Visible to Intel only — GUID: GUID-94A50590-0FF8-4B80-9F45-34AA69D398F2
Visible to Intel only — GUID: GUID-94A50590-0FF8-4B80-9F45-34AA69D398F2
?larrb2
Provides limited bisection to locate eigenvalues for more accuracy.
Syntax
call slarrb2( n, d, lld, ifirst, ilast, rtol1, rtol2, offset, w, wgap, werr, work, iwork, pivmin, lgpvmn, lgspdm, twist, info )
call dlarrb2( n, d, lld, ifirst, ilast, rtol1, rtol2, offset, w, wgap, werr, work, iwork, pivmin, lgpvmn, lgspdm, twist, info )
Description
Given the relatively robust representation (RRR) LDLT, ?larrb2 does "limited" bisection to refine the eigenvalues of LDLT, w( ifirst - offset ) through w( ilast - offset ), to more accuracy. Initial guesses for these eigenvalues are input in w, the corresponding estimate of the error in these guesses and their gaps are input in werr and wgap, respectively. During bisection, intervals [left, right] are maintained by storing their mid-points and semi-widths in the arrays w and werr respectively.
There are very few minor differences between larrb from LAPACK and this current subroutine ?larrb2. The most important reason for creating this nearly identical copy is profiling: in the ScaLAPACK MRRR algorithm, eigenvalue computation using ?larrb2 is used for refinement in the construction of the representation tree, as opposed to the initial computation of the eigenvalues for the root RRR which uses ?larrb. When profiling, this allows an easy quantification of refinement work vs. computing eigenvalues of the root.
Input Parameters
- n
-
INTEGER
The order of the matrix.
- d
-
REAL for slarrb2
DOUBLE PRECISION for dlarrb2
Array of size n.
The n diagonal elements of the diagonal matrix D.
- lld
-
REAL for slarrb2
DOUBLE PRECISION for dlarrb2
Array of size n-1.
The (n-1) elements li*li*d(i).
- ifirst
-
INTEGER
The index of the first eigenvalue to be computed.
- ilast
-
INTEGER
The index of the last eigenvalue to be computed.
- rtol1, rtol2
-
REAL for slarrb2
DOUBLE PRECISION for dlarrb2
Tolerance for the convergence of the bisection intervals.
An interval [left, right] has converged if right - left < max (rtol1 * gap, rtol2 * max(|left|, |right|)) where gap is the (estimated) distance to the nearest eigenvalue.
- offset
-
INTEGER
Offset for the arrays w, wgap and werr, i.e., the ifirst - offset through ilast - offset elements of these arrays are to be used.
- w
-
REAL for slarrb2
DOUBLE PRECISION for dlarrb2
Array of size n
On input, w( ifirst - offset ) through w( ilast- offset ) are estimates of the eigenvalues of LDLT indexed ifirst through ilast.
- wgap
-
REAL for slarrb2
DOUBLE PRECISION for dlarrb2
Array of size n-1.
On input, the (estimated) gaps between consecutive eigenvalues of LDLT, i.e., wgap(I - offset) is the gap between eigenvalues I and I + 1. Note that if ifirst = ilast then wgap(ifirst - offset) must be set to zero.
- werr
-
REAL for slarrb2
DOUBLE PRECISION for dlarrb2
Array of size n.
On input, werr( ifirst - offset ) through werr( ilast - offset ) are the errors in the estimates of the corresponding elements in w.
- work
-
REAL for slarrb2
DOUBLE PRECISION for dlarrb2
(workspace) array of size 4*n.
Workspace.
- iwork
-
(workspace) INTEGER array of size 2*n.
Workspace.
- pivmin
-
REAL for slarrb2
DOUBLE PRECISION for dlarrb2
The minimum pivot in the Sturm sequence.
- lgpvmn
-
REAL for slarrb2
DOUBLE PRECISION for dlarrb2
Logarithm of pivmin, precomputed.
- lgspdm
-
REAL for slarrb2
DOUBLE PRECISION for dlarrb2
Logarithm of the spectral diameter, precomputed.
- twist
-
INTEGER
The twist index for the twisted factorization that is used for the negcount.
twist = n: Compute negcount from LDLT - λI = L+D+L+T
twist = 1: Compute negcount from LDLT - λI = U-D-U-T
twist = r, 1 < r < n: Compute negcount from LDLT - λI = Nr Δr NrT
OUTPUT Parameters
- w
-
On output, the eigenvalue estimates in w are refined.
- wgap
-
On output, the eigenvalue gaps in wgap are refined.
- werr
-
On output, the errors in werr are refined.
- info
-
INTEGER
Error flag.