| auxOTHERcomputational(3) | LAPACK | auxOTHERcomputational(3) |
auxOTHERcomputational - auxiliary Computational routines
character *1 function chla_transtype (TRANS)
CHLA_TRANSTYPE subroutine dbdsdc (UPLO, COMPQ, N, D, E, U, LDU,
VT, LDVT, Q, IQ, WORK, IWORK, INFO)
DBDSDC subroutine dbdsqr (UPLO, N, NCVT, NRU, NCC, D, E, VT,
LDVT, U, LDU, C, LDC, WORK, INFO)
DBDSQR subroutine ddisna (JOB, M, N, D, SEP, INFO)
DDISNA subroutine dlaed0 (ICOMPQ, QSIZ, N, D, E, Q, LDQ, QSTORE,
LDQS, WORK, IWORK, INFO)
DLAED0 used by sstedc. Computes all eigenvalues and corresponding
eigenvectors of an unreduced symmetric tridiagonal matrix using the divide
and conquer method. subroutine dlaed1 (N, D, Q, LDQ, INDXQ, RHO,
CUTPNT, WORK, IWORK, INFO)
DLAED1 used by sstedc. Computes the updated eigensystem of a diagonal
matrix after modification by a rank-one symmetric matrix. Used when the
original matrix is tridiagonal. subroutine dlaed2 (K, N, N1, D, Q,
LDQ, INDXQ, RHO, Z, DLAMDA, W, Q2, INDX, INDXC, INDXP, COLTYP, INFO)
DLAED2 used by sstedc. Merges eigenvalues and deflates secular
equation. Used when the original matrix is tridiagonal. subroutine
dlaed3 (K, N, N1, D, Q, LDQ, RHO, DLAMDA, Q2, INDX, CTOT, W, S, INFO)
DLAED3 used by sstedc. Finds the roots of the secular equation and
updates the eigenvectors. Used when the original matrix is tridiagonal.
subroutine dlaed4 (N, I, D, Z, DELTA, RHO, DLAM, INFO)
DLAED4 used by sstedc. Finds a single root of the secular equation.
subroutine dlaed5 (I, D, Z, DELTA, RHO, DLAM)
DLAED5 used by sstedc. Solves the 2-by-2 secular equation. subroutine
dlaed6 (KNITER, ORGATI, RHO, D, Z, FINIT, TAU, INFO)
DLAED6 used by sstedc. Computes one Newton step in solution of the
secular equation. subroutine dlaed7 (ICOMPQ, N, QSIZ, TLVLS, CURLVL,
CURPBM, D, Q, LDQ, INDXQ, RHO, CUTPNT, QSTORE, QPTR, PRMPTR, PERM, GIVPTR,
GIVCOL, GIVNUM, WORK, IWORK, INFO)
DLAED7 used by sstedc. Computes the updated eigensystem of a diagonal
matrix after modification by a rank-one symmetric matrix. Used when the
original matrix is dense. subroutine dlaed8 (ICOMPQ, K, N, QSIZ, D,
Q, LDQ, INDXQ, RHO, CUTPNT, Z, DLAMDA, Q2, LDQ2, W, PERM, GIVPTR, GIVCOL,
GIVNUM, INDXP, INDX, INFO)
DLAED8 used by sstedc. Merges eigenvalues and deflates secular
equation. Used when the original matrix is dense. subroutine dlaed9
(K, KSTART, KSTOP, N, D, Q, LDQ, RHO, DLAMDA, W, S, LDS, INFO)
DLAED9 used by sstedc. Finds the roots of the secular equation and
updates the eigenvectors. Used when the original matrix is dense. subroutine
dlaeda (N, TLVLS, CURLVL, CURPBM, PRMPTR, PERM, GIVPTR, GIVCOL,
GIVNUM, Q, QPTR, Z, ZTEMP, INFO)
DLAEDA used by sstedc. Computes the Z vector determining the rank-one
modification of the diagonal matrix. Used when the original matrix is dense.
subroutine dlagtf (N, A, LAMBDA, B, C, TOL, D, IN, INFO)
DLAGTF computes an LU factorization of a matrix T-λI, where T is
a general tridiagonal matrix, and λ a scalar, using partial pivoting
with row interchanges. subroutine dlamrg (N1, N2, A, DTRD1, DTRD2,
INDEX)
DLAMRG creates a permutation list to merge the entries of two
independently sorted sets into a single set sorted in ascending order.
subroutine dlartgs (X, Y, SIGMA, CS, SN)
DLARTGS generates a plane rotation designed to introduce a bulge in
implicit QR iteration for the bidiagonal SVD problem. subroutine
dlasq1 (N, D, E, WORK, INFO)
DLASQ1 computes the singular values of a real square bidiagonal matrix.
Used by sbdsqr. subroutine dlasq2 (N, Z, INFO)
DLASQ2 computes all the eigenvalues of the symmetric positive definite
tridiagonal matrix associated with the qd Array Z to high relative accuracy.
Used by sbdsqr and sstegr. subroutine dlasq3 (I0, N0, Z, PP, DMIN,
SIGMA, DESIG, QMAX, NFAIL, ITER, NDIV, IEEE, TTYPE, DMIN1, DMIN2, DN, DN1,
DN2, G, TAU)
DLASQ3 checks for deflation, computes a shift and calls dqds. Used by
sbdsqr. subroutine dlasq4 (I0, N0, Z, PP, N0IN, DMIN, DMIN1, DMIN2,
DN, DN1, DN2, TAU, TTYPE, G)
DLASQ4 computes an approximation to the smallest eigenvalue using
values of d from the previous transform. Used by sbdsqr. subroutine
dlasq5 (I0, N0, Z, PP, TAU, SIGMA, DMIN, DMIN1, DMIN2, DN, DNM1,
DNM2, IEEE, EPS)
DLASQ5 computes one dqds transform in ping-pong form. Used by sbdsqr
and sstegr. subroutine dlasq6 (I0, N0, Z, PP, DMIN, DMIN1, DMIN2, DN,
DNM1, DNM2)
DLASQ6 computes one dqd transform in ping-pong form. Used by sbdsqr and
sstegr. subroutine dlasrt (ID, N, D, INFO)
DLASRT sorts numbers in increasing or decreasing order. subroutine
dstebz (RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL, D, E, M, NSPLIT, W,
IBLOCK, ISPLIT, WORK, IWORK, INFO)
DSTEBZ subroutine dstedc (COMPZ, N, D, E, Z, LDZ, WORK, LWORK,
IWORK, LIWORK, INFO)
DSTEDC subroutine dsteqr (COMPZ, N, D, E, Z, LDZ, WORK, INFO)
DSTEQR subroutine dsterf (N, D, E, INFO)
DSTERF integer function iladiag (DIAG)
ILADIAG integer function ilaprec (PREC)
ILAPREC integer function ilatrans (TRANS)
ILATRANS integer function ilauplo (UPLO)
ILAUPLO subroutine sbdsdc (UPLO, COMPQ, N, D, E, U, LDU, VT,
LDVT, Q, IQ, WORK, IWORK, INFO)
SBDSDC subroutine sbdsqr (UPLO, N, NCVT, NRU, NCC, D, E, VT,
LDVT, U, LDU, C, LDC, WORK, INFO)
SBDSQR subroutine sdisna (JOB, M, N, D, SEP, INFO)
SDISNA subroutine slaed0 (ICOMPQ, QSIZ, N, D, E, Q, LDQ, QSTORE,
LDQS, WORK, IWORK, INFO)
SLAED0 used by sstedc. Computes all eigenvalues and corresponding
eigenvectors of an unreduced symmetric tridiagonal matrix using the divide
and conquer method. subroutine slaed1 (N, D, Q, LDQ, INDXQ, RHO,
CUTPNT, WORK, IWORK, INFO)
SLAED1 used by sstedc. Computes the updated eigensystem of a diagonal
matrix after modification by a rank-one symmetric matrix. Used when the
original matrix is tridiagonal. subroutine slaed2 (K, N, N1, D, Q,
LDQ, INDXQ, RHO, Z, DLAMDA, W, Q2, INDX, INDXC, INDXP, COLTYP, INFO)
SLAED2 used by sstedc. Merges eigenvalues and deflates secular
equation. Used when the original matrix is tridiagonal. subroutine
slaed3 (K, N, N1, D, Q, LDQ, RHO, DLAMDA, Q2, INDX, CTOT, W, S, INFO)
SLAED3 used by sstedc. Finds the roots of the secular equation and
updates the eigenvectors. Used when the original matrix is tridiagonal.
subroutine slaed4 (N, I, D, Z, DELTA, RHO, DLAM, INFO)
SLAED4 used by sstedc. Finds a single root of the secular equation.
subroutine slaed5 (I, D, Z, DELTA, RHO, DLAM)
SLAED5 used by sstedc. Solves the 2-by-2 secular equation. subroutine
slaed6 (KNITER, ORGATI, RHO, D, Z, FINIT, TAU, INFO)
SLAED6 used by sstedc. Computes one Newton step in solution of the
secular equation. subroutine slaed7 (ICOMPQ, N, QSIZ, TLVLS, CURLVL,
CURPBM, D, Q, LDQ, INDXQ, RHO, CUTPNT, QSTORE, QPTR, PRMPTR, PERM, GIVPTR,
GIVCOL, GIVNUM, WORK, IWORK, INFO)
SLAED7 used by sstedc. Computes the updated eigensystem of a diagonal
matrix after modification by a rank-one symmetric matrix. Used when the
original matrix is dense. subroutine slaed8 (ICOMPQ, K, N, QSIZ, D,
Q, LDQ, INDXQ, RHO, CUTPNT, Z, DLAMDA, Q2, LDQ2, W, PERM, GIVPTR, GIVCOL,
GIVNUM, INDXP, INDX, INFO)
SLAED8 used by sstedc. Merges eigenvalues and deflates secular
equation. Used when the original matrix is dense. subroutine slaed9
(K, KSTART, KSTOP, N, D, Q, LDQ, RHO, DLAMDA, W, S, LDS, INFO)
SLAED9 used by sstedc. Finds the roots of the secular equation and
updates the eigenvectors. Used when the original matrix is dense. subroutine
slaeda (N, TLVLS, CURLVL, CURPBM, PRMPTR, PERM, GIVPTR, GIVCOL,
GIVNUM, Q, QPTR, Z, ZTEMP, INFO)
SLAEDA used by sstedc. Computes the Z vector determining the rank-one
modification of the diagonal matrix. Used when the original matrix is dense.
subroutine slagtf (N, A, LAMBDA, B, C, TOL, D, IN, INFO)
SLAGTF computes an LU factorization of a matrix T-λI, where T is
a general tridiagonal matrix, and λ a scalar, using partial pivoting
with row interchanges. subroutine slamrg (N1, N2, A, STRD1, STRD2,
INDEX)
SLAMRG creates a permutation list to merge the entries of two
independently sorted sets into a single set sorted in ascending order.
subroutine slartgs (X, Y, SIGMA, CS, SN)
SLARTGS generates a plane rotation designed to introduce a bulge in
implicit QR iteration for the bidiagonal SVD problem. subroutine
slasq1 (N, D, E, WORK, INFO)
SLASQ1 computes the singular values of a real square bidiagonal matrix.
Used by sbdsqr. subroutine slasq2 (N, Z, INFO)
SLASQ2 computes all the eigenvalues of the symmetric positive definite
tridiagonal matrix associated with the qd Array Z to high relative accuracy.
Used by sbdsqr and sstegr. subroutine slasq3 (I0, N0, Z, PP, DMIN,
SIGMA, DESIG, QMAX, NFAIL, ITER, NDIV, IEEE, TTYPE, DMIN1, DMIN2, DN, DN1,
DN2, G, TAU)
SLASQ3 checks for deflation, computes a shift and calls dqds. Used by
sbdsqr. subroutine slasq4 (I0, N0, Z, PP, N0IN, DMIN, DMIN1, DMIN2,
DN, DN1, DN2, TAU, TTYPE, G)
SLASQ4 computes an approximation to the smallest eigenvalue using
values of d from the previous transform. Used by sbdsqr. subroutine
slasq5 (I0, N0, Z, PP, TAU, SIGMA, DMIN, DMIN1, DMIN2, DN, DNM1,
DNM2, IEEE, EPS)
SLASQ5 computes one dqds transform in ping-pong form. Used by sbdsqr and
sstegr. subroutine slasq6 (I0, N0, Z, PP, DMIN, DMIN1, DMIN2,
DN, DNM1, DNM2)
SLASQ6 computes one dqd transform in ping-pong form. Used by sbdsqr and
sstegr. subroutine slasrt (ID, N, D, INFO)
SLASRT sorts numbers in increasing or decreasing order. subroutine
spttrf (N, D, E, INFO)
SPTTRF subroutine sstebz (RANGE, ORDER, N, VL, VU, IL, IU,
ABSTOL, D, E, M, NSPLIT, W, IBLOCK, ISPLIT, WORK, IWORK, INFO)
SSTEBZ subroutine sstedc (COMPZ, N, D, E, Z, LDZ, WORK, LWORK,
IWORK, LIWORK, INFO)
SSTEDC subroutine ssteqr (COMPZ, N, D, E, Z, LDZ, WORK, INFO)
SSTEQR subroutine ssterf (N, D, E, INFO)
SSTERF
This is the group of auxiliary Computational routines
CHLA_TRANSTYPE
Purpose:
This subroutine translates from a BLAST-specified integer constant to
the character string specifying a transposition operation.
CHLA_TRANSTYPE returns an CHARACTER*1. If CHLA_TRANSTYPE is 'X',
then input is not an integer indicating a transposition operator.
Otherwise CHLA_TRANSTYPE returns the constant value corresponding to
TRANS.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
DBDSDC
Purpose:
DBDSDC computes the singular value decomposition (SVD) of a real
N-by-N (upper or lower) bidiagonal matrix B: B = U * S * VT,
using a divide and conquer method, where S is a diagonal matrix
with non-negative diagonal elements (the singular values of B), and
U and VT are orthogonal matrices of left and right singular vectors,
respectively. DBDSDC can be used to compute all singular values,
and optionally, singular vectors or singular vectors in compact form.
This code makes very mild assumptions about floating point
arithmetic. It will work on machines with a guard digit in
add/subtract, or on those binary machines without guard digits
which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
It could conceivably fail on hexadecimal or decimal machines
without guard digits, but we know of none. See DLASD3 for details.
The code currently calls DLASDQ if singular values only are desired.
However, it can be slightly modified to compute singular values
using the divide and conquer method.
Parameters
UPLO is CHARACTER*1
= 'U': B is upper bidiagonal.
= 'L': B is lower bidiagonal.
COMPQ
COMPQ is CHARACTER*1
Specifies whether singular vectors are to be computed
as follows:
= 'N': Compute singular values only;
= 'P': Compute singular values and compute singular
vectors in compact form;
= 'I': Compute singular values and singular vectors.
N
N is INTEGER
The order of the matrix B. N >= 0.
