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| pure subroutine, public | la_lapack_eigv_tridiag2::la_slaebz (ijob, nitmax, n, mmax, minp, nbmin, abstol, reltol, pivmin, d, e, e2, nval, ab, c, mout, nab, work, iwork, info) |
| | 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 sufficiently small intervals whose union contains the same eigenvalues as the union of the original intervals. The input intervals are (AB(j,1),AB(j,2)], j=1,...,MINP. The output interval (AB(j,1),AB(j,2)] will contain eigenvalues NAB(j,1)+1,...,NAB(j,2), where 1 <= j <= MOUT. IJOB=3: It performs a binary search in each input interval (AB(j,1),AB(j,2)] for a point w(j) such that N(w(j))=NVAL(j), and uses C(j) as the starting point of the search. If such a w(j) is found, then on output AB(j,1)=AB(j,2)=w. If no such w(j) is found, then on output (AB(j,1),AB(j,2)] will be a small interval containing the point where N(w) jumps through NVAL(j), unless that point lies outside the initial interval. Note that the intervals are in all cases half-open 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.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_dlaebz (ijob, nitmax, n, mmax, minp, nbmin, abstol, reltol, pivmin, d, e, e2, nval, ab, c, mout, nab, work, iwork, info) |
| | DLAEBZ: 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 sufficiently small intervals whose union contains the same eigenvalues as the union of the original intervals. The input intervals are (AB(j,1),AB(j,2)], j=1,...,MINP. The output interval (AB(j,1),AB(j,2)] will contain eigenvalues NAB(j,1)+1,...,NAB(j,2), where 1 <= j <= MOUT. IJOB=3: It performs a binary search in each input interval (AB(j,1),AB(j,2)] for a point w(j) such that N(w(j))=NVAL(j), and uses C(j) as the starting point of the search. If such a w(j) is found, then on output AB(j,1)=AB(j,2)=w. If no such w(j) is found, then on output (AB(j,1),AB(j,2)] will be a small interval containing the point where N(w) jumps through NVAL(j), unless that point lies outside the initial interval. Note that the intervals are in all cases half-open 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.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_qlaebz (ijob, nitmax, n, mmax, minp, nbmin, abstol, reltol, pivmin, d, e, e2, nval, ab, c, mout, nab, work, iwork, info) |
| | QLAEBZ: 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 sufficiently small intervals whose union contains the same eigenvalues as the union of the original intervals. The input intervals are (AB(j,1),AB(j,2)], j=1,...,MINP. The output interval (AB(j,1),AB(j,2)] will contain eigenvalues NAB(j,1)+1,...,NAB(j,2), where 1 <= j <= MOUT. IJOB=3: It performs a binary search in each input interval (AB(j,1),AB(j,2)] for a point w(j) such that N(w(j))=NVAL(j), and uses C(j) as the starting point of the search. If such a w(j) is found, then on output AB(j,1)=AB(j,2)=w. If no such w(j) is found, then on output (AB(j,1),AB(j,2)] will be a small interval containing the point where N(w) jumps through NVAL(j), unless that point lies outside the initial interval. Note that the intervals are in all cases half-open 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.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_slaeda (n, tlvls, curlvl, curpbm, prmptr, perm, givptr, givcol, givnum, q, qptr, z, ztemp, info) |
| | SLAEDA: computes the Z vector corresponding to the merge step in the CURLVLth step of the merge process with TLVLS steps for the CURPBMth problem.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_dlaeda (n, tlvls, curlvl, curpbm, prmptr, perm, givptr, givcol, givnum, q, qptr, z, ztemp, info) |
| | DLAEDA: computes the Z vector corresponding to the merge step in the CURLVLth step of the merge process with TLVLS steps for the CURPBMth problem.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_qlaeda (n, tlvls, curlvl, curpbm, prmptr, perm, givptr, givcol, givnum, q, qptr, z, ztemp, info) |
| | QLAEDA: computes the Z vector corresponding to the merge step in the CURLVLth step of the merge process with TLVLS steps for the CURPBMth problem.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_slamrg (n1, n2, a, strd1, strd2, index) |
