computation
* of small eigenvalues, the matrix should be scaled to be
* not much smaller than that, either.
*
* See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal
* Matrix", Report CS41, Computer Science Dept., Stanford
* University, July 21, 1966
*
* Note: the arguments are, in general, *not* checked for unreasonable
* values.
*
* Arguments
* =========
*
* IJOB (input) INTEGER
* Specifies what is to be done:
* = 1: Compute NAB for the initial intervals.
* = 2: Perform bisection iteration to find eigenvalues of T.
* = 3: Perform bisection iteration to invert N(w), i.e.,
* to find a point which has a specified number of
* eigenvalues of T to its left.
* Other values will cause SLAEBZ to return with INFO=-1.
*
* NITMAX (input) INTEGER
* The maximum number of "levels" of bisection to be
* performed, i.e., an interval of width W will not be made
* smaller than 2^(-NITMAX) * W. If not all intervals
* have converged after NITMAX iterations, then INFO is set
* to the number of non-converged intervals.
*
* N (input) INTEGER
* The dimension n of the tridiagonal matrix T. It must be at
* least 1.
*
* MMAX (input) INTEGER
* The maximum number of intervals. If more than MMAX intervals
* are generated, then SLAEBZ will quit with INFO=MMAX+1.
*
* MINP (input) INTEGER
* The initial number of intervals. It may not be greater than
* MMAX.
*
* NBMIN (input) INTEGER
* The smallest number of intervals that should be processed
* using a vector loop. If zero, then only the scalar loop
* will be used.
*
* ABSTOL (input) REAL
* The minimum (absolute) width of an interval. When an
* interval is narrower than ABSTOL, or than RELTOL times the
* larger (in magnitude) endpoint, then it is considered to be
* sufficiently small, i.e., converged. This must be at least
* zero.
*
* RELTOL (input) REAL
* The minimum relative width of an interval. When an interval
* is narrower than ABSTOL, or than RELTOL times the larger (in
* magnitude) endpoint, then it is considered to be
* sufficiently small, i.e., converged. Note: this should
* always be at least radix*machine epsilon.
*
* PIVMIN (input) REAL
* The minimum absolute value of a "pivot" in the Sturm
* sequence loop. This *must* be at least max |e(j)**2| *
* safe_min and at least safe_min, where safe_min is at least
* the smallest number that can divide one without overflow.
*
* D (input) REAL array, dimension (N)
* The diagonal elements of the tridiagonal matrix T.
*
* E (input) REAL array, dimension (N)
* The offdiagonal elements of the tridiagonal matrix T in
* positions 1 through N-1. E(N) is arbitrary.
*
* E2 (input) REAL array, dimension (N)
* The squares of the offdiagonal elements of the tridiagonal
* matrix T. E2(N) is ignored.
*
* NVAL (input/output) INTEGER array, dimension (MINP)
* If IJOB=1 or 2, not referenced.
* If IJOB=3, the desired values of N(w). The elements of NVAL
* will be reordered to correspond with the intervals in AB.
* Thus, NVAL(j) on output will not, in general be the same as
* NVAL(j) on input, but it will correspond with the interval
* (AB(j,1),AB(j,2)] on output.
*
* AB (input/output) REAL array, dimension (MMAX,2)
* The endpoints of the intervals. AB(j,1) is a(j), the left
* endpoint of the j-th interval, and AB(j,2) is b(j), the
* right endpoint of the j-th interval. The input intervals
* will, in general, be modified, split, and reordered by the
* calculation.
*
* C (input/output) REAL array, dimension (MMAX)
* If IJOB=1, ignored.
* If IJOB=2, workspace.
* If IJOB=3, then on input C(j) should be initialized to the
* first search point in the binary search.
*
* MOUT (output) INTEGER
* If IJOB=1, the number of eigenvalues in the intervals.
* If IJOB=2 or 3, the number of intervals output.
* If IJOB=3, MOUT will equal MINP.
*
* NAB (input/output) INTEGER array, dimension (MMAX,2)
* If IJOB=1, then on output NAB(i,j) will be set to N(AB(i,j)).
* If IJOB=2, then on input, NAB(i,j) should be set. It must
* satisfy the condition:
* N(AB(i,1)) <= NAB(i,1) <= NAB(i,2) <= N(AB(i,2)),
* which means that in interval i only eigenvalues
* NAB(i,1)+1,...,NAB(i,2) will be considered. Usually,
* NAB(i,j)=N(AB(i,j)), from a previous call to SLAEBZ with
* IJOB=1.