D
D is DOUBLE PRECISION array, dimension (N)
On entry, the n diagonal elements of the bidiagonal matrix B.
On exit, if INFO=0, the singular values of B.
E
E is DOUBLE PRECISION array, dimension (N-1)
On entry, the elements of E contain the offdiagonal
elements of the bidiagonal matrix whose SVD is desired.
On exit, E has been destroyed.
U
U is DOUBLE PRECISION array, dimension (LDU,N)
If COMPQ = 'I', then:
On exit, if INFO = 0, U contains the left singular vectors
of the bidiagonal matrix.
For other values of COMPQ, U is not referenced.
LDU
LDU is INTEGER
The leading dimension of the array U. LDU >= 1.
If singular vectors are desired, then LDU >= max( 1, N ).
VT
VT is DOUBLE PRECISION array, dimension (LDVT,N)
If COMPQ = 'I', then:
On exit, if INFO = 0, VT**T contains the right singular
vectors of the bidiagonal matrix.
For other values of COMPQ, VT is not referenced.
LDVT
LDVT is INTEGER
The leading dimension of the array VT. LDVT >= 1.
If singular vectors are desired, then LDVT >= max( 1, N ).
Q
Q is DOUBLE PRECISION array, dimension (LDQ)
If COMPQ = 'P', then:
On exit, if INFO = 0, Q and IQ contain the left
and right singular vectors in a compact form,
requiring O(N log N) space instead of 2*N**2.
In particular, Q contains all the DOUBLE PRECISION data in
LDQ >= N*(11 + 2*SMLSIZ + 8*INT(LOG_2(N/(SMLSIZ+1))))
words of memory, where SMLSIZ is returned by ILAENV and
is equal to the maximum size of the subproblems at the
bottom of the computation tree (usually about 25).
For other values of COMPQ, Q is not referenced.
IQ
IQ is INTEGER array, dimension (LDIQ)
If COMPQ = 'P', then:
On exit, if INFO = 0, Q and IQ contain the left
and right singular vectors in a compact form,
requiring O(N log N) space instead of 2*N**2.
In particular, IQ contains all INTEGER data in
LDIQ >= N*(3 + 3*INT(LOG_2(N/(SMLSIZ+1))))
words of memory, where SMLSIZ is returned by ILAENV and
is equal to the maximum size of the subproblems at the
bottom of the computation tree (usually about 25).
For other values of COMPQ, IQ is not referenced.
WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
If COMPQ = 'N' then LWORK >= (4 * N).
If COMPQ = 'P' then LWORK >= (6 * N).
If COMPQ = 'I' then LWORK >= (3 * N**2 + 4 * N).
IWORK
IWORK is INTEGER array, dimension (8*N)
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: The algorithm failed to compute a singular value.
The update process of divide and conquer failed.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
Contributors:
DBDSQR
Purpose:
DBDSQR computes the singular values and, optionally, the right and/or
left singular vectors from the singular value decomposition (SVD) of
a real N-by-N (upper or lower) bidiagonal matrix B using the implicit
zero-shift QR algorithm. The SVD of B has the form
B = Q * S * P**T
where S is the diagonal matrix of singular values, Q is an orthogonal
matrix of left singular vectors, and P is an orthogonal matrix of
right singular vectors. If left singular vectors are requested, this
subroutine actually returns U*Q instead of Q, and, if right singular
vectors are requested, this subroutine returns P**T*VT instead of
P**T, for given real input matrices U and VT. When U and VT are the
orthogonal matrices that reduce a general matrix A to bidiagonal
form: A = U*B*VT, as computed by DGEBRD, then
A = (U*Q) * S * (P**T*VT)
is the SVD of A. Optionally, the subroutine may also compute Q**T*C
for a given real input matrix C.
See "Computing Small Singular Values of Bidiagonal Matrices With
Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan,
LAPACK Working Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11,
no. 5, pp. 873-912, Sept 1990) and
"Accurate singular values and differential qd algorithms," by
B. Parlett and V. Fernando, Technical Report CPAM-554, Mathematics
Department, University of California at Berkeley, July 1992
for a detailed description of the algorithm.
Parameters
UPLO is CHARACTER*1
= 'U': B is upper bidiagonal;
= 'L': B is lower bidiagonal.
N
N is INTEGER
The order of the matrix B. N >= 0.
NCVT
NCVT is INTEGER
The number of columns of the matrix VT. NCVT >= 0.
NRU
NRU is INTEGER
The number of rows of the matrix U. NRU >= 0.
NCC
NCC is INTEGER
The number of columns of the matrix C. NCC >= 0.
D
D is DOUBLE PRECISION array, dimension (N)
On entry, the n diagonal elements of the bidiagonal matrix B.
On exit, if INFO=0, the singular values of B in decreasing
order.
E
E is DOUBLE PRECISION array, dimension (N-1)
On entry, the N-1 offdiagonal elements of the bidiagonal
matrix B.
On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E
will contain the diagonal and superdiagonal elements of a
bidiagonal matrix orthogonally equivalent to the one given
as input.
VT
VT is DOUBLE PRECISION array, dimension (LDVT, NCVT)
On entry, an N-by-NCVT matrix VT.
On exit, VT is overwritten by P**T * VT.
Not referenced if NCVT = 0.
LDVT
LDVT is INTEGER
The leading dimension of the array VT.
LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0.
U
U is DOUBLE PRECISION array, dimension (LDU, N)
On entry, an NRU-by-N matrix U.
On exit, U is overwritten by U * Q.
Not referenced if NRU = 0.
LDU
LDU is INTEGER
The leading dimension of the array U. LDU >= max(1,NRU).
C
C is DOUBLE PRECISION array, dimension (LDC, NCC)
On entry, an N-by-NCC matrix C.
On exit, C is overwritten by Q**T * C.
Not referenced if NCC = 0.
LDC
LDC is INTEGER
The leading dimension of the array C.
LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0.
WORK
WORK is DOUBLE PRECISION array, dimension (4*(N-1))
INFO
INFO is INTEGER
= 0: successful exit
< 0: If INFO = -i, the i-th argument had an illegal value
> 0:
if NCVT = NRU = NCC = 0,
= 1, a split was marked by a positive value in E
= 2, current block of Z not diagonalized after 30*N
iterations (in inner while loop)
= 3, termination criterion of outer while loop not met
(program created more than N unreduced blocks)
else NCVT = NRU = NCC = 0,
the algorithm did not converge; D and E contain the
elements of a bidiagonal matrix which is orthogonally
similar to the input matrix B; if INFO = i, i
elements of E have not converged to zero.
Internal Parameters:
TOLMUL DOUBLE PRECISION, default = max(10,min(100,EPS**(-1/8)))
TOLMUL controls the convergence criterion of the QR loop.
If it is positive, TOLMUL*EPS is the desired relative
precision in the computed singular values.
If it is negative, abs(TOLMUL*EPS*sigma_max) is the
desired absolute accuracy in the computed singular
values (corresponds to relative accuracy
abs(TOLMUL*EPS) in the largest singular value.
abs(TOLMUL) should be between 1 and 1/EPS, and preferably
between 10 (for fast convergence) and .1/EPS
(for there to be some accuracy in the results).
Default is to lose at either one eighth or 2 of the
available decimal digits in each computed singular value
(whichever is smaller).
MAXITR INTEGER, default = 6
MAXITR controls the maximum number of passes of the
algorithm through its inner loop. The algorithms stops
(and so fails to converge) if the number of passes
through the inner loop exceeds MAXITR*N**2.
Note:
Bug report from Cezary Dendek.
On March 23rd 2017, the INTEGER variable MAXIT = MAXITR*N**2 is
removed since it can overflow pretty easily (for N larger or equal
than 18,919). We instead use MAXITDIVN = MAXITR*N.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
DDISNA
Purpose:
DDISNA computes the reciprocal condition numbers for the eigenvectors
of a real symmetric or complex Hermitian matrix or for the left or
right singular vectors of a general m-by-n matrix. The reciprocal
condition number is the 'gap' between the corresponding eigenvalue or
singular value and the nearest other one.
The bound on the error, measured by angle in radians, in the I-th
computed vector is given by
DLAMCH( 'E' ) * ( ANORM / SEP( I ) )
where ANORM = 2-norm(A) = max( abs( D(j) ) ). SEP(I) is not allowed
to be smaller than DLAMCH( 'E' )*ANORM in order to limit the size of
the error bound.
DDISNA may also be used to compute error bounds for eigenvectors of
the generalized symmetric definite eigenproblem.
Parameters
JOB is CHARACTER*1
Specifies for which problem the reciprocal condition numbers
should be computed:
= 'E': the eigenvectors of a symmetric/Hermitian matrix;
= 'L': the left singular vectors of a general matrix;
= 'R': the right singular vectors of a general matrix.
M
M is INTEGER
The number of rows of the matrix. M >= 0.
N
N is INTEGER
If JOB = 'L' or 'R', the number of columns of the matrix,
in which case N >= 0. Ignored if JOB = 'E'.
D
D is DOUBLE PRECISION array, dimension (M) if JOB = 'E'
dimension (min(M,N)) if JOB = 'L' or 'R'
The eigenvalues (if JOB = 'E') or singular values (if JOB =
'L' or 'R') of the matrix, in either increasing or decreasing
order. If singular values, they must be non-negative.
SEP
SEP is DOUBLE PRECISION array, dimension (M) if JOB = 'E'
dimension (min(M,N)) if JOB = 'L' or 'R'
The reciprocal condition numbers of the vectors.
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
DLAED0 used by sstedc. Computes all eigenvalues and corresponding eigenvectors of an unreduced symmetric tridiagonal matrix using the divide and conquer method.
Purpose:
DLAED0 computes all eigenvalues and corresponding eigenvectors of a
symmetric tridiagonal matrix using the divide and conquer method.
Parameters
ICOMPQ is INTEGER
= 0: Compute eigenvalues only.
= 1: Compute eigenvectors of original dense symmetric matrix
also. On entry, Q contains the orthogonal matrix used
to reduce the original matrix to tridiagonal form.
= 2: Compute eigenvalues and eigenvectors of tridiagonal
matrix.
QSIZ
QSIZ is INTEGER
The dimension of the orthogonal matrix used to reduce
the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.
N
N is INTEGER
The dimension of the symmetric tridiagonal matrix. N >= 0.
D
D is DOUBLE PRECISION array, dimension (N)
On entry, the main diagonal of the tridiagonal matrix.
On exit, its eigenvalues.
E
E is DOUBLE PRECISION array, dimension (N-1)
The off-diagonal elements of the tridiagonal matrix.
On exit, E has been destroyed.
Q
Q is DOUBLE PRECISION array, dimension (LDQ, N)
On entry, Q must contain an N-by-N orthogonal matrix.
If ICOMPQ = 0 Q is not referenced.
If ICOMPQ = 1 On entry, Q is a subset of the columns of the
orthogonal matrix used to reduce the full
matrix to tridiagonal form corresponding to
the subset of the full matrix which is being
decomposed at this time.
If ICOMPQ = 2 On entry, Q will be the identity matrix.
On exit, Q contains the eigenvectors of the
tridiagonal matrix.
LDQ
LDQ is INTEGER
The leading dimension of the array Q. If eigenvectors are
desired, then LDQ >= max(1,N). In any case, LDQ >= 1.
QSTORE
QSTORE is DOUBLE PRECISION array, dimension (LDQS, N)
Referenced only when ICOMPQ = 1. Used to store parts of
the eigenvector matrix when the updating matrix multiplies
take place.
LDQS
LDQS is INTEGER
The leading dimension of the array QSTORE. If ICOMPQ = 1,
then LDQS >= max(1,N). In any case, LDQS >= 1.
WORK
WORK is DOUBLE PRECISION array,
If ICOMPQ = 0 or 1, the dimension of WORK must be at least
1 + 3*N + 2*N*lg N + 3*N**2
( lg( N ) = smallest integer k
such that 2^k >= N )
If ICOMPQ = 2, the dimension of WORK must be at least
4*N + N**2.
IWORK
IWORK is INTEGER array,
If ICOMPQ = 0 or 1, the dimension of IWORK must be at least
6 + 6*N + 5*N*lg N.
( lg( N ) = smallest integer k
such that 2^k >= N )
If ICOMPQ = 2, the dimension of IWORK must be at least
3 + 5*N.
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: The algorithm failed to compute an eigenvalue while
working on the submatrix lying in rows and columns
INFO/(N+1) through mod(INFO,N+1).
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
Contributors:
DLAED1 used by sstedc. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is tridiagonal.
Purpose:
DLAED1 computes the updated eigensystem of a diagonal
matrix after modification by a rank-one symmetric matrix. This
routine is used only for the eigenproblem which requires all
eigenvalues and eigenvectors of a tridiagonal matrix. DLAED7 handles
the case in which eigenvalues only or eigenvalues and eigenvectors
of a full symmetric matrix (which was reduced to tridiagonal form)
are desired.
T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)
where Z = Q**T*u, u is a vector of length N with ones in the
CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
The eigenvectors of the original matrix are stored in Q, and the
eigenvalues are in D. The algorithm consists of three stages:
The first stage consists of deflating the size of the problem
when there are multiple eigenvalues or if there is a zero in
the Z vector. For each such occurrence the dimension of the
secular equation problem is reduced by one. This stage is
performed by the routine DLAED2.
The second stage consists of calculating the updated
eigenvalues. This is done by finding the roots of the secular
equation via the routine DLAED4 (as called by DLAED3).
This routine also calculates the eigenvectors of the current
problem.
The final stage consists of computing the updated eigenvectors
directly using the updated eigenvalues. The eigenvectors for
the current problem are multiplied with the eigenvectors from
the overall problem.
Parameters
N is INTEGER
The dimension of the symmetric tridiagonal matrix. N >= 0.
D
D is DOUBLE PRECISION array, dimension (N)
On entry, the eigenvalues of the rank-1-perturbed matrix.
On exit, the eigenvalues of the repaired matrix.
Q
Q is DOUBLE PRECISION array, dimension (LDQ,N)
On entry, the eigenvectors of the rank-1-perturbed matrix.
On exit, the eigenvectors of the repaired tridiagonal matrix.
LDQ
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N).
INDXQ
INDXQ is INTEGER array, dimension (N)
On entry, the permutation which separately sorts the two
subproblems in D into ascending order.
On exit, the permutation which will reintegrate the
subproblems back into sorted order,
i.e. D( INDXQ( I = 1, N ) ) will be in ascending order.
RHO
RHO is DOUBLE PRECISION
The subdiagonal entry used to create the rank-1 modification.
CUTPNT
CUTPNT is INTEGER
The location of the last eigenvalue in the leading sub-matrix.
min(1,N) <= CUTPNT <= N/2.
WORK
WORK is DOUBLE PRECISION array, dimension (4*N + N**2)
IWORK
IWORK is INTEGER array, dimension (4*N)
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = 1, an eigenvalue did not converge
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
Contributors:
DLAED2 used by sstedc. Merges eigenvalues and deflates secular equation. Used when the original matrix is tridiagonal.