| | SLAMRG: will create a permutation list which will merge the elements of A (which is composed of two independently sorted sets) into a single set which is sorted in ascending order.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_dlamrg (n1, n2, a, dtrd1, dtrd2, index) |
| | DLAMRG: will create a permutation list which will merge the elements of A (which is composed of two independently sorted sets) into a single set which is sorted in ascending order.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_qlamrg (n1, n2, a, qtrd1, qtrd2, index) |
| | QLAMRG: will create a permutation list which will merge the elements of A (which is composed of two independently sorted sets) into a single set which is sorted in ascending order.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_slarra (n, d, e, e2, spltol, tnrm, nsplit, isplit, info) |
| | Compute the splitting points with threshold SPLTOL. SLARRA: sets any "small" off-diagonal elements to zero.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_dlarra (n, d, e, e2, spltol, tnrm, nsplit, isplit, info) |
| | Compute the splitting points with threshold SPLTOL. DLARRA: sets any "small" off-diagonal elements to zero.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_qlarra (n, d, e, e2, spltol, tnrm, nsplit, isplit, info) |
| | Compute the splitting points with threshold SPLTOL. QLARRA: sets any "small" off-diagonal elements to zero.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_slarrc (jobt, n, vl, vu, d, e, pivmin, eigcnt, lcnt, rcnt, info) |
| | Find the number of eigenvalues of the symmetric tridiagonal matrix T that are in the interval (VL,VU] if JOBT = 'T', and of L D L^T if JOBT = 'L'.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_dlarrc (jobt, n, vl, vu, d, e, pivmin, eigcnt, lcnt, rcnt, info) |
| | Find the number of eigenvalues of the symmetric tridiagonal matrix T that are in the interval (VL,VU] if JOBT = 'T', and of L D L^T if JOBT = 'L'.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_qlarrc (jobt, n, vl, vu, d, e, pivmin, eigcnt, lcnt, rcnt, info) |
| | Find the number of eigenvalues of the symmetric tridiagonal matrix T that are in the interval (VL,VU] if JOBT = 'T', and of L D L^T if JOBT = 'L'.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_slarrd (range, order, n, vl, vu, il, iu, gers, reltol, d, e, e2, pivmin, nsplit, isplit, m, w, werr, wl, wu, iblock, indexw, work, iwork, info) |
| | SLARRD: computes the eigenvalues of a symmetric tridiagonal matrix T to suitable accuracy. This is an auxiliary code to be called from SSTEMR. The user may ask for all eigenvalues, all eigenvalues in the half-open interval (VL, VU], or the IL-th through IU-th eigenvalues. To avoid overflow, the matrix must be scaled so that its largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest accuracy, it should not be much smaller than that. See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal
Matrix", Report CS41, Computer Science Dept., Stanford University, July 21, 1966.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_dlarrd (range, order, n, vl, vu, il, iu, gers, reltol, d, e, e2, pivmin, nsplit, isplit, m, w, werr, wl, wu, iblock, indexw, work, iwork, info) |
| | DLARRD: computes the eigenvalues of a symmetric tridiagonal matrix T to suitable accuracy. This is an auxiliary code to be called from DSTEMR. The user may ask for all eigenvalues, all eigenvalues in the half-open interval (VL, VU], or the IL-th through IU-th eigenvalues. To avoid overflow, the matrix must be scaled so that its largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest accuracy, it should not be much smaller than that. See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal
Matrix", Report CS41, Computer Science Dept., Stanford University, July 21, 1966.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_qlarrd (range, order, n, vl, vu, il, iu, gers, reltol, d, e, e2, pivmin, nsplit, isplit, m, w, werr, wl, wu, iblock, indexw, work, iwork, info) |
