slaebz (l)  Linux Man Pages
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
NAME
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 wSYNOPSIS
 SUBROUTINE SLAEBZ(
 IJOB, NITMAX, N, MMAX, MINP, NBMIN, ABSTOL, RELTOL, PIVMIN, D, E, E2, NVAL, AB, C, MOUT, NAB, WORK, IWORK, INFO )
 INTEGER IJOB, INFO, MINP, MMAX, MOUT, N, NBMIN, NITMAX
 REAL ABSTOL, PIVMIN, RELTOL
 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
Note that the intervals are in all cases halfopen 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 nonconverged 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 N1. 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 jth interval, and AB(j,2) is b(j), the right endpoint of the jth 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.
= 1MMAX: The last INFO intervals did not converge.
= MMAX+1: More than MMAX intervals were generated.
FURTHER DETAILS
(a) finding eigenvalues. In this case, SLAEBZ should have one or
(b) finding an interval (aaq,baq] containing eigenvalues w(f),...,w(l).