Purpose:
DLAED2 merges the two sets of eigenvalues together into a single
sorted set. Then it tries to deflate the size of the problem.
There are two ways in which deflation can occur: when two or more
eigenvalues are close together or if there is a tiny entry in the
Z vector. For each such occurrence the order of the related secular
equation problem is reduced by one.
Parameters
K is INTEGER
The number of non-deflated eigenvalues, and the order of the
related secular equation. 0 <= K <=N.
N
N is INTEGER
The dimension of the symmetric tridiagonal matrix. N >= 0.
N1
N1 is INTEGER
The location of the last eigenvalue in the leading sub-matrix.
min(1,N) <= N1 <= N/2.
D
D is DOUBLE PRECISION array, dimension (N)
On entry, D contains the eigenvalues of the two submatrices to
be combined.
On exit, D contains the trailing (N-K) updated eigenvalues
(those which were deflated) sorted into increasing order.
Q
Q is DOUBLE PRECISION array, dimension (LDQ, N)
On entry, Q contains the eigenvectors of two submatrices in
the two square blocks with corners at (1,1), (N1,N1)
and (N1+1, N1+1), (N,N).
On exit, Q contains the trailing (N-K) updated eigenvectors
(those which were deflated) in its last N-K columns.
LDQ
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N).
INDXQ
INDXQ is INTEGER array, dimension (N)
The permutation which separately sorts the two sub-problems
in D into ascending order. Note that elements in the second
half of this permutation must first have N1 added to their
values. Destroyed on exit.
RHO
RHO is DOUBLE PRECISION
On entry, the off-diagonal element associated with the rank-1
cut which originally split the two submatrices which are now
being recombined.
On exit, RHO has been modified to the value required by
DLAED3.
Z
Z is DOUBLE PRECISION array, dimension (N)
On entry, Z contains the updating vector (the last
row of the first sub-eigenvector matrix and the first row of
the second sub-eigenvector matrix).
On exit, the contents of Z have been destroyed by the updating
process.
DLAMDA
DLAMDA is DOUBLE PRECISION array, dimension (N)
A copy of the first K eigenvalues which will be used by
DLAED3 to form the secular equation.
W
W is DOUBLE PRECISION array, dimension (N)
The first k values of the final deflation-altered z-vector
which will be passed to DLAED3.
Q2
Q2 is DOUBLE PRECISION array, dimension (N1**2+(N-N1)**2)
A copy of the first K eigenvectors which will be used by
DLAED3 in a matrix multiply (DGEMM) to solve for the new
eigenvectors.
INDX
INDX is INTEGER array, dimension (N)
The permutation used to sort the contents of DLAMDA into
ascending order.
INDXC
INDXC is INTEGER array, dimension (N)
The permutation used to arrange the columns of the deflated
Q matrix into three groups: the first group contains non-zero
elements only at and above N1, the second contains
non-zero elements only below N1, and the third is dense.
INDXP
INDXP is INTEGER array, dimension (N)
The permutation used to place deflated values of D at the end
of the array. INDXP(1:K) points to the nondeflated D-values
and INDXP(K+1:N) points to the deflated eigenvalues.
COLTYP
COLTYP is INTEGER array, dimension (N)
During execution, a label which will indicate which of the
following types a column in the Q2 matrix is:
1 : non-zero in the upper half only;
2 : dense;
3 : non-zero in the lower half only;
4 : deflated.
On exit, COLTYP(i) is the number of columns of type i,
for i=1 to 4 only.
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
Contributors:
DLAED3 used by sstedc. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is tridiagonal.
Purpose:
DLAED3 finds the roots of the secular equation, as defined by the
values in D, W, and RHO, between 1 and K. It makes the
appropriate calls to DLAED4 and then updates the eigenvectors by
multiplying the matrix of eigenvectors of the pair of eigensystems
being combined by the matrix of eigenvectors of the K-by-K system
which is solved here.
This code makes very mild assumptions about floating point
arithmetic. It will work on machines with a guard digit in
add/subtract, or on those binary machines without guard digits
which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
It could conceivably fail on hexadecimal or decimal machines
without guard digits, but we know of none.
Parameters
K is INTEGER
The number of terms in the rational function to be solved by
DLAED4. K >= 0.
N
N is INTEGER
The number of rows and columns in the Q matrix.
N >= K (deflation may result in N>K).
N1
N1 is INTEGER
The location of the last eigenvalue in the leading submatrix.
min(1,N) <= N1 <= N/2.
D
D is DOUBLE PRECISION array, dimension (N)
D(I) contains the updated eigenvalues for
1 <= I <= K.
Q
Q is DOUBLE PRECISION array, dimension (LDQ,N)
Initially the first K columns are used as workspace.
On output the columns 1 to K contain
the updated eigenvectors.
LDQ
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N).
RHO
RHO is DOUBLE PRECISION
The value of the parameter in the rank one update equation.
RHO >= 0 required.
DLAMDA
DLAMDA is DOUBLE PRECISION array, dimension (K)
The first K elements of this array contain the old roots
of the deflated updating problem. These are the poles
of the secular equation. May be changed on output by
having lowest order bit set to zero on Cray X-MP, Cray Y-MP,
Cray-2, or Cray C-90, as described above.
Q2
Q2 is DOUBLE PRECISION array, dimension (LDQ2*N)
The first K columns of this matrix contain the non-deflated
eigenvectors for the split problem.
INDX
INDX is INTEGER array, dimension (N)
The permutation used to arrange the columns of the deflated
Q matrix into three groups (see DLAED2).
The rows of the eigenvectors found by DLAED4 must be likewise
permuted before the matrix multiply can take place.
CTOT
CTOT is INTEGER array, dimension (4)
A count of the total number of the various types of columns
in Q, as described in INDX. The fourth column type is any
column which has been deflated.
W
W is DOUBLE PRECISION array, dimension (K)
The first K elements of this array contain the components
of the deflation-adjusted updating vector. Destroyed on
output.
S
S is DOUBLE PRECISION array, dimension (N1 + 1)*K
Will contain the eigenvectors of the repaired matrix which
will be multiplied by the previously accumulated eigenvectors
to update the system.
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = 1, an eigenvalue did not converge
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
Contributors:
DLAED4 used by sstedc. Finds a single root of the secular equation.
Purpose:
This subroutine computes the I-th updated eigenvalue of a symmetric
rank-one modification to a diagonal matrix whose elements are
given in the array d, and that
D(i) < D(j) for i < j
and that RHO > 0. This is arranged by the calling routine, and is
no loss in generality. The rank-one modified system is thus
diag( D ) + RHO * Z * Z_transpose.
where we assume the Euclidean norm of Z is 1.
The method consists of approximating the rational functions in the
secular equation by simpler interpolating rational functions.
Parameters
N is INTEGER
The length of all arrays.
I
I is INTEGER
The index of the eigenvalue to be computed. 1 <= I <= N.
D
D is DOUBLE PRECISION array, dimension (N)
The original eigenvalues. It is assumed that they are in
order, D(I) < D(J) for I < J.
Z
Z is DOUBLE PRECISION array, dimension (N)
The components of the updating vector.
DELTA
DELTA is DOUBLE PRECISION array, dimension (N)
If N > 2, DELTA contains (D(j) - lambda_I) in its j-th
component. If N = 1, then DELTA(1) = 1. If N = 2, see DLAED5
for detail. The vector DELTA contains the information necessary
to construct the eigenvectors by DLAED3 and DLAED9.
RHO
RHO is DOUBLE PRECISION
The scalar in the symmetric updating formula.
DLAM
DLAM is DOUBLE PRECISION
The computed lambda_I, the I-th updated eigenvalue.
INFO
INFO is INTEGER
= 0: successful exit
> 0: if INFO = 1, the updating process failed.
Internal Parameters:
Logical variable ORGATI (origin-at-i?) is used for distinguishing
whether D(i) or D(i+1) is treated as the origin.
ORGATI = .true. origin at i
ORGATI = .false. origin at i+1
Logical variable SWTCH3 (switch-for-3-poles?) is for noting
if we are working with THREE poles!
MAXIT is the maximum number of iterations allowed for each
eigenvalue.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
Contributors:
DLAED5 used by sstedc. Solves the 2-by-2 secular equation.
Purpose:
This subroutine computes the I-th eigenvalue of a symmetric rank-one
modification of a 2-by-2 diagonal matrix
diag( D ) + RHO * Z * transpose(Z) .
The diagonal elements in the array D are assumed to satisfy
D(i) < D(j) for i < j .
We also assume RHO > 0 and that the Euclidean norm of the vector
Z is one.
Parameters
I is INTEGER
The index of the eigenvalue to be computed. I = 1 or I = 2.
D
D is DOUBLE PRECISION array, dimension (2)
The original eigenvalues. We assume D(1) < D(2).
Z
Z is DOUBLE PRECISION array, dimension (2)
The components of the updating vector.
DELTA
DELTA is DOUBLE PRECISION array, dimension (2)
The vector DELTA contains the information necessary
to construct the eigenvectors.
RHO
RHO is DOUBLE PRECISION
The scalar in the symmetric updating formula.
DLAM
DLAM is DOUBLE PRECISION
The computed lambda_I, the I-th updated eigenvalue.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
Contributors:
DLAED6 used by sstedc. Computes one Newton step in solution of the secular equation.
Purpose:
DLAED6 computes the positive or negative root (closest to the origin)
of
z(1) z(2) z(3)
f(x) = rho + --------- + ---------- + ---------
d(1)-x d(2)-x d(3)-x
It is assumed that
if ORGATI = .true. the root is between d(2) and d(3);
otherwise it is between d(1) and d(2)
This routine will be called by DLAED4 when necessary. In most cases,
the root sought is the smallest in magnitude, though it might not be
in some extremely rare situations.
Parameters
KNITER is INTEGER
Refer to DLAED4 for its significance.
ORGATI
ORGATI is LOGICAL
If ORGATI is true, the needed root is between d(2) and
d(3); otherwise it is between d(1) and d(2). See
DLAED4 for further details.
RHO
RHO is DOUBLE PRECISION
Refer to the equation f(x) above.
D
D is DOUBLE PRECISION array, dimension (3)
D satisfies d(1) < d(2) < d(3).
Z
Z is DOUBLE PRECISION array, dimension (3)
Each of the elements in z must be positive.
FINIT
FINIT is DOUBLE PRECISION
The value of f at 0. It is more accurate than the one
evaluated inside this routine (if someone wants to do
so).
TAU
TAU is DOUBLE PRECISION
The root of the equation f(x).
INFO
INFO is INTEGER
= 0: successful exit
> 0: if INFO = 1, failure to converge
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
Further Details:
10/02/03: This version has a few statements commented out for thread
safety (machine parameters are computed on each entry). SJH.
05/10/06: Modified from a new version of Ren-Cang Li, use
Gragg-Thornton-Warner cubic convergent scheme for better stability.
Contributors:
DLAED7 used by sstedc. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.
Purpose:
DLAED7 computes the updated eigensystem of a diagonal
matrix after modification by a rank-one symmetric matrix. This
routine is used only for the eigenproblem which requires all
eigenvalues and optionally eigenvectors of a dense symmetric matrix
that has been reduced to tridiagonal form. DLAED1 handles
the case in which all eigenvalues and eigenvectors of a symmetric
tridiagonal matrix are desired.
T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)
where Z = Q**Tu, u is a vector of length N with ones in the
CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
The eigenvectors of the original matrix are stored in Q, and the
eigenvalues are in D. The algorithm consists of three stages:
The first stage consists of deflating the size of the problem
when there are multiple eigenvalues or if there is a zero in
the Z vector. For each such occurrence the dimension of the
secular equation problem is reduced by one. This stage is
performed by the routine DLAED8.
The second stage consists of calculating the updated
eigenvalues. This is done by finding the roots of the secular
equation via the routine DLAED4 (as called by DLAED9).
This routine also calculates the eigenvectors of the current
problem.
The final stage consists of computing the updated eigenvectors
directly using the updated eigenvalues. The eigenvectors for
the current problem are multiplied with the eigenvectors from
the overall problem.
Parameters
ICOMPQ is INTEGER
= 0: Compute eigenvalues only.
= 1: Compute eigenvectors of original dense symmetric matrix
also. On entry, Q contains the orthogonal matrix used
to reduce the original matrix to tridiagonal form.
N
N is INTEGER
The dimension of the symmetric tridiagonal matrix. N >= 0.
QSIZ
QSIZ is INTEGER
The dimension of the orthogonal matrix used to reduce
the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.
TLVLS
TLVLS is INTEGER
The total number of merging levels in the overall divide and
conquer tree.
CURLVL
CURLVL is INTEGER
The current level in the overall merge routine,
0 <= CURLVL <= TLVLS.
CURPBM
CURPBM is INTEGER
The current problem in the current level in the overall
merge routine (counting from upper left to lower right).
D
D is DOUBLE PRECISION array, dimension (N)
On entry, the eigenvalues of the rank-1-perturbed matrix.
On exit, the eigenvalues of the repaired matrix.
Q
Q is DOUBLE PRECISION array, dimension (LDQ, N)
On entry, the eigenvectors of the rank-1-perturbed matrix.
On exit, the eigenvectors of the repaired tridiagonal matrix.
LDQ
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N).
INDXQ
INDXQ is INTEGER array, dimension (N)
The permutation which will reintegrate the subproblem just
solved back into sorted order, i.e., D( INDXQ( I = 1, N ) )
will be in ascending order.
RHO
RHO is DOUBLE PRECISION
The subdiagonal element used to create the rank-1
modification.
CUTPNT
CUTPNT is INTEGER
Contains the location of the last eigenvalue in the leading
sub-matrix. min(1,N) <= CUTPNT <= N.
QSTORE
QSTORE is DOUBLE PRECISION array, dimension (N**2+1)
Stores eigenvectors of submatrices encountered during
divide and conquer, packed together. QPTR points to
beginning of the submatrices.
QPTR
QPTR is INTEGER array, dimension (N+2)
List of indices pointing to beginning of submatrices stored
in QSTORE. The submatrices are numbered starting at the
bottom left of the divide and conquer tree, from left to
right and bottom to top.
PRMPTR
PRMPTR is INTEGER array, dimension (N lg N)
Contains a list of pointers which indicate where in PERM a
level's permutation is stored. PRMPTR(i+1) - PRMPTR(i)
indicates the size of the permutation and also the size of
the full, non-deflated problem.
PERM
PERM is INTEGER array, dimension (N lg N)
Contains the permutations (from deflation and sorting) to be
applied to each eigenblock.
GIVPTR
GIVPTR is INTEGER array, dimension (N lg N)
Contains a list of pointers which indicate where in GIVCOL a
level's Givens rotations are stored. GIVPTR(i+1) - GIVPTR(i)
indicates the number of Givens rotations.
GIVCOL
GIVCOL is INTEGER array, dimension (2, N lg N)
Each pair of numbers indicates a pair of columns to take place
in a Givens rotation.