| | QLARRD: computes the eigenvalues of a symmetric tridiagonal matrix T to suitable accuracy. This is an auxiliary code to be called from QSTEMR. The user may ask for all eigenvalues, all eigenvalues in the half-open interval (VL, VU], or the IL-th through IU-th eigenvalues. To avoid overflow, the matrix must be scaled so that its largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest accuracy, it should not be much smaller than that. See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal
Matrix", Report CS41, Computer Science Dept., Stanford University, July 21, 1966.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_slarrj (n, d, e2, ifirst, ilast, rtol, offset, w, werr, work, iwork, pivmin, spdiam, info) |
| | Given the initial eigenvalue approximations of T, SLARRJ: does bisection to refine the eigenvalues of 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 in WERR. During bisection, intervals [left, right] are maintained by storing their mid-points and semi-widths in the arrays W and WERR respectively.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_dlarrj (n, d, e2, ifirst, ilast, rtol, offset, w, werr, work, iwork, pivmin, spdiam, info) |
| | Given the initial eigenvalue approximations of T, DLARRJ: does bisection to refine the eigenvalues of 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 in WERR. During bisection, intervals [left, right] are maintained by storing their mid-points and semi-widths in the arrays W and WERR respectively.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_qlarrj (n, d, e2, ifirst, ilast, rtol, offset, w, werr, work, iwork, pivmin, spdiam, info) |
| | Given the initial eigenvalue approximations of T, QLARRJ: does bisection to refine the eigenvalues of 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 in WERR. During bisection, intervals [left, right] are maintained by storing their mid-points and semi-widths in the arrays W and WERR respectively.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_slarrk (n, iw, gl, gu, d, e2, pivmin, reltol, w, werr, info) |
| | SLARRK: computes one eigenvalue of a symmetric tridiagonal matrix T to suitable accuracy. This is an auxiliary code to be called from SSTEMR. To avoid overflow, the matrix must be scaled so that its largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest accuracy, it should not be much smaller than that. See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal
Matrix", Report CS41, Computer Science Dept., Stanford University, July 21, 1966.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_dlarrk (n, iw, gl, gu, d, e2, pivmin, reltol, w, werr, info) |
| | DLARRK: computes one eigenvalue of a symmetric tridiagonal matrix T to suitable accuracy. This is an auxiliary code to be called from DSTEMR. To avoid overflow, the matrix must be scaled so that its largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest accuracy, it should not be much smaller than that. See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal
Matrix", Report CS41, Computer Science Dept., Stanford University, July 21, 1966.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_qlarrk (n, iw, gl, gu, d, e2, pivmin, reltol, w, werr, info) |
| | QLARRK: computes one eigenvalue of a symmetric tridiagonal matrix T to suitable accuracy. This is an auxiliary code to be called from QSTEMR. To avoid overflow, the matrix must be scaled so that its largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest accuracy, it should not be much smaller than that. See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal
Matrix", Report CS41, Computer Science Dept., Stanford University, July 21, 1966.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_slarrr (n, d, e, info) |
| | Perform tests to decide whether the symmetric tridiagonal matrix T warrants expensive computations which guarantee high relative accuracy in the eigenvalues.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_dlarrr (n, d, e, info) |
| | Perform tests to decide whether the symmetric tridiagonal matrix T warrants expensive computations which guarantee high relative accuracy in the eigenvalues.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_qlarrr (n, d, e, info) |
| | Perform tests to decide whether the symmetric tridiagonal matrix T warrants expensive computations which guarantee high relative accuracy in the eigenvalues.
