larrb#

Functions

void slarrb(
    const INT  n,
    const f32* D,
    const f32* lld,
    const INT  ifirst,
    const INT  ilast,
    const f32  rtol1,
    const f32  rtol2,
    const INT  offset,
          f32* W,
          f32* wgap,
          f32* werr,
          f32* work,
          INT* iwork,
    const f32  pivmin,
    const f32  spdiam,
    const INT  twist,
          INT* info
);
void slarrb(const INT n, const f32 *D, const f32 *lld, const INT ifirst, const INT ilast, const f32 rtol1, const f32 rtol2, const INT offset, f32 *W, f32 *wgap, f32 *werr, f32 *work, INT *iwork, const f32 pivmin, const f32 spdiam, const INT twist, INT *info)#

Given the relatively robust representation(RRR) L D L^T, SLARRB does “limited” bisection to refine the eigenvalues of L D L^T, 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.

Parameters

in
n

The order of the matrix.

in
D

Double precision array, dimension (N). The N diagonal elements of the diagonal matrix D.

in
lld

Double precision array, dimension (N-1). The (N-1) elements L(i)*L(i)*D(i).

in
ifirst

The index of the first eigenvalue to be computed (0-based).

in
ilast

The index of the last eigenvalue to be computed (0-based).

in
rtol1

Tolerance for the convergence of the bisection intervals.

in
rtol2

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.

in
offset

Offset for the arrays W, WGAP and WERR, i.e., the IFIRST-OFFSET through ILAST-OFFSET elements of these arrays are to be used.

inout
W

Double precision array, dimension (N). On input, W( IFIRST-OFFSET ) through W( ILAST-OFFSET ) are estimates of the eigenvalues of L D L^T indexed IFIRST through ILAST. On output, these estimates are refined.

inout
wgap

Double precision array, dimension (N-1). On input, the (estimated) gaps between consecutive eigenvalues of L D L^T. Note that if IFIRST = ILAST then WGAP(IFIRST-OFFSET) must be set to ZERO. On output, these gaps are refined.

inout
werr

Double precision array, dimension (N). On input, WERR( IFIRST-OFFSET ) through WERR( ILAST-OFFSET ) are the errors in the estimates of the corresponding elements in W. On output, these errors are refined.

out
work

Double precision array, dimension (2*N). Workspace.

out
iwork

Integer array, dimension (2*N). Workspace.

in
pivmin

The minimum pivot in the Sturm sequence.

in
spdiam

The spectral diameter of the matrix.

in
twist

The twist index for the twisted factorization that is used for the negcount. TWIST = N-1: Compute negcount from L D L^T - LAMBDA I = L+ D+ L+^T TWIST = 0: Compute negcount from L D L^T - LAMBDA I = U- D- U-^T TWIST = R: Compute negcount from L D L^T - LAMBDA I = N(r) D(r) N(r)

out
info

Error flag.

Functions

void dlarrb(
    const INT  n,
    const f64* D,
    const f64* lld,
    const INT  ifirst,
    const INT  ilast,
    const f64  rtol1,
    const f64  rtol2,
    const INT  offset,
          f64* W,
          f64* wgap,
          f64* werr,
          f64* work,
          INT* iwork,
    const f64  pivmin,
    const f64  spdiam,
    const INT  twist,
          INT* info
);
void dlarrb(const INT n, const f64 *D, const f64 *lld, const INT ifirst, const INT ilast, const f64 rtol1, const f64 rtol2, const INT offset, f64 *W, f64 *wgap, f64 *werr, f64 *work, INT *iwork, const f64 pivmin, const f64 spdiam, const INT twist, INT *info)#

Given the relatively robust representation(RRR) L D L^T, DLARRB does “limited” bisection to refine the eigenvalues of L D L^T, 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.

Parameters

in
n

The order of the matrix.

in
D

Double precision array, dimension (N). The N diagonal elements of the diagonal matrix D.

in
lld

Double precision array, dimension (N-1). The (N-1) elements L(i)*L(i)*D(i).

in
ifirst

The index of the first eigenvalue to be computed (0-based).

in
ilast

The index of the last eigenvalue to be computed (0-based).

in
rtol1

Tolerance for the convergence of the bisection intervals.

in
rtol2

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.

in
offset

Offset for the arrays W, WGAP and WERR, i.e., the IFIRST-OFFSET through ILAST-OFFSET elements of these arrays are to be used.

inout
W

Double precision array, dimension (N). On input, W( IFIRST-OFFSET ) through W( ILAST-OFFSET ) are estimates of the eigenvalues of L D L^T indexed IFIRST through ILAST. On output, these estimates are refined.

inout
wgap

Double precision array, dimension (N-1). On input, the (estimated) gaps between consecutive eigenvalues of L D L^T. Note that if IFIRST = ILAST then WGAP(IFIRST-OFFSET) must be set to ZERO. On output, these gaps are refined.

inout
werr

Double precision array, dimension (N). On input, WERR( IFIRST-OFFSET ) through WERR( ILAST-OFFSET ) are the errors in the estimates of the corresponding elements in W. On output, these errors are refined.

out
work

Double precision array, dimension (2*N). Workspace.

out
iwork

Integer array, dimension (2*N). Workspace.

in
pivmin

The minimum pivot in the Sturm sequence.

in
spdiam

The spectral diameter of the matrix.

in
twist

The twist index for the twisted factorization that is used for the negcount. TWIST = N-1: Compute negcount from L D L^T - LAMBDA I = L+ D+ L+^T TWIST = 0: Compute negcount from L D L^T - LAMBDA I = U- D- U-^T TWIST = R: Compute negcount from L D L^T - LAMBDA I = N(r) D(r) N(r)

out
info

Error flag.

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