GIVNUM
GIVNUM is DOUBLE PRECISION array, dimension (2, N lg N)
Each number indicates the S value to be used in the
corresponding Givens rotation.
WORK
WORK is DOUBLE PRECISION array, dimension (3*N+2*QSIZ*N)
IWORK
IWORK is INTEGER array, dimension (4*N)
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = 1, an eigenvalue did not converge
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
Contributors:
DLAED8 used by sstedc. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.
Purpose:
DLAED8 merges the two sets of eigenvalues together into a single
sorted set. Then it tries to deflate the size of the problem.
There are two ways in which deflation can occur: when two or more
eigenvalues are close together or if there is a tiny element in the
Z vector. For each such occurrence the order of the related secular
equation problem is reduced by one.
Parameters
ICOMPQ is INTEGER
= 0: Compute eigenvalues only.
= 1: Compute eigenvectors of original dense symmetric matrix
also. On entry, Q contains the orthogonal matrix used
to reduce the original matrix to tridiagonal form.
K
K is INTEGER
The number of non-deflated eigenvalues, and the order of the
related secular equation.
N
N is INTEGER
The dimension of the symmetric tridiagonal matrix. N >= 0.
QSIZ
QSIZ is INTEGER
The dimension of the orthogonal matrix used to reduce
the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.
D
D is DOUBLE PRECISION array, dimension (N)
On entry, the eigenvalues of the two submatrices to be
combined. On exit, the trailing (N-K) updated eigenvalues
(those which were deflated) sorted into increasing order.
Q
Q is DOUBLE PRECISION array, dimension (LDQ,N)
If ICOMPQ = 0, Q is not referenced. Otherwise,
on entry, Q contains the eigenvectors of the partially solved
system which has been previously updated in matrix
multiplies with other partially solved eigensystems.
On exit, Q contains the trailing (N-K) updated eigenvectors
(those which were deflated) in its last N-K columns.
LDQ
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N).
INDXQ
INDXQ is INTEGER array, dimension (N)
The permutation which separately sorts the two sub-problems
in D into ascending order. Note that elements in the second
half of this permutation must first have CUTPNT added to
their values in order to be accurate.
RHO
RHO is DOUBLE PRECISION
On entry, the off-diagonal element associated with the rank-1
cut which originally split the two submatrices which are now
being recombined.
On exit, RHO has been modified to the value required by
DLAED3.
CUTPNT
CUTPNT is INTEGER
The location of the last eigenvalue in the leading
sub-matrix. min(1,N) <= CUTPNT <= N.
Z
Z is DOUBLE PRECISION array, dimension (N)
On entry, Z contains the updating vector (the last row of
the first sub-eigenvector matrix and the first row of the
second sub-eigenvector matrix).
On exit, the contents of Z are destroyed by the updating
process.
DLAMDA
DLAMDA is DOUBLE PRECISION array, dimension (N)
A copy of the first K eigenvalues which will be used by
DLAED3 to form the secular equation.
Q2
Q2 is DOUBLE PRECISION array, dimension (LDQ2,N)
If ICOMPQ = 0, Q2 is not referenced. Otherwise,
a copy of the first K eigenvectors which will be used by
DLAED7 in a matrix multiply (DGEMM) to update the new
eigenvectors.
LDQ2
LDQ2 is INTEGER
The leading dimension of the array Q2. LDQ2 >= max(1,N).
W
W is DOUBLE PRECISION array, dimension (N)
The first k values of the final deflation-altered z-vector and
will be passed to DLAED3.
PERM
PERM is INTEGER array, dimension (N)
The permutations (from deflation and sorting) to be applied
to each eigenblock.
GIVPTR
GIVPTR is INTEGER
The number of Givens rotations which took place in this
subproblem.
GIVCOL
GIVCOL is INTEGER array, dimension (2, N)
Each pair of numbers indicates a pair of columns to take place
in a Givens rotation.
GIVNUM
GIVNUM is DOUBLE PRECISION array, dimension (2, N)
Each number indicates the S value to be used in the
corresponding Givens rotation.
INDXP
INDXP is INTEGER array, dimension (N)
The permutation used to place deflated values of D at the end
of the array. INDXP(1:K) points to the nondeflated D-values
and INDXP(K+1:N) points to the deflated eigenvalues.
INDX
INDX is INTEGER array, dimension (N)
The permutation used to sort the contents of D into ascending
order.
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
Contributors:
DLAED9 used by sstedc. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is dense.
Purpose:
DLAED9 finds the roots of the secular equation, as defined by the
values in D, Z, and RHO, between KSTART and KSTOP. It makes the
appropriate calls to DLAED4 and then stores the new matrix of
eigenvectors for use in calculating the next level of Z vectors.
Parameters
K is INTEGER
The number of terms in the rational function to be solved by
DLAED4. K >= 0.
KSTART
KSTART is INTEGER
KSTOP
KSTOP is INTEGER
The updated eigenvalues Lambda(I), KSTART <= I <= KSTOP
are to be computed. 1 <= KSTART <= KSTOP <= K.
N
N is INTEGER
The number of rows and columns in the Q matrix.
N >= K (delation may result in N > K).
D
D is DOUBLE PRECISION array, dimension (N)
D(I) contains the updated eigenvalues
for KSTART <= I <= KSTOP.
Q
Q is DOUBLE PRECISION array, dimension (LDQ,N)
LDQ
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max( 1, N ).
RHO
RHO is DOUBLE PRECISION
The value of the parameter in the rank one update equation.
RHO >= 0 required.
DLAMDA
DLAMDA is DOUBLE PRECISION array, dimension (K)
The first K elements of this array contain the old roots
of the deflated updating problem. These are the poles
of the secular equation.
W
W is DOUBLE PRECISION array, dimension (K)
The first K elements of this array contain the components
of the deflation-adjusted updating vector.
S
S is DOUBLE PRECISION array, dimension (LDS, K)
Will contain the eigenvectors of the repaired matrix which
will be stored for subsequent Z vector calculation and
multiplied by the previously accumulated eigenvectors
to update the system.
LDS
LDS is INTEGER
The leading dimension of S. LDS >= max( 1, K ).
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = 1, an eigenvalue did not converge
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
Contributors:
DLAEDA used by sstedc. Computes the Z vector determining the rank-one modification of the diagonal matrix. Used when the original matrix is dense.
Purpose:
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.
Parameters
N is INTEGER
The dimension of the symmetric tridiagonal matrix. N >= 0.
TLVLS
TLVLS is INTEGER
The total number of merging levels in the overall divide and
conquer tree.
CURLVL
CURLVL is INTEGER
The current level in the overall merge routine,
0 <= curlvl <= tlvls.
CURPBM
CURPBM is INTEGER
The current problem in the current level in the overall
merge routine (counting from upper left to lower right).
PRMPTR
PRMPTR is INTEGER array, dimension (N lg N)
Contains a list of pointers which indicate where in PERM a
level's permutation is stored. PRMPTR(i+1) - PRMPTR(i)
indicates the size of the permutation and incidentally the
size of the full, non-deflated problem.
PERM
PERM is INTEGER array, dimension (N lg N)
Contains the permutations (from deflation and sorting) to be
applied to each eigenblock.
GIVPTR
GIVPTR is INTEGER array, dimension (N lg N)
Contains a list of pointers which indicate where in GIVCOL a
level's Givens rotations are stored. GIVPTR(i+1) - GIVPTR(i)
indicates the number of Givens rotations.
GIVCOL
GIVCOL is INTEGER array, dimension (2, N lg N)
Each pair of numbers indicates a pair of columns to take place
in a Givens rotation.
GIVNUM
GIVNUM is DOUBLE PRECISION array, dimension (2, N lg N)
Each number indicates the S value to be used in the
corresponding Givens rotation.
Q
Q is DOUBLE PRECISION array, dimension (N**2)
Contains the square eigenblocks from previous levels, the
starting positions for blocks are given by QPTR.
QPTR
QPTR is INTEGER array, dimension (N+2)
Contains a list of pointers which indicate where in Q an
eigenblock is stored. SQRT( QPTR(i+1) - QPTR(i) ) indicates
the size of the block.
Z
Z is DOUBLE PRECISION array, dimension (N)
On output this vector contains the updating vector (the last
row of the first sub-eigenvector matrix and the first row of
the second sub-eigenvector matrix).
ZTEMP
ZTEMP is DOUBLE PRECISION array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
Contributors:
DLAGTF computes an LU factorization of a matrix T-λI, where T is a general tridiagonal matrix, and λ a scalar, using partial pivoting with row interchanges.
Purpose:
DLAGTF factorizes the matrix (T - lambda*I), where T is an n by n
tridiagonal matrix and lambda is a scalar, as
T - lambda*I = PLU,
where P is a permutation matrix, L is a unit lower tridiagonal matrix
with at most one non-zero sub-diagonal elements per column and U is
an upper triangular matrix with at most two non-zero super-diagonal
elements per column.
The factorization is obtained by Gaussian elimination with partial
pivoting and implicit row scaling.
The parameter LAMBDA is included in the routine so that DLAGTF may
be used, in conjunction with DLAGTS, to obtain eigenvectors of T by
inverse iteration.
Parameters
N is INTEGER
The order of the matrix T.
A
A is DOUBLE PRECISION array, dimension (N)
On entry, A must contain the diagonal elements of T.
On exit, A is overwritten by the n diagonal elements of the
upper triangular matrix U of the factorization of T.
LAMBDA
LAMBDA is DOUBLE PRECISION
On entry, the scalar lambda.
B
B is DOUBLE PRECISION array, dimension (N-1)
On entry, B must contain the (n-1) super-diagonal elements of
T.
On exit, B is overwritten by the (n-1) super-diagonal
elements of the matrix U of the factorization of T.
C
C is DOUBLE PRECISION array, dimension (N-1)
On entry, C must contain the (n-1) sub-diagonal elements of
T.
On exit, C is overwritten by the (n-1) sub-diagonal elements
of the matrix L of the factorization of T.
TOL
TOL is DOUBLE PRECISION
On entry, a relative tolerance used to indicate whether or
not the matrix (T - lambda*I) is nearly singular. TOL should
normally be chose as approximately the largest relative error
in the elements of T. For example, if the elements of T are
correct to about 4 significant figures, then TOL should be
set to about 5*10**(-4). If TOL is supplied as less than eps,
where eps is the relative machine precision, then the value
eps is used in place of TOL.
D
D is DOUBLE PRECISION array, dimension (N-2)
On exit, D is overwritten by the (n-2) second super-diagonal
elements of the matrix U of the factorization of T.
IN
IN is INTEGER array, dimension (N)
On exit, IN contains details of the permutation matrix P. If
an interchange occurred at the kth step of the elimination,
then IN(k) = 1, otherwise IN(k) = 0. The element IN(n)
returns the smallest positive integer j such that
abs( u(j,j) ) <= norm( (T - lambda*I)(j) )*TOL,
where norm( A(j) ) denotes the sum of the absolute values of
the jth row of the matrix A. If no such j exists then IN(n)
is returned as zero. If IN(n) is returned as positive, then a
diagonal element of U is small, indicating that
(T - lambda*I) is singular or nearly singular,
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -k, the kth argument had an illegal value
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
DLAMRG creates a permutation list to merge the entries of two independently sorted sets into a single set sorted in ascending order.
Purpose:
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.
Parameters
N1 is INTEGER
N2
N2 is INTEGER
These arguments contain the respective lengths of the two
sorted lists to be merged.
A
A is DOUBLE PRECISION array, dimension (N1+N2)
The first N1 elements of A contain a list of numbers which
are sorted in either ascending or descending order. Likewise
for the final N2 elements.
DTRD1
DTRD1 is INTEGER
DTRD2
DTRD2 is INTEGER
These are the strides to be taken through the array A.
Allowable strides are 1 and -1. They indicate whether a
subset of A is sorted in ascending (DTRDx = 1) or descending
(DTRDx = -1) order.
INDEX
INDEX is INTEGER array, dimension (N1+N2)
On exit this array will contain a permutation such that
if B( I ) = A( INDEX( I ) ) for I=1,N1+N2, then B will be
sorted in ascending order.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
DLARTGS generates a plane rotation designed to introduce a bulge in implicit QR iteration for the bidiagonal SVD problem.
Purpose:
DLARTGS generates a plane rotation designed to introduce a bulge in
Golub-Reinsch-style implicit QR iteration for the bidiagonal SVD
problem. X and Y are the top-row entries, and SIGMA is the shift.
The computed CS and SN define a plane rotation satisfying
[ CS SN ] . [ X^2 - SIGMA ] = [ R ],
[ -SN CS ] [ X * Y ] [ 0 ]
with R nonnegative. If X^2 - SIGMA and X * Y are 0, then the
rotation is by PI/2.
Parameters
X is DOUBLE PRECISION
The (1,1) entry of an upper bidiagonal matrix.
Y
Y is DOUBLE PRECISION
The (1,2) entry of an upper bidiagonal matrix.
SIGMA
SIGMA is DOUBLE PRECISION
The shift.
CS
CS is DOUBLE PRECISION
The cosine of the rotation.
SN
SN is DOUBLE PRECISION
The sine of the rotation.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
DLASQ1 computes the singular values of a real square bidiagonal matrix. Used by sbdsqr.
Purpose:
DLASQ1 computes the singular values of a real N-by-N bidiagonal
matrix with diagonal D and off-diagonal E. The singular values
are computed to high relative accuracy, in the absence of
denormalization, underflow and overflow. The algorithm was first
presented in
"Accurate singular values and differential qd algorithms" by K. V.
Fernando and B. N. Parlett, Numer. Math., Vol-67, No. 2, pp. 191-230,
1994,
and the present implementation is described in "An implementation of
the dqds Algorithm (Positive Case)", LAPACK Working Note.
Parameters
N is INTEGER
The number of rows and columns in the matrix. N >= 0.
D
D is DOUBLE PRECISION array, dimension (N)
On entry, D contains the diagonal elements of the
bidiagonal matrix whose SVD is desired. On normal exit,
D contains the singular values in decreasing order.
E
E is DOUBLE PRECISION array, dimension (N)
On entry, elements E(1:N-1) contain the off-diagonal elements
of the bidiagonal matrix whose SVD is desired.
On exit, E is overwritten.
WORK
WORK is DOUBLE PRECISION array, dimension (4*N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: the algorithm failed
= 1, a split was marked by a positive value in E
= 2, current block of Z not diagonalized after 100*N
iterations (in inner while loop) On exit D and E
represent a matrix with the same singular values
which the calling subroutine could use to finish the
computation, or even feed back into DLASQ1
= 3, termination criterion of outer while loop not met
(program created more than N unreduced blocks)
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
DLASQ2 computes all the eigenvalues of the symmetric positive definite tridiagonal matrix associated with the qd Array Z to high relative accuracy. Used by sbdsqr and sstegr.