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| pure integer(ilp) function, public | la_lapack_eigv_tridiag2::la_slaneg (n, d, lld, sigma, pivmin, r) |
| | SLANEG: computes the Sturm count, the number of negative pivots encountered while factoring tridiagonal T - sigma I = L D L^T. This implementation works directly on the factors without forming the tridiagonal matrix T. The Sturm count is also the number of eigenvalues of T less than sigma. This routine is called from SLARRB. The current routine does not use the PIVMIN parameter but rather requires IEEE-754 propagation of Infinities and NaNs. This routine also has no input range restrictions but does require default exception handling such that x/0 produces Inf when x is non-zero, and Inf/Inf produces NaN. For more information, see: Marques, Riedy, and Voemel, "Benefits of IEEE-754 Features in
Modern Symmetric Tridiagonal Eigensolvers," SIAM Journal on Scientific Computing, v28, n5, 2006. DOI 10.1137/050641624 (Tech report version in LAWN 172 with the same title.)
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| pure integer(ilp) function, public | la_lapack_eigv_tridiag2::la_dlaneg (n, d, lld, sigma, pivmin, r) |
| | DLANEG: computes the Sturm count, the number of negative pivots encountered while factoring tridiagonal T - sigma I = L D L^T. This implementation works directly on the factors without forming the tridiagonal matrix T. The Sturm count is also the number of eigenvalues of T less than sigma. This routine is called from DLARRB. The current routine does not use the PIVMIN parameter but rather requires IEEE-754 propagation of Infinities and NaNs. This routine also has no input range restrictions but does require default exception handling such that x/0 produces Inf when x is non-zero, and Inf/Inf produces NaN. For more information, see: Marques, Riedy, and Voemel, "Benefits of IEEE-754 Features in
Modern Symmetric Tridiagonal Eigensolvers," SIAM Journal on Scientific Computing, v28, n5, 2006. DOI 10.1137/050641624 (Tech report version in LAWN 172 with the same title.)
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| pure integer(ilp) function, public | la_lapack_eigv_tridiag2::la_qlaneg (n, d, lld, sigma, pivmin, r) |
| | QLANEG: computes the Sturm count, the number of negative pivots encountered while factoring tridiagonal T - sigma I = L D L^T. This implementation works directly on the factors without forming the tridiagonal matrix T. The Sturm count is also the number of eigenvalues of T less than sigma. This routine is called from QLARRB. The current routine does not use the PIVMIN parameter but rather requires IEEE-754 propagation of Infinities and NaNs. This routine also has no input range restrictions but does require default exception handling such that x/0 produces Inf when x is non-zero, and Inf/Inf produces NaN. For more information, see: Marques, Riedy, and Voemel, "Benefits of IEEE-754 Features in
Modern Symmetric Tridiagonal Eigensolvers," SIAM Journal on Scientific Computing, v28, n5, 2006. DOI 10.1137/050641624 (Tech report version in LAWN 172 with the same title.)
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_slar1v (n, b1, bn, lambda, d, l, ld, lld, pivmin, gaptol, z, wantnc, negcnt, ztz, mingma, r, isuppz, nrminv, resid, rqcorr, work) |
| | SLAR1V: 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.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_dlar1v (n, b1, bn, lambda, d, l, ld, lld, pivmin, gaptol, z, wantnc, negcnt, ztz, mingma, r, isuppz, nrminv, resid, rqcorr, work) |
| | DLAR1V: 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.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_qlar1v (n, b1, bn, lambda, d, l, ld, lld, pivmin, gaptol, z, wantnc, negcnt, ztz, mingma, r, isuppz, nrminv, resid, rqcorr, work) |
