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---
:name: dlaebz
:md5sum: 667b9d1d21f6f805330766d1f74f173b
:category: :subroutine
:arguments:
- ijob:
:type: integer
:intent: input
- nitmax:
:type: integer
:intent: input
- n:
:type: integer
:intent: input
- mmax:
:type: integer
:intent: input
- minp:
:type: integer
:intent: input
- nbmin:
:type: integer
:intent: input
- abstol:
:type: doublereal
:intent: input
- reltol:
:type: doublereal
:intent: input
- pivmin:
:type: doublereal
:intent: input
- d:
:type: doublereal
:intent: input
:dims:
- n
- e:
:type: doublereal
:intent: input
:dims:
- n
- e2:
:type: doublereal
:intent: input
:dims:
- n
- nval:
:type: integer
:intent: input/output
:dims:
- "(ijob==1||ijob==2) ? 0 : ijob==3 ? minp : 0"
- ab:
:type: doublereal
:intent: input/output
:dims:
- mmax
- "2"
- c:
:type: doublereal
:intent: input/output
:dims:
- "ijob==1 ? 0 : (ijob==2||ijob==3) ? mmax : 0"
- mout:
:type: integer
:intent: output
- nab:
:type: integer
:intent: input/output
:dims:
- mmax
- "2"
- work:
:type: doublereal
:intent: workspace
:dims:
- mmax
- iwork:
:type: integer
:intent: workspace
:dims:
- mmax
- info:
:type: integer
:intent: output
:substitutions: {}
:fortran_help: " SUBROUTINE DLAEBZ( IJOB, NITMAX, N, MMAX, MINP, NBMIN, ABSTOL, RELTOL, PIVMIN, D, E, E2, NVAL, AB, C, MOUT, NAB, WORK, IWORK, INFO )\n\n\
* Purpose\n\
* =======\n\
*\n\
* DLAEBZ contains the iteration loops which compute and use the\n\
* function N(w), which is the count of eigenvalues of a symmetric\n\
* tridiagonal matrix T less than or equal to its argument w. It\n\
* performs a choice of two types of loops:\n\
*\n\
* IJOB=1, followed by\n\
* IJOB=2: It takes as input a list of intervals and returns a list of\n\
* sufficiently small intervals whose union contains the same\n\
* eigenvalues as the union of the original intervals.\n\
* The input intervals are (AB(j,1),AB(j,2)], j=1,...,MINP.\n\
* The output interval (AB(j,1),AB(j,2)] will contain\n\
* eigenvalues NAB(j,1)+1,...,NAB(j,2), where 1 <= j <= MOUT.\n\
*\n\
* IJOB=3: It performs a binary search in each input interval\n\
* (AB(j,1),AB(j,2)] for a point w(j) such that\n\
* N(w(j))=NVAL(j), and uses C(j) as the starting point of\n\
* the search. If such a w(j) is found, then on output\n\
* AB(j,1)=AB(j,2)=w. If no such w(j) is found, then on output\n\
* (AB(j,1),AB(j,2)] will be a small interval containing the\n\
* point where N(w) jumps through NVAL(j), unless that point\n\
* lies outside the initial interval.\n\
*\n\
* Note that the intervals are in all cases half-open intervals,\n\
* i.e., of the form (a,b] , which includes b but not a .\n\
*\n\
* To avoid underflow, the matrix should be scaled so that its largest\n\
* element is no greater than overflow**(1/2) * underflow**(1/4)\n\
* in absolute value. To assure the most accurate computation\n\
* of small eigenvalues, the matrix should be scaled to be\n\
* not much smaller than that, either.\n\
*\n\
* See W. Kahan \"Accurate Eigenvalues of a Symmetric Tridiagonal\n\
* Matrix\", Report CS41, Computer Science Dept., Stanford\n\
* University, July 21, 1966\n\
*\n\
* Note: the arguments are, in general, *not* checked for unreasonable\n\
* values.\n\
*\n\n\
* Arguments\n\
* =========\n\
*\n\
* IJOB (input) INTEGER\n\
* Specifies what is to be done:\n\
* = 1: Compute NAB for the initial intervals.\n\
* = 2: Perform bisection iteration to find eigenvalues of T.\n\
* = 3: Perform bisection iteration to invert N(w), i.e.,\n\