Purpose:
DLASQ2 computes all the eigenvalues of the symmetric positive
definite tridiagonal matrix associated with the qd array Z to high
relative accuracy are computed to high relative accuracy, in the
absence of denormalization, underflow and overflow.
To see the relation of Z to the tridiagonal matrix, let L be a
unit lower bidiagonal matrix with subdiagonals Z(2,4,6,,..) and
let U be an upper bidiagonal matrix with 1's above and diagonal
Z(1,3,5,,..). The tridiagonal is L*U or, if you prefer, the
symmetric tridiagonal to which it is similar.
Note : DLASQ2 defines a logical variable, IEEE, which is true
on machines which follow ieee-754 floating-point standard in their
handling of infinities and NaNs, and false otherwise. This variable
is passed to DLASQ3.
Parameters
N is INTEGER
The number of rows and columns in the matrix. N >= 0.
Z
Z is DOUBLE PRECISION array, dimension ( 4*N )
On entry Z holds the qd array. On exit, entries 1 to N hold
the eigenvalues in decreasing order, Z( 2*N+1 ) holds the
trace, and Z( 2*N+2 ) holds the sum of the eigenvalues. If
N > 2, then Z( 2*N+3 ) holds the iteration count, Z( 2*N+4 )
holds NDIVS/NIN^2, and Z( 2*N+5 ) holds the percentage of
shifts that failed.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if the i-th argument is a scalar and had an illegal
value, then INFO = -i, if the i-th argument is an
array and the j-entry had an illegal value, then
INFO = -(i*100+j)
> 0: the algorithm failed
= 1, a split was marked by a positive value in E
= 2, current block of Z not diagonalized after 100*N
iterations (in inner while loop). On exit Z holds
a qd array with the same eigenvalues as the given Z.
= 3, termination criterion of outer while loop not met
(program created more than N unreduced blocks)
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
Further Details:
Local Variables: I0:N0 defines a current unreduced segment of Z.
The shifts are accumulated in SIGMA. Iteration count is in ITER.
Ping-pong is controlled by PP (alternates between 0 and 1).
DLASQ3 checks for deflation, computes a shift and calls dqds. Used by sbdsqr.
Purpose:
DLASQ3 checks for deflation, computes a shift (TAU) and calls dqds.
In case of failure it changes shifts, and tries again until output
is positive.
Parameters
I0 is INTEGER
First index.
N0
N0 is INTEGER
Last index.
Z
Z is DOUBLE PRECISION array, dimension ( 4*N0 )
Z holds the qd array.
PP
PP is INTEGER
PP=0 for ping, PP=1 for pong.
PP=2 indicates that flipping was applied to the Z array
and that the initial tests for deflation should not be
performed.
DMIN
DMIN is DOUBLE PRECISION
Minimum value of d.
SIGMA
SIGMA is DOUBLE PRECISION
Sum of shifts used in current segment.
DESIG
DESIG is DOUBLE PRECISION
Lower order part of SIGMA
QMAX
QMAX is DOUBLE PRECISION
Maximum value of q.
NFAIL
NFAIL is INTEGER
Increment NFAIL by 1 each time the shift was too big.
ITER
ITER is INTEGER
Increment ITER by 1 for each iteration.
NDIV
NDIV is INTEGER
Increment NDIV by 1 for each division.
IEEE
IEEE is LOGICAL
Flag for IEEE or non IEEE arithmetic (passed to DLASQ5).
TTYPE
TTYPE is INTEGER
Shift type.
DMIN1
DMIN1 is DOUBLE PRECISION
DMIN2
DMIN2 is DOUBLE PRECISION
DN
DN is DOUBLE PRECISION
DN1
DN1 is DOUBLE PRECISION
DN2
DN2 is DOUBLE PRECISION
G
G is DOUBLE PRECISION
TAU
TAU is DOUBLE PRECISION
These are passed as arguments in order to save their values
between calls to DLASQ3.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
DLASQ4 computes an approximation to the smallest eigenvalue using values of d from the previous transform. Used by sbdsqr.
Purpose:
DLASQ4 computes an approximation TAU to the smallest eigenvalue
using values of d from the previous transform.
Parameters
I0 is INTEGER
First index.
N0
N0 is INTEGER
Last index.
Z
Z is DOUBLE PRECISION array, dimension ( 4*N0 )
Z holds the qd array.
PP
PP is INTEGER
PP=0 for ping, PP=1 for pong.
N0IN
N0IN is INTEGER
The value of N0 at start of EIGTEST.
DMIN
DMIN is DOUBLE PRECISION
Minimum value of d.
DMIN1
DMIN1 is DOUBLE PRECISION
Minimum value of d, excluding D( N0 ).
DMIN2
DMIN2 is DOUBLE PRECISION
Minimum value of d, excluding D( N0 ) and D( N0-1 ).
DN
DN is DOUBLE PRECISION
d(N)
DN1
DN1 is DOUBLE PRECISION
d(N-1)
DN2
DN2 is DOUBLE PRECISION
d(N-2)
TAU
TAU is DOUBLE PRECISION
This is the shift.
TTYPE
TTYPE is INTEGER
Shift type.
G
G is DOUBLE PRECISION
G is passed as an argument in order to save its value between
calls to DLASQ4.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
Further Details:
CNST1 = 9/16
DLASQ5 computes one dqds transform in ping-pong form. Used by sbdsqr and sstegr.
Purpose:
DLASQ5 computes one dqds transform in ping-pong form, one
version for IEEE machines another for non IEEE machines.
Parameters
I0 is INTEGER
First index.
N0
N0 is INTEGER
Last index.
Z
Z is DOUBLE PRECISION array, dimension ( 4*N )
Z holds the qd array. EMIN is stored in Z(4*N0) to avoid
an extra argument.
PP
PP is INTEGER
PP=0 for ping, PP=1 for pong.
TAU
TAU is DOUBLE PRECISION
This is the shift.
SIGMA
SIGMA is DOUBLE PRECISION
This is the accumulated shift up to this step.
DMIN
DMIN is DOUBLE PRECISION
Minimum value of d.
DMIN1
DMIN1 is DOUBLE PRECISION
Minimum value of d, excluding D( N0 ).
DMIN2
DMIN2 is DOUBLE PRECISION
Minimum value of d, excluding D( N0 ) and D( N0-1 ).
DN
DN is DOUBLE PRECISION
d(N0), the last value of d.
DNM1
DNM1 is DOUBLE PRECISION
d(N0-1).
DNM2
DNM2 is DOUBLE PRECISION
d(N0-2).
IEEE
IEEE is LOGICAL
Flag for IEEE or non IEEE arithmetic.
EPS
EPS is DOUBLE PRECISION
This is the value of epsilon used.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
DLASQ6 computes one dqd transform in ping-pong form. Used by sbdsqr and sstegr.
Purpose:
DLASQ6 computes one dqd (shift equal to zero) transform in
ping-pong form, with protection against underflow and overflow.
Parameters
I0 is INTEGER
First index.
N0
N0 is INTEGER
Last index.
Z
Z is DOUBLE PRECISION array, dimension ( 4*N )
Z holds the qd array. EMIN is stored in Z(4*N0) to avoid
an extra argument.
PP
PP is INTEGER
PP=0 for ping, PP=1 for pong.
DMIN
DMIN is DOUBLE PRECISION
Minimum value of d.
DMIN1
DMIN1 is DOUBLE PRECISION
Minimum value of d, excluding D( N0 ).
DMIN2
DMIN2 is DOUBLE PRECISION
Minimum value of d, excluding D( N0 ) and D( N0-1 ).
DN
DN is DOUBLE PRECISION
d(N0), the last value of d.
DNM1
DNM1 is DOUBLE PRECISION
d(N0-1).
DNM2
DNM2 is DOUBLE PRECISION
d(N0-2).
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
DLASRT sorts numbers in increasing or decreasing order.
Purpose:
Sort the numbers in D in increasing order (if ID = 'I') or
in decreasing order (if ID = 'D' ).
Use Quick Sort, reverting to Insertion sort on arrays of
size <= 20. Dimension of STACK limits N to about 2**32.
Parameters
ID is CHARACTER*1
= 'I': sort D in increasing order;
= 'D': sort D in decreasing order.
N
N is INTEGER
The length of the array D.
D
D is DOUBLE PRECISION array, dimension (N)
On entry, the array to be sorted.
On exit, D has been sorted into increasing order
(D(1) <= ... <= D(N) ) or into decreasing order
(D(1) >= ... >= D(N) ), depending on ID.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
DSTEBZ
Purpose:
DSTEBZ computes the eigenvalues of a symmetric tridiagonal
matrix T. 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.
Parameters
RANGE is CHARACTER*1
= 'A': ("All") all eigenvalues will be found.
= 'V': ("Value") all eigenvalues in the half-open interval
(VL, VU] will be found.
= 'I': ("Index") the IL-th through IU-th eigenvalues (of the
entire matrix) will be found.
ORDER
ORDER is CHARACTER*1
= 'B': ("By Block") the eigenvalues will be grouped by
split-off block (see IBLOCK, ISPLIT) and
ordered from smallest to largest within
the block.
= 'E': ("Entire matrix")
the eigenvalues for the entire matrix
will be ordered from smallest to
largest.
N
N is INTEGER
The order of the tridiagonal matrix T. N >= 0.
VL
VL is DOUBLE PRECISION
If RANGE='V', the lower bound of the interval to
be searched for eigenvalues. Eigenvalues less than or equal
to VL, or greater than VU, will not be returned. VL < VU.
Not referenced if RANGE = 'A' or 'I'.
VU
VU is DOUBLE PRECISION
If RANGE='V', the upper bound of the interval to
be searched for eigenvalues. Eigenvalues less than or equal
to VL, or greater than VU, will not be returned. VL < VU.
Not referenced if RANGE = 'A' or 'I'.
IL
IL is INTEGER
If RANGE='I', the index of the
smallest eigenvalue to be returned.
1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
Not referenced if RANGE = 'A' or 'V'.
IU
IU is INTEGER
If RANGE='I', the index of the
largest eigenvalue to be returned.
1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
Not referenced if RANGE = 'A' or 'V'.
ABSTOL
ABSTOL is DOUBLE PRECISION
The absolute tolerance for the eigenvalues. An eigenvalue
(or cluster) is considered to be located if it has been
determined to lie in an interval whose width is ABSTOL or
less. If ABSTOL is less than or equal to zero, then ULP*|T|
will be used, where |T| means the 1-norm of T.
Eigenvalues will be computed most accurately when ABSTOL is
set to twice the underflow threshold 2*DLAMCH('S'), not zero.
D
D is DOUBLE PRECISION array, dimension (N)
The n diagonal elements of the tridiagonal matrix T.
E
E is DOUBLE PRECISION array, dimension (N-1)
The (n-1) off-diagonal elements of the tridiagonal matrix T.
M
M is INTEGER
The actual number of eigenvalues found. 0 <= M <= N.
(See also the description of INFO=2,3.)
NSPLIT
NSPLIT is INTEGER
The number of diagonal blocks in the matrix T.
1 <= NSPLIT <= N.
W
W is DOUBLE PRECISION array, dimension (N)
On exit, the first M elements of W will contain the
eigenvalues. (DSTEBZ may use the remaining N-M elements as
workspace.)
IBLOCK
IBLOCK is INTEGER array, dimension (N)
At each row/column j where E(j) is zero or small, the
matrix T is considered to split into a block diagonal
matrix. On exit, if INFO = 0, IBLOCK(i) specifies to which
block (from 1 to the number of blocks) the eigenvalue W(i)
belongs. (DSTEBZ may use the remaining N-M elements as
workspace.)
ISPLIT
ISPLIT is INTEGER array, dimension (N)
The splitting points, at which T breaks up into submatrices.
The first submatrix consists of rows/columns 1 to ISPLIT(1),
the second of rows/columns ISPLIT(1)+1 through ISPLIT(2),
etc., and the NSPLIT-th consists of rows/columns
ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.
(Only the first NSPLIT elements will actually be used, but
since the user cannot know a priori what value NSPLIT will
have, N words must be reserved for ISPLIT.)
WORK
WORK is DOUBLE PRECISION array, dimension (4*N)
IWORK
IWORK is INTEGER array, dimension (3*N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: some or all of the eigenvalues failed to converge or
were not computed:
=1 or 3: Bisection failed to converge for some
eigenvalues; these eigenvalues are flagged by a
negative block number. The effect is that the
eigenvalues may not be as accurate as the
absolute and relative tolerances. This is
generally caused by unexpectedly inaccurate
arithmetic.
=2 or 3: RANGE='I' only: Not all of the eigenvalues
IL:IU were found.
Effect: M < IU+1-IL
Cause: non-monotonic arithmetic, causing the
Sturm sequence to be non-monotonic.
Cure: recalculate, using RANGE='A', and pick
out eigenvalues IL:IU. In some cases,
increasing the PARAMETER "FUDGE" may
make things work.
= 4: RANGE='I', and the Gershgorin interval
initially used was too small. No eigenvalues
were computed.
Probable cause: your machine has sloppy
floating-point arithmetic.
Cure: Increase the PARAMETER "FUDGE",
recompile, and try again.
Internal Parameters:
RELFAC DOUBLE PRECISION, default = 2.0e0
The relative tolerance. An interval (a,b] lies within
"relative tolerance" if b-a < RELFAC*ulp*max(|a|,|b|),
where "ulp" is the machine precision (distance from 1 to
the next larger floating point number.)
FUDGE DOUBLE PRECISION, default = 2
A "fudge factor" to widen the Gershgorin intervals. Ideally,
a value of 1 should work, but on machines with sloppy
arithmetic, this needs to be larger. The default for
publicly released versions should be large enough to handle
the worst machine around. Note that this has no effect
on accuracy of the solution.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
DSTEDC
Purpose:
DSTEDC computes all eigenvalues and, optionally, eigenvectors of a
symmetric tridiagonal matrix using the divide and conquer method.
The eigenvectors of a full or band real symmetric matrix can also be
found if DSYTRD or DSPTRD or DSBTRD has been used to reduce this
matrix to tridiagonal form.
This code makes very mild assumptions about floating point
arithmetic. It will work on machines with a guard digit in
add/subtract, or on those binary machines without guard digits
which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
It could conceivably fail on hexadecimal or decimal machines
without guard digits, but we know of none. See DLAED3 for details.
Parameters
COMPZ is CHARACTER*1
= 'N': Compute eigenvalues only.
= 'I': Compute eigenvectors of tridiagonal matrix also.
= 'V': Compute eigenvectors of original dense symmetric
matrix also. On entry, Z contains the orthogonal
matrix used to reduce the original matrix to
tridiagonal form.
N
N is INTEGER
The dimension of the symmetric tridiagonal matrix. N >= 0.
D
D is DOUBLE PRECISION array, dimension (N)
On entry, the diagonal elements of the tridiagonal matrix.
On exit, if INFO = 0, the eigenvalues in ascending order.
E
E is DOUBLE PRECISION array, dimension (N-1)
On entry, the subdiagonal elements of the tridiagonal matrix.