| | QLAR1V: 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.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_slarrb (n, d, lld, ifirst, ilast, rtol1, rtol2, offset, w, wgap, werr, work, iwork, pivmin, spdiam, twist, 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.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_dlarrb (n, d, lld, ifirst, ilast, rtol1, rtol2, offset, w, wgap, werr, work, iwork, pivmin, spdiam, twist, 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.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_qlarrb (n, d, lld, ifirst, ilast, rtol1, rtol2, offset, w, wgap, werr, work, iwork, pivmin, spdiam, twist, info) |
| | Given the relatively robust representation(RRR) L D L^T, QLARRB: 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.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_slarrf (n, d, l, ld, clstrt, clend, w, wgap, werr, spdiam, clgapl, clgapr, pivmin, sigma, dplus, lplus, work, info) |
| | Given the initial representation L D L^T and its cluster of close eigenvalues (in a relative measure), W( CLSTRT ), W( CLSTRT+1 ), ... W( CLEND ), SLARRF: finds a new relatively robust representation L D L^T - SIGMA I = L(+) D(+) L(+)^T such that at least one of the eigenvalues of L(+) D(+) L(+)^T is relatively isolated.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_dlarrf (n, d, l, ld, clstrt, clend, w, wgap, werr, spdiam, clgapl, clgapr, pivmin, sigma, dplus, lplus, work, info) |
| | Given the initial representation L D L^T and its cluster of close eigenvalues (in a relative measure), W( CLSTRT ), W( CLSTRT+1 ), ... W( CLEND ), DLARRF: finds a new relatively robust representation L D L^T - SIGMA I = L(+) D(+) L(+)^T such that at least one of the eigenvalues of L(+) D(+) L(+)^T is relatively isolated.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_qlarrf (n, d, l, ld, clstrt, clend, w, wgap, werr, spdiam, clgapl, clgapr, pivmin, sigma, dplus, lplus, work, info) |
| | Given the initial representation L D L^T and its cluster of close eigenvalues (in a relative measure), W( CLSTRT ), W( CLSTRT+1 ), ... W( CLEND ), QLARRF: finds a new relatively robust representation L D L^T - SIGMA I = L(+) D(+) L(+)^T such that at least one of the eigenvalues of L(+) D(+) L(+)^T is relatively isolated.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_slarrv (n, vl, vu, d, l, pivmin, isplit, m, dol, dou, minrgp, rtol1, rtol2, w, werr, wgap, iblock, indexw, gers, z, ldz, isuppz, work, iwork, info) |
| | SLARRV: computes the eigenvectors of the tridiagonal matrix T = L D L**T given L, D and APPROXIMATIONS to the eigenvalues of L D L**T. The input eigenvalues should have been computed by SLARRE.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_dlarrv (n, vl, vu, d, l, pivmin, isplit, m, dol, dou, minrgp, rtol1, rtol2, w, werr, wgap, iblock, indexw, gers, z, ldz, isuppz, work, iwork, info) |
| | DLARRV: computes the eigenvectors of the tridiagonal matrix T = L D L**T given L, D and APPROXIMATIONS to the eigenvalues of L D L**T. The input eigenvalues should have been computed by DLARRE.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_qlarrv (n, vl, vu, d, l, pivmin, isplit, m, dol, dou, minrgp, rtol1, rtol2, w, werr, wgap, iblock, indexw, gers, z, ldz, isuppz, work, iwork, info) |
| | QLARRV: computes the eigenvectors of the tridiagonal matrix T = L D L**T given L, D and APPROXIMATIONS to the eigenvalues of L D L**T. The input eigenvalues should have been computed by QLARRE.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_slarre (range, n, vl, vu, il, iu, d, e, e2, rtol1, rtol2, spltol, nsplit, isplit, m, w, werr, wgap, iblock, indexw, gers, pivmin, work, iwork, info) |
| | To find the desired eigenvalues of a given real symmetric tridiagonal matrix T, SLARRE: sets any "small" off-diagonal elements to zero, and for each unreduced block T_i, it finds (a) a suitable shift at one end of the block's spectrum, (b) the base representation, T_i - sigma_i I = L_i D_i L_i^T, and (c) eigenvalues of each L_i D_i L_i^T. The representations and eigenvalues found are then used by SSTEMR to compute the eigenvectors of T. The accuracy varies depending on whether bisection is used to find a few eigenvalues or the dqds algorithm (subroutine SLASQ2) to conpute all and then discard any unwanted one. As an added benefit, SLARRE also outputs the n Gerschgorin intervals for the matrices L_i D_i L_i^T.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_dlarre (range, n, vl, vu, il, iu, d, e, e2, rtol1, rtol2, spltol, nsplit, isplit, m, w, werr, wgap, iblock, indexw, gers, pivmin, work, iwork, info) |