* On output, NAB(i,j) will contain
* max(na(k),min(nb(k),N(AB(i,j)))), where k is the index of
* the input interval that the output interval
* (AB(j,1),AB(j,2)] came from, and na(k) and nb(k) are the
* the input values of NAB(k,1) and NAB(k,2).
* If IJOB=3, then on output, NAB(i,j) contains N(AB(i,j)),
* unless N(w) > NVAL(i) for all search points w , in which
* case NAB(i,1) will not be modified, i.e., the output
* value will be the same as the input value (modulo
* reorderings -- see NVAL and AB), or unless N(w) < NVAL(i)
* for all search points w , in which case NAB(i,2) will
* not be modified. Normally, NAB should be set to some
* distinctive value(s) before SLAEBZ is called.
*
* WORK (workspace) REAL array, dimension (MMAX)
* Workspace.
*
* IWORK (workspace) INTEGER array, dimension (MMAX)
* Workspace.
*
* INFO (output) INTEGER
* = 0: All intervals converged.
* = 1--MMAX: The last INFO intervals did not converge.
* = MMAX+1: More than MMAX intervals were generated.
*
* Further Details
* ===============
*
* This routine is intended to be called only by other LAPACK
* routines, thus the interface is less user-friendly. It is intended
* for two purposes:
*
* (a) finding eigenvalues. In this case, SLAEBZ should have one or
* more initial intervals set up in AB, and SLAEBZ should be called
* with IJOB=1. This sets up NAB, and also counts the eigenvalues.
* Intervals with no eigenvalues would usually be thrown out at
* this point. Also, if not all the eigenvalues in an interval i
* are desired, NAB(i,1) can be increased or NAB(i,2) decreased.
* For example, set NAB(i,1)=NAB(i,2)-1 to get the largest
* eigenvalue. SLAEBZ is then called with IJOB=2 and MMAX
* no smaller than the value of MOUT returned by the call with
* IJOB=1. After this (IJOB=2) call, eigenvalues NAB(i,1)+1
* through NAB(i,2) are approximately AB(i,1) (or AB(i,2)) to the
* tolerance specified by ABSTOL and RELTOL.
*
* (b) finding an interval (a',b'] containing eigenvalues w(f),...,w(l).
* In this case, start with a Gershgorin interval (a,b). Set up
* AB to contain 2 search intervals, both initially (a,b). One
* NVAL element should contain f-1 and the other should contain l
* , while C should contain a and b, resp. NAB(i,1) should be -1
* and NAB(i,2) should be N+1, to flag an error if the desired
* interval does not lie in (a,b). SLAEBZ is then called with
* IJOB=3. On exit, if w(f-1) < w(f), then one of the intervals --
* j -- will have AB(j,1)=AB(j,2) and NAB(j,1)=NAB(j,2)=f-1, while
* if, to the specified tolerance, w(f-k)=...=w(f+r), k > 0 and r
* >= 0, then the interval will have N(AB(j,1))=NAB(j,1)=f-k and
* N(AB(j,2))=NAB(j,2)=f+r. The cases w(l) < w(l+1) and
* w(l-r)=...=w(l+k) are handled similarly.
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, TWO, HALF
PARAMETER ( ZERO = 0.0E0, TWO = 2.0E0,
$ HALF = 1.0E0 / TWO )
* ..
* .. Local Scalars ..
INTEGER ITMP1, ITMP2, J, JI, JIT, JP, KF, KFNEW, KL,
$ KLNEW
REAL TMP1, TMP2
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. Executable Statements ..
*
* Check for Errors
*
INFO = 0
IF( IJOB.LT.1 .OR. IJOB.GT.3 ) THEN
INFO = -1
RETURN
END IF
*
* Initialize NAB
*
IF( IJOB.EQ.1 ) THEN
*
* Compute the number of eigenvalues in the initial intervals.
*
MOUT = 0
CDIR$ NOVECTOR
DO 30 JI = 1, MINP
DO 20 JP = 1, 2
TMP1 = D( 1 ) - AB( JI, JP )
IF( ABS( TMP1 ).LT.PIVMIN )
$ TMP1 = -PIVMIN
NAB( JI, JP ) = 0
IF( TMP1.LE.ZERO )
$ NAB( JI, JP ) = 1
*
DO 10 J = 2, N
TMP1 = D( J ) - E2( J-1 ) / TMP1 - AB( JI, JP )
IF( ABS( TMP1 ).LT.PIVMIN )
$ TMP1 = -PIVMIN
IF( TMP1.LE.ZERO )
$ NAB( JI, JP ) = NAB( JI, JP ) + 1
10 CONTINUE
20 CONTINUE
MOUT = MOUT + NAB( JI, 2 ) - NAB( JI, 1 )
30 CONTINUE
RETURN
END IF
*
* Initialize for loop
*
* KF and KL have the following meaning:
* Intervals 1,...,KF-1 have converged.