On exit, E has been destroyed.
Z
Z is DOUBLE PRECISION array, dimension (LDZ,N)
On entry, if COMPZ = 'V', then Z contains the orthogonal
matrix used in the reduction to tridiagonal form.
On exit, if INFO = 0, then if COMPZ = 'V', Z contains the
orthonormal eigenvectors of the original symmetric matrix,
and if COMPZ = 'I', Z contains the orthonormal eigenvectors
of the symmetric tridiagonal matrix.
If COMPZ = 'N', then Z is not referenced.
LDZ
LDZ is INTEGER
The leading dimension of the array Z. LDZ >= 1.
If eigenvectors are desired, then LDZ >= max(1,N).
WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK.
If COMPZ = 'N' or N <= 1 then LWORK must be at least 1.
If COMPZ = 'V' and N > 1 then LWORK must be at least
( 1 + 3*N + 2*N*lg N + 4*N**2 ),
where lg( N ) = smallest integer k such
that 2**k >= N.
If COMPZ = 'I' and N > 1 then LWORK must be at least
( 1 + 4*N + N**2 ).
Note that for COMPZ = 'I' or 'V', then if N is less than or
equal to the minimum divide size, usually 25, then LWORK need
only be max(1,2*(N-1)).
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
IWORK
IWORK is INTEGER array, dimension (MAX(1,LIWORK))
On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
LIWORK
LIWORK is INTEGER
The dimension of the array IWORK.
If COMPZ = 'N' or N <= 1 then LIWORK must be at least 1.
If COMPZ = 'V' and N > 1 then LIWORK must be at least
( 6 + 6*N + 5*N*lg N ).
If COMPZ = 'I' and N > 1 then LIWORK must be at least
( 3 + 5*N ).
Note that for COMPZ = 'I' or 'V', then if N is less than or
equal to the minimum divide size, usually 25, then LIWORK
need only be 1.
If LIWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the IWORK array,
returns this value as the first entry of the IWORK array, and
no error message related to LIWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: The algorithm failed to compute an eigenvalue while
working on the submatrix lying in rows and columns
INFO/(N+1) through mod(INFO,N+1).
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
Contributors:
DSTEQR
Purpose:
DSTEQR computes all eigenvalues and, optionally, eigenvectors of a
symmetric tridiagonal matrix using the implicit QL or QR method.
The eigenvectors of a full or band symmetric matrix can also be found
if DSYTRD or DSPTRD or DSBTRD has been used to reduce this matrix to
tridiagonal form.
Parameters
COMPZ is CHARACTER*1
= 'N': Compute eigenvalues only.
= 'V': Compute eigenvalues and eigenvectors of the original
symmetric matrix. On entry, Z must contain the
orthogonal matrix used to reduce the original matrix
to tridiagonal form.
= 'I': Compute eigenvalues and eigenvectors of the
tridiagonal matrix. Z is initialized to the identity
matrix.
N
N is INTEGER
The order of the matrix. N >= 0.
D
D is DOUBLE PRECISION array, dimension (N)
On entry, the diagonal elements of the tridiagonal matrix.
On exit, if INFO = 0, the eigenvalues in ascending order.
E
E is DOUBLE PRECISION array, dimension (N-1)
On entry, the (n-1) subdiagonal elements of the tridiagonal
matrix.
On exit, E has been destroyed.
Z
Z is DOUBLE PRECISION array, dimension (LDZ, N)
On entry, if COMPZ = 'V', then Z contains the orthogonal
matrix used in the reduction to tridiagonal form.
On exit, if INFO = 0, then if COMPZ = 'V', Z contains the
orthonormal eigenvectors of the original symmetric matrix,
and if COMPZ = 'I', Z contains the orthonormal eigenvectors
of the symmetric tridiagonal matrix.
If COMPZ = 'N', then Z is not referenced.
LDZ
LDZ is INTEGER
The leading dimension of the array Z. LDZ >= 1, and if
eigenvectors are desired, then LDZ >= max(1,N).
WORK
WORK is DOUBLE PRECISION array, dimension (max(1,2*N-2))
If COMPZ = 'N', then WORK is not referenced.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: the algorithm has failed to find all the eigenvalues in
a total of 30*N iterations; if INFO = i, then i
elements of E have not converged to zero; on exit, D
and E contain the elements of a symmetric tridiagonal
matrix which is orthogonally similar to the original
matrix.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
DSTERF
Purpose:
DSTERF computes all eigenvalues of a symmetric tridiagonal matrix
using the Pal-Walker-Kahan variant of the QL or QR algorithm.
Parameters
N is INTEGER
The order of the matrix. N >= 0.
D
D is DOUBLE PRECISION array, dimension (N)
On entry, the n diagonal elements of the tridiagonal matrix.
On exit, if INFO = 0, the eigenvalues in ascending order.
E
E is DOUBLE PRECISION array, dimension (N-1)
On entry, the (n-1) subdiagonal elements of the tridiagonal
matrix.
On exit, E has been destroyed.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: the algorithm failed to find all of the eigenvalues in
a total of 30*N iterations; if INFO = i, then i
elements of E have not converged to zero.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
ILADIAG
Purpose:
This subroutine translated from a character string specifying if a
matrix has unit diagonal or not to the relevant BLAST-specified
integer constant.
ILADIAG returns an INTEGER. If ILADIAG < 0, then the input is not a
character indicating a unit or non-unit diagonal. Otherwise ILADIAG
returns the constant value corresponding to DIAG.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
ILAPREC
Purpose:
This subroutine translated from a character string specifying an
intermediate precision to the relevant BLAST-specified integer
constant.
ILAPREC returns an INTEGER. If ILAPREC < 0, then the input is not a
character indicating a supported intermediate precision. Otherwise
ILAPREC returns the constant value corresponding to PREC.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
ILATRANS
Purpose:
This subroutine translates from a character string specifying a
transposition operation to the relevant BLAST-specified integer
constant.
ILATRANS returns an INTEGER. If ILATRANS < 0, then the input is not
a character indicating a transposition operator. Otherwise ILATRANS
returns the constant value corresponding to TRANS.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
ILAUPLO
Purpose:
This subroutine translated from a character string specifying a
upper- or lower-triangular matrix to the relevant BLAST-specified
integer constant.
ILAUPLO returns an INTEGER. If ILAUPLO < 0, then the input is not
a character indicating an upper- or lower-triangular matrix.
Otherwise ILAUPLO returns the constant value corresponding to UPLO.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
SBDSDC
Purpose:
SBDSDC computes the singular value decomposition (SVD) of a real
N-by-N (upper or lower) bidiagonal matrix B: B = U * S * VT,
using a divide and conquer method, where S is a diagonal matrix
with non-negative diagonal elements (the singular values of B), and
U and VT are orthogonal matrices of left and right singular vectors,
respectively. SBDSDC can be used to compute all singular values,
and optionally, singular vectors or singular vectors in compact form.
This code makes very mild assumptions about floating point
arithmetic. It will work on machines with a guard digit in
add/subtract, or on those binary machines without guard digits
which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
It could conceivably fail on hexadecimal or decimal machines
without guard digits, but we know of none. See SLASD3 for details.
The code currently calls SLASDQ if singular values only are desired.
However, it can be slightly modified to compute singular values
using the divide and conquer method.
Parameters
UPLO is CHARACTER*1
= 'U': B is upper bidiagonal.
= 'L': B is lower bidiagonal.
COMPQ
COMPQ is CHARACTER*1
Specifies whether singular vectors are to be computed
as follows:
= 'N': Compute singular values only;
= 'P': Compute singular values and compute singular
vectors in compact form;
= 'I': Compute singular values and singular vectors.
N
N is INTEGER
The order of the matrix B. N >= 0.
D
D is REAL array, dimension (N)
On entry, the n diagonal elements of the bidiagonal matrix B.
On exit, if INFO=0, the singular values of B.
E
E is REAL array, dimension (N-1)
On entry, the elements of E contain the offdiagonal
elements of the bidiagonal matrix whose SVD is desired.
On exit, E has been destroyed.
U
U is REAL array, dimension (LDU,N)
If COMPQ = 'I', then:
On exit, if INFO = 0, U contains the left singular vectors
of the bidiagonal matrix.
For other values of COMPQ, U is not referenced.
LDU
LDU is INTEGER
The leading dimension of the array U. LDU >= 1.
If singular vectors are desired, then LDU >= max( 1, N ).
VT
VT is REAL array, dimension (LDVT,N)
If COMPQ = 'I', then:
On exit, if INFO = 0, VT**T contains the right singular
vectors of the bidiagonal matrix.
For other values of COMPQ, VT is not referenced.
LDVT
LDVT is INTEGER
The leading dimension of the array VT. LDVT >= 1.
If singular vectors are desired, then LDVT >= max( 1, N ).
Q
Q is REAL array, dimension (LDQ)
If COMPQ = 'P', then:
On exit, if INFO = 0, Q and IQ contain the left
and right singular vectors in a compact form,
requiring O(N log N) space instead of 2*N**2.
In particular, Q contains all the REAL data in
LDQ >= N*(11 + 2*SMLSIZ + 8*INT(LOG_2(N/(SMLSIZ+1))))
words of memory, where SMLSIZ is returned by ILAENV and
is equal to the maximum size of the subproblems at the
bottom of the computation tree (usually about 25).
For other values of COMPQ, Q is not referenced.
IQ
IQ is INTEGER array, dimension (LDIQ)
If COMPQ = 'P', then:
On exit, if INFO = 0, Q and IQ contain the left
and right singular vectors in a compact form,
requiring O(N log N) space instead of 2*N**2.
In particular, IQ contains all INTEGER data in
LDIQ >= N*(3 + 3*INT(LOG_2(N/(SMLSIZ+1))))
words of memory, where SMLSIZ is returned by ILAENV and
is equal to the maximum size of the subproblems at the
bottom of the computation tree (usually about 25).
For other values of COMPQ, IQ is not referenced.
WORK
WORK is REAL array, dimension (MAX(1,LWORK))
If COMPQ = 'N' then LWORK >= (4 * N).
If COMPQ = 'P' then LWORK >= (6 * N).
If COMPQ = 'I' then LWORK >= (3 * N**2 + 4 * N).
IWORK
IWORK is INTEGER array, dimension (8*N)
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: The algorithm failed to compute a singular value.
The update process of divide and conquer failed.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
Contributors:
SBDSQR
Purpose:
SBDSQR computes the singular values and, optionally, the right and/or
left singular vectors from the singular value decomposition (SVD) of
a real N-by-N (upper or lower) bidiagonal matrix B using the implicit
zero-shift QR algorithm. The SVD of B has the form
B = Q * S * P**T
where S is the diagonal matrix of singular values, Q is an orthogonal
matrix of left singular vectors, and P is an orthogonal matrix of
right singular vectors. If left singular vectors are requested, this
subroutine actually returns U*Q instead of Q, and, if right singular
vectors are requested, this subroutine returns P**T*VT instead of
P**T, for given real input matrices U and VT. When U and VT are the
orthogonal matrices that reduce a general matrix A to bidiagonal
form: A = U*B*VT, as computed by SGEBRD, then
A = (U*Q) * S * (P**T*VT)
is the SVD of A. Optionally, the subroutine may also compute Q**T*C
for a given real input matrix C.
See "Computing Small Singular Values of Bidiagonal Matrices With
Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan,
LAPACK Working Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11,
no. 5, pp. 873-912, Sept 1990) and
"Accurate singular values and differential qd algorithms," by
B. Parlett and V. Fernando, Technical Report CPAM-554, Mathematics
Department, University of California at Berkeley, July 1992
for a detailed description of the algorithm.
Parameters
UPLO is CHARACTER*1
= 'U': B is upper bidiagonal;
= 'L': B is lower bidiagonal.
N
N is INTEGER
The order of the matrix B. N >= 0.
NCVT
NCVT is INTEGER
The number of columns of the matrix VT. NCVT >= 0.
NRU
NRU is INTEGER
The number of rows of the matrix U. NRU >= 0.
NCC
NCC is INTEGER
The number of columns of the matrix C. NCC >= 0.
D
D is REAL array, dimension (N)
On entry, the n diagonal elements of the bidiagonal matrix B.
On exit, if INFO=0, the singular values of B in decreasing
order.
E
E is REAL array, dimension (N-1)
On entry, the N-1 offdiagonal elements of the bidiagonal
matrix B.
On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E
will contain the diagonal and superdiagonal elements of a
bidiagonal matrix orthogonally equivalent to the one given
as input.
VT
VT is REAL array, dimension (LDVT, NCVT)
On entry, an N-by-NCVT matrix VT.
On exit, VT is overwritten by P**T * VT.
Not referenced if NCVT = 0.
LDVT
LDVT is INTEGER
The leading dimension of the array VT.
LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0.
U
U is REAL array, dimension (LDU, N)
On entry, an NRU-by-N matrix U.
On exit, U is overwritten by U * Q.
Not referenced if NRU = 0.
LDU
LDU is INTEGER
The leading dimension of the array U. LDU >= max(1,NRU).
C
C is REAL array, dimension (LDC, NCC)
On entry, an N-by-NCC matrix C.
On exit, C is overwritten by Q**T * C.
Not referenced if NCC = 0.
LDC
LDC is INTEGER
The leading dimension of the array C.
LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0.
WORK
WORK is REAL array, dimension (4*N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: If INFO = -i, the i-th argument had an illegal value
> 0:
if NCVT = NRU = NCC = 0,
= 1, a split was marked by a positive value in E
= 2, current block of Z not diagonalized after 30*N
iterations (in inner while loop)
= 3, termination criterion of outer while loop not met
(program created more than N unreduced blocks)
else NCVT = NRU = NCC = 0,
the algorithm did not converge; D and E contain the
elements of a bidiagonal matrix which is orthogonally
similar to the input matrix B; if INFO = i, i
elements of E have not converged to zero.
Internal Parameters:
TOLMUL REAL, default = max(10,min(100,EPS**(-1/8)))
TOLMUL controls the convergence criterion of the QR loop.
If it is positive, TOLMUL*EPS is the desired relative
precision in the computed singular values.
If it is negative, abs(TOLMUL*EPS*sigma_max) is the
desired absolute accuracy in the computed singular
values (corresponds to relative accuracy
abs(TOLMUL*EPS) in the largest singular value.
abs(TOLMUL) should be between 1 and 1/EPS, and preferably
between 10 (for fast convergence) and .1/EPS
(for there to be some accuracy in the results).
Default is to lose at either one eighth or 2 of the
available decimal digits in each computed singular value
(whichever is smaller).
MAXITR INTEGER, default = 6
MAXITR controls the maximum number of passes of the
algorithm through its inner loop. The algorithms stops
(and so fails to converge) if the number of passes
through the inner loop exceeds MAXITR*N**2.
Note:
Bug report from Cezary Dendek.
