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LAPACK
3.7.0
LAPACK: Linear Algebra PACKage
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subroutine clar1v | ( | integer | N, |
integer | B1, | ||
integer | BN, | ||
real | LAMBDA, | ||
real, dimension( * ) | D, | ||
real, dimension( * ) | L, | ||
real, dimension( * ) | LD, | ||
real, dimension( * ) | LLD, | ||
real | PIVMIN, | ||
real | GAPTOL, | ||
complex, dimension( * ) | Z, | ||
logical | WANTNC, | ||
integer | NEGCNT, | ||
real | ZTZ, | ||
real | MINGMA, | ||
integer | R, | ||
integer, dimension( * ) | ISUPPZ, | ||
real | NRMINV, | ||
real | RESID, | ||
real | RQCORR, | ||
real, dimension( * ) | WORK | ||
) |
CLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI.
Download CLAR1V + dependencies [TGZ] [ZIP] [TXT]
CLAR1V computes the (scaled) r-th column of the inverse of the sumbmatrix in rows B1 through BN of the tridiagonal matrix L D L**T - sigma I. When sigma is close to an eigenvalue, the computed vector is an accurate eigenvector. Usually, r corresponds to the index where the eigenvector is largest in magnitude. The following steps accomplish this computation : (a) Stationary qd transform, L D L**T - sigma I = L(+) D(+) L(+)**T, (b) Progressive qd transform, L D L**T - sigma I = U(-) D(-) U(-)**T, (c) Computation of the diagonal elements of the inverse of L D L**T - sigma I by combining the above transforms, and choosing r as the index where the diagonal of the inverse is (one of the) largest in magnitude. (d) Computation of the (scaled) r-th column of the inverse using the twisted factorization obtained by combining the top part of the the stationary and the bottom part of the progressive transform.
[in] | N | N is INTEGER The order of the matrix L D L**T. |
[in] | B1 | B1 is INTEGER First index of the submatrix of L D L**T. |
[in] | BN | BN is INTEGER Last index of the submatrix of L D L**T. |
[in] | LAMBDA | LAMBDA is REAL The shift. In order to compute an accurate eigenvector, LAMBDA should be a good approximation to an eigenvalue of L D L**T. |
[in] | L | L is REAL array, dimension (N-1) The (n-1) subdiagonal elements of the unit bidiagonal matrix L, in elements 1 to N-1. |
[in] | D | D is REAL array, dimension (N) The n diagonal elements of the diagonal matrix D. |
[in] | LD | LD is REAL array, dimension (N-1) The n-1 elements L(i)*D(i). |
[in] | LLD | LLD is REAL array, dimension (N-1) The n-1 elements L(i)*L(i)*D(i). |
[in] | PIVMIN | PIVMIN is REAL The minimum pivot in the Sturm sequence. |
[in] | GAPTOL | GAPTOL is REAL Tolerance that indicates when eigenvector entries are negligible w.r.t. their contribution to the residual. |
[in,out] | Z | Z is COMPLEX array, dimension (N) On input, all entries of Z must be set to 0. On output, Z contains the (scaled) r-th column of the inverse. The scaling is such that Z(R) equals 1. |
[in] | WANTNC | WANTNC is LOGICAL Specifies whether NEGCNT has to be computed. |
[out] | NEGCNT | NEGCNT is INTEGER If WANTNC is .TRUE. then NEGCNT = the number of pivots < pivmin in the matrix factorization L D L**T, and NEGCNT = -1 otherwise. |
[out] | ZTZ | ZTZ is REAL The square of the 2-norm of Z. |
[out] | MINGMA | MINGMA is REAL The reciprocal of the largest (in magnitude) diagonal element of the inverse of L D L**T - sigma I. |
[in,out] | R | R is INTEGER The twist index for the twisted factorization used to compute Z. On input, 0 <= R <= N. If R is input as 0, R is set to the index where (L D L**T - sigma I)^{-1} is largest in magnitude. If 1 <= R <= N, R is unchanged. On output, R contains the twist index used to compute Z. Ideally, R designates the position of the maximum entry in the eigenvector. |
[out] | ISUPPZ | ISUPPZ is INTEGER array, dimension (2) The support of the vector in Z, i.e., the vector Z is nonzero only in elements ISUPPZ(1) through ISUPPZ( 2 ). |
[out] | NRMINV | NRMINV is REAL NRMINV = 1/SQRT( ZTZ ) |
[out] | RESID | RESID is REAL The residual of the FP vector. RESID = ABS( MINGMA )/SQRT( ZTZ ) |
[out] | RQCORR | RQCORR is REAL The Rayleigh Quotient correction to LAMBDA. RQCORR = MINGMA*TMP |
[out] | WORK | WORK is REAL array, dimension (4*N) |
Definition at line 232 of file clar1v.f.