* to find a point which has a specified number of\n\
* eigenvalues of T to its left.\n\
* Other values will cause DLAEBZ to return with INFO=-1.\n\
*\n\
* NITMAX (input) INTEGER\n\
* The maximum number of \"levels\" of bisection to be\n\
* performed, i.e., an interval of width W will not be made\n\
* smaller than 2^(-NITMAX) * W. If not all intervals\n\
* have converged after NITMAX iterations, then INFO is set\n\
* to the number of non-converged intervals.\n\
*\n\
* N (input) INTEGER\n\
* The dimension n of the tridiagonal matrix T. It must be at\n\
* least 1.\n\
*\n\
* MMAX (input) INTEGER\n\
* The maximum number of intervals. If more than MMAX intervals\n\
* are generated, then DLAEBZ will quit with INFO=MMAX+1.\n\
*\n\
* MINP (input) INTEGER\n\
* The initial number of intervals. It may not be greater than\n\
* MMAX.\n\
*\n\
* NBMIN (input) INTEGER\n\
* The smallest number of intervals that should be processed\n\
* using a vector loop. If zero, then only the scalar loop\n\
* will be used.\n\
*\n\
* ABSTOL (input) DOUBLE PRECISION\n\
* The minimum (absolute) width of an interval. When an\n\
* interval is narrower than ABSTOL, or than RELTOL times the\n\
* larger (in magnitude) endpoint, then it is considered to be\n\
* sufficiently small, i.e., converged. This must be at least\n\
* zero.\n\
*\n\
* RELTOL (input) DOUBLE PRECISION\n\
* The minimum relative width of an interval. When an interval\n\
* is narrower than ABSTOL, or than RELTOL times the larger (in\n\
* magnitude) endpoint, then it is considered to be\n\
* sufficiently small, i.e., converged. Note: this should\n\
* always be at least radix*machine epsilon.\n\
*\n\
* PIVMIN (input) DOUBLE PRECISION\n\
* The minimum absolute value of a \"pivot\" in the Sturm\n\
* sequence loop. This *must* be at least max |e(j)**2| *\n\
* safe_min and at least safe_min, where safe_min is at least\n\
* the smallest number that can divide one without overflow.\n\
*\n\
* D (input) DOUBLE PRECISION array, dimension (N)\n\
* The diagonal elements of the tridiagonal matrix T.\n\
*\n\
* E (input) DOUBLE PRECISION array, dimension (N)\n\
* The offdiagonal elements of the tridiagonal matrix T in\n\
* positions 1 through N-1. E(N) is arbitrary.\n\
*\n\
* E2 (input) DOUBLE PRECISION array, dimension (N)\n\
* The squares of the offdiagonal elements of the tridiagonal\n\
* matrix T. E2(N) is ignored.\n\
*\n\
* NVAL (input/output) INTEGER array, dimension (MINP)\n\
* If IJOB=1 or 2, not referenced.\n\
* If IJOB=3, the desired values of N(w). The elements of NVAL\n\
* will be reordered to correspond with the intervals in AB.\n\
* Thus, NVAL(j) on output will not, in general be the same as\n\
* NVAL(j) on input, but it will correspond with the interval\n\
* (AB(j,1),AB(j,2)] on output.\n\
*\n\
* AB (input/output) DOUBLE PRECISION array, dimension (MMAX,2)\n\
* The endpoints of the intervals. AB(j,1) is a(j), the left\n\
* endpoint of the j-th interval, and AB(j,2) is b(j), the\n\
* right endpoint of the j-th interval. The input intervals\n\
* will, in general, be modified, split, and reordered by the\n\
* calculation.\n\
*\n\
* C (input/output) DOUBLE PRECISION array, dimension (MMAX)\n\
* If IJOB=1, ignored.\n\
* If IJOB=2, workspace.\n\
* If IJOB=3, then on input C(j) should be initialized to the\n\
* first search point in the binary search.\n\
*\n\
* MOUT (output) INTEGER\n\
* If IJOB=1, the number of eigenvalues in the intervals.\n\
* If IJOB=2 or 3, the number of intervals output.\n\