On March 23rd 2017, the INTEGER variable MAXIT = MAXITR*N**2 is
removed since it can overflow pretty easily (for N larger or equal
than 18,919). We instead use MAXITDIVN = MAXITR*N.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
SDISNA
Purpose:
SDISNA computes the reciprocal condition numbers for the eigenvectors
of a real symmetric or complex Hermitian matrix or for the left or
right singular vectors of a general m-by-n matrix. The reciprocal
condition number is the 'gap' between the corresponding eigenvalue or
singular value and the nearest other one.
The bound on the error, measured by angle in radians, in the I-th
computed vector is given by
SLAMCH( 'E' ) * ( ANORM / SEP( I ) )
where ANORM = 2-norm(A) = max( abs( D(j) ) ). SEP(I) is not allowed
to be smaller than SLAMCH( 'E' )*ANORM in order to limit the size of
the error bound.
SDISNA may also be used to compute error bounds for eigenvectors of
the generalized symmetric definite eigenproblem.
Parameters
JOB is CHARACTER*1
Specifies for which problem the reciprocal condition numbers
should be computed:
= 'E': the eigenvectors of a symmetric/Hermitian matrix;
= 'L': the left singular vectors of a general matrix;
= 'R': the right singular vectors of a general matrix.
M
M is INTEGER
The number of rows of the matrix. M >= 0.
N
N is INTEGER
If JOB = 'L' or 'R', the number of columns of the matrix,
in which case N >= 0. Ignored if JOB = 'E'.
D
D is REAL array, dimension (M) if JOB = 'E'
dimension (min(M,N)) if JOB = 'L' or 'R'
The eigenvalues (if JOB = 'E') or singular values (if JOB =
'L' or 'R') of the matrix, in either increasing or decreasing
order. If singular values, they must be non-negative.
SEP
SEP is REAL array, dimension (M) if JOB = 'E'
dimension (min(M,N)) if JOB = 'L' or 'R'
The reciprocal condition numbers of the vectors.
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
SLAED0 used by sstedc. Computes all eigenvalues and corresponding eigenvectors of an unreduced symmetric tridiagonal matrix using the divide and conquer method.
Purpose:
SLAED0 computes all eigenvalues and corresponding eigenvectors of a
symmetric tridiagonal matrix using the divide and conquer method.
Parameters
ICOMPQ is INTEGER
= 0: Compute eigenvalues only.
= 1: Compute eigenvectors of original dense symmetric matrix
also. On entry, Q contains the orthogonal matrix used
to reduce the original matrix to tridiagonal form.
= 2: Compute eigenvalues and eigenvectors of tridiagonal
matrix.
QSIZ
QSIZ is INTEGER
The dimension of the orthogonal matrix used to reduce
the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.
N
N is INTEGER
The dimension of the symmetric tridiagonal matrix. N >= 0.
D
D is REAL array, dimension (N)
On entry, the main diagonal of the tridiagonal matrix.
On exit, its eigenvalues.
E
E is REAL array, dimension (N-1)
The off-diagonal elements of the tridiagonal matrix.
On exit, E has been destroyed.
Q
Q is REAL array, dimension (LDQ, N)
On entry, Q must contain an N-by-N orthogonal matrix.
If ICOMPQ = 0 Q is not referenced.
If ICOMPQ = 1 On entry, Q is a subset of the columns of the
orthogonal matrix used to reduce the full
matrix to tridiagonal form corresponding to
the subset of the full matrix which is being
decomposed at this time.
If ICOMPQ = 2 On entry, Q will be the identity matrix.
On exit, Q contains the eigenvectors of the
tridiagonal matrix.
LDQ
LDQ is INTEGER
The leading dimension of the array Q. If eigenvectors are
desired, then LDQ >= max(1,N). In any case, LDQ >= 1.
QSTORE
QSTORE is REAL array, dimension (LDQS, N)
Referenced only when ICOMPQ = 1. Used to store parts of
the eigenvector matrix when the updating matrix multiplies
take place.
LDQS
LDQS is INTEGER
The leading dimension of the array QSTORE. If ICOMPQ = 1,
then LDQS >= max(1,N). In any case, LDQS >= 1.
WORK
WORK is REAL array,
If ICOMPQ = 0 or 1, the dimension of WORK must be at least
1 + 3*N + 2*N*lg N + 3*N**2
( lg( N ) = smallest integer k
such that 2^k >= N )
If ICOMPQ = 2, the dimension of WORK must be at least
4*N + N**2.
IWORK
IWORK is INTEGER array,
If ICOMPQ = 0 or 1, the dimension of IWORK must be at least
6 + 6*N + 5*N*lg N.
( lg( N ) = smallest integer k
such that 2^k >= N )
If ICOMPQ = 2, the dimension of IWORK must be at least
3 + 5*N.
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: The algorithm failed to compute an eigenvalue while
working on the submatrix lying in rows and columns
INFO/(N+1) through mod(INFO,N+1).
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
Contributors:
SLAED1 used by sstedc. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is tridiagonal.
Purpose:
SLAED1 computes the updated eigensystem of a diagonal
matrix after modification by a rank-one symmetric matrix. This
routine is used only for the eigenproblem which requires all
eigenvalues and eigenvectors of a tridiagonal matrix. SLAED7 handles
the case in which eigenvalues only or eigenvalues and eigenvectors
of a full symmetric matrix (which was reduced to tridiagonal form)
are desired.
T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)
where Z = Q**T*u, u is a vector of length N with ones in the
CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
The eigenvectors of the original matrix are stored in Q, and the
eigenvalues are in D. The algorithm consists of three stages:
The first stage consists of deflating the size of the problem
when there are multiple eigenvalues or if there is a zero in
the Z vector. For each such occurrence the dimension of the
secular equation problem is reduced by one. This stage is
performed by the routine SLAED2.
The second stage consists of calculating the updated
eigenvalues. This is done by finding the roots of the secular
equation via the routine SLAED4 (as called by SLAED3).
This routine also calculates the eigenvectors of the current
problem.
The final stage consists of computing the updated eigenvectors
directly using the updated eigenvalues. The eigenvectors for
the current problem are multiplied with the eigenvectors from
the overall problem.
Parameters
N is INTEGER
The dimension of the symmetric tridiagonal matrix. N >= 0.
D
D is REAL array, dimension (N)
On entry, the eigenvalues of the rank-1-perturbed matrix.
On exit, the eigenvalues of the repaired matrix.
Q
Q is REAL array, dimension (LDQ,N)
On entry, the eigenvectors of the rank-1-perturbed matrix.
On exit, the eigenvectors of the repaired tridiagonal matrix.
LDQ
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N).
INDXQ
INDXQ is INTEGER array, dimension (N)
On entry, the permutation which separately sorts the two
subproblems in D into ascending order.
On exit, the permutation which will reintegrate the
subproblems back into sorted order,
i.e. D( INDXQ( I = 1, N ) ) will be in ascending order.
RHO
RHO is REAL
The subdiagonal entry used to create the rank-1 modification.
CUTPNT
CUTPNT is INTEGER
The location of the last eigenvalue in the leading sub-matrix.
min(1,N) <= CUTPNT <= N/2.
WORK
WORK is REAL array, dimension (4*N + N**2)
IWORK
IWORK is INTEGER array, dimension (4*N)
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = 1, an eigenvalue did not converge
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
Contributors:
SLAED2 used by sstedc. Merges eigenvalues and deflates secular equation. Used when the original matrix is tridiagonal.
Purpose:
SLAED2 merges the two sets of eigenvalues together into a single
sorted set. Then it tries to deflate the size of the problem.
There are two ways in which deflation can occur: when two or more
eigenvalues are close together or if there is a tiny entry in the
Z vector. For each such occurrence the order of the related secular
equation problem is reduced by one.
Parameters
K is INTEGER
The number of non-deflated eigenvalues, and the order of the
related secular equation. 0 <= K <=N.
N
N is INTEGER
The dimension of the symmetric tridiagonal matrix. N >= 0.
N1
N1 is INTEGER
The location of the last eigenvalue in the leading sub-matrix.
min(1,N) <= N1 <= N/2.
D
D is REAL array, dimension (N)
On entry, D contains the eigenvalues of the two submatrices to
be combined.
On exit, D contains the trailing (N-K) updated eigenvalues
(those which were deflated) sorted into increasing order.
Q
Q is REAL array, dimension (LDQ, N)
On entry, Q contains the eigenvectors of two submatrices in
the two square blocks with corners at (1,1), (N1,N1)
and (N1+1, N1+1), (N,N).
On exit, Q contains the trailing (N-K) updated eigenvectors
(those which were deflated) in its last N-K columns.
LDQ
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N).
INDXQ
INDXQ is INTEGER array, dimension (N)
The permutation which separately sorts the two sub-problems
in D into ascending order. Note that elements in the second
half of this permutation must first have N1 added to their
values. Destroyed on exit.
RHO
RHO is REAL
On entry, the off-diagonal element associated with the rank-1
cut which originally split the two submatrices which are now
being recombined.
On exit, RHO has been modified to the value required by
SLAED3.
Z
Z is REAL array, dimension (N)
On entry, Z contains the updating vector (the last
row of the first sub-eigenvector matrix and the first row of
the second sub-eigenvector matrix).
On exit, the contents of Z have been destroyed by the updating
process.
DLAMDA
DLAMDA is REAL array, dimension (N)
A copy of the first K eigenvalues which will be used by
SLAED3 to form the secular equation.
W
W is REAL array, dimension (N)
The first k values of the final deflation-altered z-vector
which will be passed to SLAED3.
Q2
Q2 is REAL array, dimension (N1**2+(N-N1)**2)
A copy of the first K eigenvectors which will be used by
SLAED3 in a matrix multiply (SGEMM) to solve for the new
eigenvectors.
INDX
INDX is INTEGER array, dimension (N)
The permutation used to sort the contents of DLAMDA into
ascending order.
INDXC
INDXC is INTEGER array, dimension (N)
The permutation used to arrange the columns of the deflated
Q matrix into three groups: the first group contains non-zero
elements only at and above N1, the second contains
non-zero elements only below N1, and the third is dense.
INDXP
INDXP is INTEGER array, dimension (N)
The permutation used to place deflated values of D at the end
of the array. INDXP(1:K) points to the nondeflated D-values
and INDXP(K+1:N) points to the deflated eigenvalues.
COLTYP
COLTYP is INTEGER array, dimension (N)
During execution, a label which will indicate which of the
following types a column in the Q2 matrix is:
1 : non-zero in the upper half only;
2 : dense;
3 : non-zero in the lower half only;
4 : deflated.
On exit, COLTYP(i) is the number of columns of type i,
for i=1 to 4 only.
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
Contributors:
SLAED3 used by sstedc. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is tridiagonal.
Purpose:
SLAED3 finds the roots of the secular equation, as defined by the
values in D, W, and RHO, between 1 and K. It makes the
appropriate calls to SLAED4 and then updates the eigenvectors by
multiplying the matrix of eigenvectors of the pair of eigensystems
being combined by the matrix of eigenvectors of the K-by-K system
which is solved here.
This code makes very mild assumptions about floating point
arithmetic. It will work on machines with a guard digit in
add/subtract, or on those binary machines without guard digits
which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
It could conceivably fail on hexadecimal or decimal machines
without guard digits, but we know of none.
Parameters
K is INTEGER
The number of terms in the rational function to be solved by
SLAED4. K >= 0.
N
N is INTEGER
The number of rows and columns in the Q matrix.
N >= K (deflation may result in N>K).
N1
N1 is INTEGER
The location of the last eigenvalue in the leading submatrix.
min(1,N) <= N1 <= N/2.
D
D is REAL array, dimension (N)
D(I) contains the updated eigenvalues for
1 <= I <= K.
Q
Q is REAL array, dimension (LDQ,N)
Initially the first K columns are used as workspace.
On output the columns 1 to K contain
the updated eigenvectors.
LDQ
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N).
RHO
RHO is REAL
The value of the parameter in the rank one update equation.
RHO >= 0 required.
DLAMDA
DLAMDA is REAL array, dimension (K)
The first K elements of this array contain the old roots
of the deflated updating problem. These are the poles
of the secular equation. May be changed on output by
having lowest order bit set to zero on Cray X-MP, Cray Y-MP,
Cray-2, or Cray C-90, as described above.
Q2
Q2 is REAL array, dimension (LDQ2*N)
The first K columns of this matrix contain the non-deflated
eigenvectors for the split problem.
INDX
INDX is INTEGER array, dimension (N)
The permutation used to arrange the columns of the deflated
Q matrix into three groups (see SLAED2).
The rows of the eigenvectors found by SLAED4 must be likewise
permuted before the matrix multiply can take place.
CTOT
CTOT is INTEGER array, dimension (4)
A count of the total number of the various types of columns
in Q, as described in INDX. The fourth column type is any
column which has been deflated.
W
W is REAL array, dimension (K)
The first K elements of this array contain the components
of the deflation-adjusted updating vector. Destroyed on
output.
S
S is REAL array, dimension (N1 + 1)*K
Will contain the eigenvectors of the repaired matrix which
will be multiplied by the previously accumulated eigenvectors
to update the system.
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = 1, an eigenvalue did not converge
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
Contributors:
SLAED4 used by sstedc. Finds a single root of the secular equation.
Purpose:
This subroutine computes the I-th updated eigenvalue of a symmetric
rank-one modification to a diagonal matrix whose elements are
given in the array d, and that
D(i) < D(j) for i < j
and that RHO > 0. This is arranged by the calling routine, and is
no loss in generality. The rank-one modified system is thus
diag( D ) + RHO * Z * Z_transpose.
where we assume the Euclidean norm of Z is 1.
The method consists of approximating the rational functions in the
secular equation by simpler interpolating rational functions.
Parameters
N is INTEGER
The length of all arrays.
I
I is INTEGER
The index of the eigenvalue to be computed. 1 <= I <= N.
D
D is REAL array, dimension (N)
The original eigenvalues. It is assumed that they are in
order, D(I) < D(J) for I < J.
Z
Z is REAL array, dimension (N)
The components of the updating vector.
DELTA
DELTA is REAL array, dimension (N)
If N > 2, DELTA contains (D(j) - lambda_I) in its j-th
component. If N = 1, then DELTA(1) = 1. If N = 2, see SLAED5
for detail. The vector DELTA contains the information necessary
to construct the eigenvectors by SLAED3 and SLAED9.
RHO
RHO is REAL
The scalar in the symmetric updating formula.
DLAM
DLAM is REAL
The computed lambda_I, the I-th updated eigenvalue.
INFO
INFO is INTEGER
= 0: successful exit
> 0: if INFO = 1, the updating process failed.
Internal Parameters:
Logical variable ORGATI (origin-at-i?) is used for distinguishing
whether D(i) or D(i+1) is treated as the origin.
ORGATI = .true. origin at i
ORGATI = .false. origin at i+1
Logical variable SWTCH3 (switch-for-3-poles?) is for noting
if we are working with THREE poles!
MAXIT is the maximum number of iterations allowed for each
eigenvalue.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
Contributors:
SLAED5 used by sstedc. Solves the 2-by-2 secular equation.