* Intervals KF,...,KL still need to be refined.
*
KF = 1
KL = MINP
*
* If IJOB=2, initialize C.
* If IJOB=3, use the user-supplied starting point.
*
IF( IJOB.EQ.2 ) THEN
DO 40 JI = 1, MINP
C( JI ) = HALF*( AB( JI, 1 )+AB( JI, 2 ) )
40 CONTINUE
END IF
*
* Iteration loop
*
DO 130 JIT = 1, NITMAX
*
* Loop over intervals
*
IF( KL-KF+1.GE.NBMIN .AND. NBMIN.GT.0 ) THEN
*
* Begin of Parallel Version of the loop
*
DO 60 JI = KF, KL
*
* Compute N(c), the number of eigenvalues less than c
*
WORK( JI ) = D( 1 ) - C( JI )
IWORK( JI ) = 0
IF( WORK( JI ).LE.PIVMIN ) THEN
IWORK( JI ) = 1
WORK( JI ) = MIN( WORK( JI ), -PIVMIN )
END IF
*
DO 50 J = 2, N
WORK( JI ) = D( J ) - E2( J-1 ) / WORK( JI ) - C( JI )
IF( WORK( JI ).LE.PIVMIN ) THEN
IWORK( JI ) = IWORK( JI ) + 1
WORK( JI ) = MIN( WORK( JI ), -PIVMIN )
END IF
50 CONTINUE
60 CONTINUE
*
IF( IJOB.LE.2 ) THEN
*
* IJOB=2: Choose all intervals containing eigenvalues.
*
KLNEW = KL
DO 70 JI = KF, KL
*
* Insure that N(w) is monotone
*
IWORK( JI ) = MIN( NAB( JI, 2 ),
$ MAX( NAB( JI, 1 ), IWORK( JI ) ) )
*
* Update the Queue -- add intervals if both halves
* contain eigenvalues.
*
IF( IWORK( JI ).EQ.NAB( JI, 2 ) ) THEN
*
* No eigenvalue in the upper interval:
* just use the lower interval.
*
AB( JI, 2 ) = C( JI )
*
ELSE IF( IWORK( JI ).EQ.NAB( JI, 1 ) ) THEN
*
* No eigenvalue in the lower interval:
* just use the upper interval.
*
AB( JI, 1 ) = C( JI )
ELSE
KLNEW = KLNEW + 1
IF( KLNEW.LE.MMAX ) THEN
*
* Eigenvalue in both intervals -- add upper to
* queue.
*
AB( KLNEW, 2 ) = AB( JI, 2 )
NAB( KLNEW, 2 ) = NAB( JI, 2 )
AB( KLNEW, 1 ) = C( JI )
NAB( KLNEW, 1 ) = IWORK( JI )
AB( JI, 2 ) = C( JI )
NAB( JI, 2 ) = IWORK( JI )
ELSE
INFO = MMAX + 1
END IF
END IF
70 CONTINUE
IF( INFO.NE.0 )
$ RETURN
KL = KLNEW
ELSE
*
* IJOB=3: Binary search. Keep only the interval containing
* w s.t. N(w) = NVAL
*
DO 80 JI = KF, KL
IF( IWORK( JI ).LE.NVAL( JI ) ) THEN
AB( JI, 1 ) = C( JI )
NAB( JI, 1 ) = IWORK( JI )
END IF
IF( IWORK( JI ).GE.NVAL( JI ) ) THEN
AB( JI, 2 ) = C( JI )
NAB( JI, 2 ) = IWORK( JI )
END IF
80 CONTINUE
END IF
*
ELSE
*
* End of Parallel Version of the loop
*
* Begin of Serial Version of the loop
*
KLNEW = KL
DO 100 JI = KF, KL
*
* Compute N(w), the number of eigenvalues less than w
*
TMP1 = C( JI )
TMP2 = D( 1 ) - TMP1
ITMP1 = 0
IF( TMP2.LE.PIVMIN ) THEN
ITMP1 = 1
TMP2 = MIN( TMP2, -PIVMIN )
END IF
*
* A series of compiler directives to defeat vectorization
* for the next loop
*
*$PL$ CMCHAR=' '
CDIR$ NEXTSCALAR
C$DIR SCALAR
CDIR$ NEXT SCALAR
CVD$L NOVECTOR
CDEC$ NOVECTOR
CVD$ NOVECTOR
*VDIR NOVECTOR
*VOCL LOOP,SCALAR
CIBM PREFER SCALAR
*$PL$ CMCHAR='*'
*
DO 90 J = 2, N
TMP2 = D( J ) - E2( J-1 ) / TMP2 - TMP1
IF( TMP2.LE.PIVMIN ) THEN
ITMP1 = ITMP1 + 1
TMP2 = MIN( TMP2, -PIVMIN )
END IF
90 CONTINUE
*
IF( IJOB.LE.2 ) THEN
*
* IJOB=2: Choose all intervals containing eigenvalues.