| | To find the desired eigenvalues of a given real symmetric tridiagonal matrix T, DLARRE: sets any "small" off-diagonal elements to zero, and for each unreduced block T_i, it finds (a) a suitable shift at one end of the block's spectrum, (b) the base representation, T_i - sigma_i I = L_i D_i L_i^T, and (c) eigenvalues of each L_i D_i L_i^T. The representations and eigenvalues found are then used by DSTEMR to compute the eigenvectors of T. The accuracy varies depending on whether bisection is used to find a few eigenvalues or the dqds algorithm (subroutine DLASQ2) to conpute all and then discard any unwanted one. As an added benefit, DLARRE also outputs the n Gerschgorin intervals for the matrices L_i D_i L_i^T.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_qlarre (range, n, vl, vu, il, iu, d, e, e2, rtol1, rtol2, spltol, nsplit, isplit, m, w, werr, wgap, iblock, indexw, gers, pivmin, work, iwork, info) |
| | To find the desired eigenvalues of a given real symmetric tridiagonal matrix T, QLARRE: sets any "small" off-diagonal elements to zero, and for each unreduced block T_i, it finds (a) a suitable shift at one end of the block's spectrum, (b) the base representation, T_i - sigma_i I = L_i D_i L_i^T, and (c) eigenvalues of each L_i D_i L_i^T. The representations and eigenvalues found are then used by QSTEMR to compute the eigenvectors of T. The accuracy varies depending on whether bisection is used to find a few eigenvalues or the dqds algorithm (subroutine QLASQ2) to conpute all and then discard any unwanted one. As an added benefit, QLARRE also outputs the n Gerschgorin intervals for the matrices L_i D_i L_i^T.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_clar1v (n, b1, bn, lambda, d, l, ld, lld, pivmin, gaptol, z, wantnc, negcnt, ztz, mingma, r, isuppz, nrminv, resid, rqcorr, work) |
| | 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.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_zlar1v (n, b1, bn, lambda, d, l, ld, lld, pivmin, gaptol, z, wantnc, negcnt, ztz, mingma, r, isuppz, nrminv, resid, rqcorr, work) |
| | ZLAR1V: 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.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_wlar1v (n, b1, bn, lambda, d, l, ld, lld, pivmin, gaptol, z, wantnc, negcnt, ztz, mingma, r, isuppz, nrminv, resid, rqcorr, work) |
| | WLAR1V: 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.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_clarrv (n, vl, vu, d, l, pivmin, isplit, m, dol, dou, minrgp, rtol1, rtol2, w, werr, wgap, iblock, indexw, gers, z, ldz, isuppz, work, iwork, info) |
| | CLARRV: computes the eigenvectors of the tridiagonal matrix T = L D L**T given L, D and APPROXIMATIONS to the eigenvalues of L D L**T. The input eigenvalues should have been computed by SLARRE.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_zlarrv (n, vl, vu, d, l, pivmin, isplit, m, dol, dou, minrgp, rtol1, rtol2, w, werr, wgap, iblock, indexw, gers, z, ldz, isuppz, work, iwork, info) |
| | ZLARRV: computes the eigenvectors of the tridiagonal matrix T = L D L**T given L, D and APPROXIMATIONS to the eigenvalues of L D L**T. The input eigenvalues should have been computed by DLARRE.
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| pure subroutine, public | la_lapack_eigv_tridiag2::la_wlarrv (n, vl, vu, d, l, pivmin, isplit, m, dol, dou, minrgp, rtol1, rtol2, w, werr, wgap, iblock, indexw, gers, z, ldz, isuppz, work, iwork, info) |
| | WLARRV: computes the eigenvectors of the tridiagonal matrix T = L D L**T given L, D and APPROXIMATIONS to the eigenvalues of L D L**T. The input eigenvalues should have been computed by QLARRE.
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1.13.2