* If IJOB=3, MOUT will equal MINP.\n\
*\n\
* NAB (input/output) INTEGER array, dimension (MMAX,2)\n\
* If IJOB=1, then on output NAB(i,j) will be set to N(AB(i,j)).\n\
* If IJOB=2, then on input, NAB(i,j) should be set. It must\n\
* satisfy the condition:\n\
* N(AB(i,1)) <= NAB(i,1) <= NAB(i,2) <= N(AB(i,2)),\n\
* which means that in interval i only eigenvalues\n\
* NAB(i,1)+1,...,NAB(i,2) will be considered. Usually,\n\
* NAB(i,j)=N(AB(i,j)), from a previous call to DLAEBZ with\n\
* IJOB=1.\n\
* On output, NAB(i,j) will contain\n\
* max(na(k),min(nb(k),N(AB(i,j)))), where k is the index of\n\
* the input interval that the output interval\n\
* (AB(j,1),AB(j,2)] came from, and na(k) and nb(k) are the\n\
* the input values of NAB(k,1) and NAB(k,2).\n\
* If IJOB=3, then on output, NAB(i,j) contains N(AB(i,j)),\n\
* unless N(w) > NVAL(i) for all search points w , in which\n\
* case NAB(i,1) will not be modified, i.e., the output\n\
* value will be the same as the input value (modulo\n\
* reorderings -- see NVAL and AB), or unless N(w) < NVAL(i)\n\
* for all search points w , in which case NAB(i,2) will\n\
* not be modified. Normally, NAB should be set to some\n\
* distinctive value(s) before DLAEBZ is called.\n\
*\n\
* WORK (workspace) DOUBLE PRECISION array, dimension (MMAX)\n\
* Workspace.\n\
*\n\
* IWORK (workspace) INTEGER array, dimension (MMAX)\n\
* Workspace.\n\
*\n\
* INFO (output) INTEGER\n\
* = 0: All intervals converged.\n\
* = 1--MMAX: The last INFO intervals did not converge.\n\
* = MMAX+1: More than MMAX intervals were generated.\n\
*\n\n\
* Further Details\n\
* ===============\n\
*\n\
* This routine is intended to be called only by other LAPACK\n\
* routines, thus the interface is less user-friendly. It is intended\n\
* for two purposes:\n\
*\n\
* (a) finding eigenvalues. In this case, DLAEBZ should have one or\n\
* more initial intervals set up in AB, and DLAEBZ should be called\n\
* with IJOB=1. This sets up NAB, and also counts the eigenvalues.\n\
* Intervals with no eigenvalues would usually be thrown out at\n\
* this point. Also, if not all the eigenvalues in an interval i\n\
* are desired, NAB(i,1) can be increased or NAB(i,2) decreased.\n\
* For example, set NAB(i,1)=NAB(i,2)-1 to get the largest\n\
* eigenvalue. DLAEBZ is then called with IJOB=2 and MMAX\n\
* no smaller than the value of MOUT returned by the call with\n\
* IJOB=1. After this (IJOB=2) call, eigenvalues NAB(i,1)+1\n\
* through NAB(i,2) are approximately AB(i,1) (or AB(i,2)) to the\n\
* tolerance specified by ABSTOL and RELTOL.\n\
*\n\
* (b) finding an interval (a',b'] containing eigenvalues w(f),...,w(l).\n\
* In this case, start with a Gershgorin interval (a,b). Set up\n\
* AB to contain 2 search intervals, both initially (a,b). One\n\
* NVAL element should contain f-1 and the other should contain l\n\
* , while C should contain a and b, resp. NAB(i,1) should be -1\n\
* and NAB(i,2) should be N+1, to flag an error if the desired\n\
* interval does not lie in (a,b). DLAEBZ is then called with\n\
* IJOB=3. On exit, if w(f-1) < w(f), then one of the intervals --\n\
* j -- will have AB(j,1)=AB(j,2) and NAB(j,1)=NAB(j,2)=f-1, while\n\
* if, to the specified tolerance, w(f-k)=...=w(f+r), k > 0 and r\n\
* >= 0, then the interval will have N(AB(j,1))=NAB(j,1)=f-k and\n\
* N(AB(j,2))=NAB(j,2)=f+r. The cases w(l) < w(l+1) and\n\
* w(l-r)=...=w(l+k) are handled similarly.\n\
*\n\
* =====================================================================\n\
*\n"
|