*
* Insure that N(w) is monotone
*
ITMP1 = MIN( NAB( JI, 2 ),
$ MAX( NAB( JI, 1 ), ITMP1 ) )
*
* Update the Queue -- add intervals if both halves
* contain eigenvalues.
*
IF( ITMP1.EQ.NAB( JI, 2 ) ) THEN
*
* No eigenvalue in the upper interval:
* just use the lower interval.
*
AB( JI, 2 ) = TMP1
*
ELSE IF( ITMP1.EQ.NAB( JI, 1 ) ) THEN
*
* No eigenvalue in the lower interval:
* just use the upper interval.
*
AB( JI, 1 ) = TMP1
ELSE IF( KLNEW.LT.MMAX ) THEN
*
* Eigenvalue in both intervals -- add upper to queue.
*
KLNEW = KLNEW + 1
AB( KLNEW, 2 ) = AB( JI, 2 )
NAB( KLNEW, 2 ) = NAB( JI, 2 )
AB( KLNEW, 1 ) = TMP1
NAB( KLNEW, 1 ) = ITMP1
AB( JI, 2 ) = TMP1
NAB( JI, 2 ) = ITMP1
ELSE
INFO = MMAX + 1
RETURN
END IF
ELSE
*
* IJOB=3: Binary search. Keep only the interval
* containing w s.t. N(w) = NVAL
*
IF( ITMP1.LE.NVAL( JI ) ) THEN
AB( JI, 1 ) = TMP1
NAB( JI, 1 ) = ITMP1
END IF
IF( ITMP1.GE.NVAL( JI ) ) THEN
AB( JI, 2 ) = TMP1
NAB( JI, 2 ) = ITMP1
END IF
END IF
100 CONTINUE
KL = KLNEW
*
* End of Serial Version of the loop
*
END IF
*
* Check for convergence
*
KFNEW = KF
DO 110 JI = KF, KL
TMP1 = ABS( AB( JI, 2 )-AB( JI, 1 ) )
TMP2 = MAX( ABS( AB( JI, 2 ) ), ABS( AB( JI, 1 ) ) )
IF( TMP1.LT.MAX( ABSTOL, PIVMIN, RELTOL*TMP2 ) .OR.
$ NAB( JI, 1 ).GE.NAB( JI, 2 ) ) THEN
*
* Converged -- Swap with position KFNEW,
* then increment KFNEW
*
IF( JI.GT.KFNEW ) THEN
TMP1 = AB( JI, 1 )
TMP2 = AB( JI, 2 )
ITMP1 = NAB( JI, 1 )
ITMP2 = NAB( JI, 2 )
AB( JI, 1 ) = AB( KFNEW, 1 )
AB( JI, 2 ) = AB( KFNEW, 2 )
NAB( JI, 1 ) = NAB( KFNEW, 1 )
NAB( JI, 2 ) = NAB( KFNEW, 2 )
AB( KFNEW, 1 ) = TMP1
AB( KFNEW, 2 ) = TMP2
NAB( KFNEW, 1 ) = ITMP1
NAB( KFNEW, 2 ) = ITMP2
IF( IJOB.EQ.3 ) THEN
ITMP1 = NVAL( JI )
NVAL( JI ) = NVAL( KFNEW )
NVAL( KFNEW ) = ITMP1
END IF
END IF
KFNEW = KFNEW + 1
END IF
110 CONTINUE
KF = KFNEW
*
* Choose Midpoints
*
DO 120 JI = KF, KL
C( JI ) = HALF*( AB( JI, 1 )+AB( JI, 2 ) )
120 CONTINUE
*
* If no more intervals to refine, quit.
*
IF( KF.GT.KL )
$ GO TO 140
130 CONTINUE
*
* Converged
*
140 CONTINUE
INFO = MAX( KL+1-KF, 0 )
MOUT = KL
*
RETURN
*
* End of SLAEBZ
*
END