Purpose:
This subroutine computes the I-th eigenvalue of a symmetric rank-one
modification of a 2-by-2 diagonal matrix
diag( D ) + RHO * Z * transpose(Z) .
The diagonal elements in the array D are assumed to satisfy
D(i) < D(j) for i < j .
We also assume RHO > 0 and that the Euclidean norm of the vector
Z is one.
Parameters
I is INTEGER
The index of the eigenvalue to be computed. I = 1 or I = 2.
D
D is REAL array, dimension (2)
The original eigenvalues. We assume D(1) < D(2).
Z
Z is REAL array, dimension (2)
The components of the updating vector.
DELTA
DELTA is REAL array, dimension (2)
The vector DELTA contains the information necessary
to construct the eigenvectors.
RHO
RHO is REAL
The scalar in the symmetric updating formula.
DLAM
DLAM is REAL
The computed lambda_I, the I-th updated eigenvalue.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
Contributors:
SLAED6 used by sstedc. Computes one Newton step in solution of the secular equation.
Purpose:
SLAED6 computes the positive or negative root (closest to the origin)
of
z(1) z(2) z(3)
f(x) = rho + --------- + ---------- + ---------
d(1)-x d(2)-x d(3)-x
It is assumed that
if ORGATI = .true. the root is between d(2) and d(3);
otherwise it is between d(1) and d(2)
This routine will be called by SLAED4 when necessary. In most cases,
the root sought is the smallest in magnitude, though it might not be
in some extremely rare situations.
Parameters
KNITER is INTEGER
Refer to SLAED4 for its significance.
ORGATI
ORGATI is LOGICAL
If ORGATI is true, the needed root is between d(2) and
d(3); otherwise it is between d(1) and d(2). See
SLAED4 for further details.
RHO
RHO is REAL
Refer to the equation f(x) above.
D
D is REAL array, dimension (3)
D satisfies d(1) < d(2) < d(3).
Z
Z is REAL array, dimension (3)
Each of the elements in z must be positive.
FINIT
FINIT is REAL
The value of f at 0. It is more accurate than the one
evaluated inside this routine (if someone wants to do
so).
TAU
TAU is REAL
The root of the equation f(x).
INFO
INFO is INTEGER
= 0: successful exit
> 0: if INFO = 1, failure to converge
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
Further Details:
10/02/03: This version has a few statements commented out for thread
safety (machine parameters are computed on each entry). SJH.
05/10/06: Modified from a new version of Ren-Cang Li, use
Gragg-Thornton-Warner cubic convergent scheme for better stability.
Contributors:
SLAED7 used by sstedc. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.
Purpose:
SLAED7 computes the updated eigensystem of a diagonal
matrix after modification by a rank-one symmetric matrix. This
routine is used only for the eigenproblem which requires all
eigenvalues and optionally eigenvectors of a dense symmetric matrix
that has been reduced to tridiagonal form. SLAED1 handles
the case in which all eigenvalues and eigenvectors of a symmetric
tridiagonal matrix are desired.
T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)
where Z = Q**Tu, u is a vector of length N with ones in the
CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
The eigenvectors of the original matrix are stored in Q, and the
eigenvalues are in D. The algorithm consists of three stages:
The first stage consists of deflating the size of the problem
when there are multiple eigenvalues or if there is a zero in
the Z vector. For each such occurrence the dimension of the
secular equation problem is reduced by one. This stage is
performed by the routine SLAED8.
The second stage consists of calculating the updated
eigenvalues. This is done by finding the roots of the secular
equation via the routine SLAED4 (as called by SLAED9).
This routine also calculates the eigenvectors of the current
problem.
The final stage consists of computing the updated eigenvectors
directly using the updated eigenvalues. The eigenvectors for
the current problem are multiplied with the eigenvectors from
the overall problem.
Parameters
ICOMPQ is INTEGER
= 0: Compute eigenvalues only.
= 1: Compute eigenvectors of original dense symmetric matrix
also. On entry, Q contains the orthogonal matrix used
to reduce the original matrix to tridiagonal form.
N
N is INTEGER
The dimension of the symmetric tridiagonal matrix. N >= 0.
QSIZ
QSIZ is INTEGER
The dimension of the orthogonal matrix used to reduce
the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.
TLVLS
TLVLS is INTEGER
The total number of merging levels in the overall divide and
conquer tree.
CURLVL
CURLVL is INTEGER
The current level in the overall merge routine,
0 <= CURLVL <= TLVLS.
CURPBM
CURPBM is INTEGER
The current problem in the current level in the overall
merge routine (counting from upper left to lower right).
D
D is REAL array, dimension (N)
On entry, the eigenvalues of the rank-1-perturbed matrix.
On exit, the eigenvalues of the repaired matrix.
Q
Q is REAL array, dimension (LDQ, N)
On entry, the eigenvectors of the rank-1-perturbed matrix.
On exit, the eigenvectors of the repaired tridiagonal matrix.
LDQ
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N).
INDXQ
INDXQ is INTEGER array, dimension (N)
The permutation which will reintegrate the subproblem just
solved back into sorted order, i.e., D( INDXQ( I = 1, N ) )
will be in ascending order.
RHO
RHO is REAL
The subdiagonal element used to create the rank-1
modification.
CUTPNT
CUTPNT is INTEGER
Contains the location of the last eigenvalue in the leading
sub-matrix. min(1,N) <= CUTPNT <= N.
QSTORE
QSTORE is REAL array, dimension (N**2+1)
Stores eigenvectors of submatrices encountered during
divide and conquer, packed together. QPTR points to
beginning of the submatrices.
QPTR
QPTR is INTEGER array, dimension (N+2)
List of indices pointing to beginning of submatrices stored
in QSTORE. The submatrices are numbered starting at the
bottom left of the divide and conquer tree, from left to
right and bottom to top.
PRMPTR
PRMPTR is INTEGER array, dimension (N lg N)
Contains a list of pointers which indicate where in PERM a
level's permutation is stored. PRMPTR(i+1) - PRMPTR(i)
indicates the size of the permutation and also the size of
the full, non-deflated problem.
PERM
PERM is INTEGER array, dimension (N lg N)
Contains the permutations (from deflation and sorting) to be
applied to each eigenblock.
GIVPTR
GIVPTR is INTEGER array, dimension (N lg N)
Contains a list of pointers which indicate where in GIVCOL a
level's Givens rotations are stored. GIVPTR(i+1) - GIVPTR(i)
indicates the number of Givens rotations.
GIVCOL
GIVCOL is INTEGER array, dimension (2, N lg N)
Each pair of numbers indicates a pair of columns to take place
in a Givens rotation.
GIVNUM
GIVNUM is REAL array, dimension (2, N lg N)
Each number indicates the S value to be used in the
corresponding Givens rotation.
WORK
WORK is REAL array, dimension (3*N+2*QSIZ*N)
IWORK
IWORK is INTEGER array, dimension (4*N)
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = 1, an eigenvalue did not converge
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
Contributors:
SLAED8 used by sstedc. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.
Purpose:
SLAED8 merges the two sets of eigenvalues together into a single
sorted set. Then it tries to deflate the size of the problem.
There are two ways in which deflation can occur: when two or more
eigenvalues are close together or if there is a tiny element in the
Z vector. For each such occurrence the order of the related secular
equation problem is reduced by one.
Parameters
ICOMPQ is INTEGER
= 0: Compute eigenvalues only.
= 1: Compute eigenvectors of original dense symmetric matrix
also. On entry, Q contains the orthogonal matrix used
to reduce the original matrix to tridiagonal form.
K
K is INTEGER
The number of non-deflated eigenvalues, and the order of the
related secular equation.
N
N is INTEGER
The dimension of the symmetric tridiagonal matrix. N >= 0.
QSIZ
QSIZ is INTEGER
The dimension of the orthogonal matrix used to reduce
the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.
D
D is REAL array, dimension (N)
On entry, the eigenvalues of the two submatrices to be
combined. On exit, the trailing (N-K) updated eigenvalues
(those which were deflated) sorted into increasing order.
Q
Q is REAL array, dimension (LDQ,N)
If ICOMPQ = 0, Q is not referenced. Otherwise,
on entry, Q contains the eigenvectors of the partially solved
system which has been previously updated in matrix
multiplies with other partially solved eigensystems.
On exit, Q contains the trailing (N-K) updated eigenvectors
(those which were deflated) in its last N-K columns.
LDQ
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N).
INDXQ
INDXQ is INTEGER array, dimension (N)
The permutation which separately sorts the two sub-problems
in D into ascending order. Note that elements in the second
half of this permutation must first have CUTPNT added to
their values in order to be accurate.
RHO
RHO is REAL
On entry, the off-diagonal element associated with the rank-1
cut which originally split the two submatrices which are now
being recombined.
On exit, RHO has been modified to the value required by
SLAED3.
CUTPNT
CUTPNT is INTEGER
The location of the last eigenvalue in the leading
sub-matrix. min(1,N) <= CUTPNT <= N.
Z
Z is REAL array, dimension (N)
On entry, Z contains the updating vector (the last row of
the first sub-eigenvector matrix and the first row of the
second sub-eigenvector matrix).
On exit, the contents of Z are destroyed by the updating
process.
DLAMDA
DLAMDA is REAL array, dimension (N)
A copy of the first K eigenvalues which will be used by
SLAED3 to form the secular equation.
Q2
Q2 is REAL array, dimension (LDQ2,N)
If ICOMPQ = 0, Q2 is not referenced. Otherwise,
a copy of the first K eigenvectors which will be used by
SLAED7 in a matrix multiply (SGEMM) to update the new
eigenvectors.
LDQ2
LDQ2 is INTEGER
The leading dimension of the array Q2. LDQ2 >= max(1,N).
W
W is REAL array, dimension (N)
The first k values of the final deflation-altered z-vector and
will be passed to SLAED3.
PERM
PERM is INTEGER array, dimension (N)
The permutations (from deflation and sorting) to be applied
to each eigenblock.
GIVPTR
GIVPTR is INTEGER
The number of Givens rotations which took place in this
subproblem.
GIVCOL
GIVCOL is INTEGER array, dimension (2, N)
Each pair of numbers indicates a pair of columns to take place
in a Givens rotation.
GIVNUM
GIVNUM is REAL array, dimension (2, N)
Each number indicates the S value to be used in the
corresponding Givens rotation.
INDXP
INDXP is INTEGER array, dimension (N)
The permutation used to place deflated values of D at the end
of the array. INDXP(1:K) points to the nondeflated D-values
and INDXP(K+1:N) points to the deflated eigenvalues.
INDX
INDX is INTEGER array, dimension (N)
The permutation used to sort the contents of D into ascending
order.
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
Contributors:
SLAED9 used by sstedc. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is dense.
Purpose:
SLAED9 finds the roots of the secular equation, as defined by the
values in D, Z, and RHO, between KSTART and KSTOP. It makes the
appropriate calls to SLAED4 and then stores the new matrix of
eigenvectors for use in calculating the next level of Z vectors.
Parameters
K is INTEGER
The number of terms in the rational function to be solved by
SLAED4. K >= 0.
KSTART
KSTART is INTEGER
KSTOP
KSTOP is INTEGER
The updated eigenvalues Lambda(I), KSTART <= I <= KSTOP
are to be computed. 1 <= KSTART <= KSTOP <= K.
N
N is INTEGER
The number of rows and columns in the Q matrix.
N >= K (delation may result in N > K).
D
D is REAL array, dimension (N)
D(I) contains the updated eigenvalues
for KSTART <= I <= KSTOP.
Q
Q is REAL array, dimension (LDQ,N)
LDQ
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max( 1, N ).
RHO
RHO is REAL
The value of the parameter in the rank one update equation.
RHO >= 0 required.
DLAMDA
DLAMDA is REAL array, dimension (K)
The first K elements of this array contain the old roots
of the deflated updating problem. These are the poles
of the secular equation.
W
W is REAL array, dimension (K)
The first K elements of this array contain the components
of the deflation-adjusted updating vector.
S
S is REAL array, dimension (LDS, K)
Will contain the eigenvectors of the repaired matrix which
will be stored for subsequent Z vector calculation and
multiplied by the previously accumulated eigenvectors
to update the system.
LDS
LDS is INTEGER
The leading dimension of S. LDS >= max( 1, K ).
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = 1, an eigenvalue did not converge
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
Contributors:
SLAEDA used by sstedc. Computes the Z vector determining the rank-one modification of the diagonal matrix. Used when the original matrix is dense.
Purpose:
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.
Parameters
N is INTEGER
The dimension of the symmetric tridiagonal matrix. N >= 0.
TLVLS
TLVLS is INTEGER
The total number of merging levels in the overall divide and
conquer tree.
CURLVL
CURLVL is INTEGER
The current level in the overall merge routine,
0 <= curlvl <= tlvls.
CURPBM
CURPBM is INTEGER
The current problem in the current level in the overall
merge routine (counting from upper left to lower right).
PRMPTR
PRMPTR is INTEGER array, dimension (N lg N)
Contains a list of pointers which indicate where in PERM a
level's permutation is stored. PRMPTR(i+1) - PRMPTR(i)
indicates the size of the permutation and incidentally the
size of the full, non-deflated problem.
PERM
PERM is INTEGER array, dimension (N lg N)
Contains the permutations (from deflation and sorting) to be
applied to each eigenblock.
GIVPTR
GIVPTR is INTEGER array, dimension (N lg N)
Contains a list of pointers which indicate where in GIVCOL a
level's Givens rotations are stored. GIVPTR(i+1) - GIVPTR(i)
indicates the number of Givens rotations.
GIVCOL
GIVCOL is INTEGER array, dimension (2, N lg N)
Each pair of numbers indicates a pair of columns to take place
in a Givens rotation.
GIVNUM
GIVNUM is REAL array, dimension (2, N lg N)
Each number indicates the S value to be used in the
corresponding Givens rotation.
Q
Q is REAL array, dimension (N**2)
Contains the square eigenblocks from previous levels, the
starting positions for blocks are given by QPTR.
QPTR
QPTR is INTEGER array, dimension (N+2)
Contains a list of pointers which indicate where in Q an
eigenblock is stored. SQRT( QPTR(i+1) - QPTR(i) ) indicates
the size of the block.
Z
Z is REAL array, dimension (N)
On output this vector contains the updating vector (the last
row of the first sub-eigenvector matrix and the first row of
the second sub-eigenvector matrix).
ZTEMP
ZTEMP is REAL array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
Contributors:
SLAGTF computes an LU factorization of a matrix T-λI, where T is a general tridiagonal matrix, and λ a scalar, using partial pivoting with row interchanges.
Purpose:
SLAGTF factorizes the matrix (T - lambda*I), where T is an n by n
tridiagonal matrix and lambda is a scalar, as
T - lambda*I = PLU,
where P is a permutation matrix, L is a unit lower tridiagonal matrix
with at most one non-zero sub-diagonal elements per column and U is
an upper triangular matrix with at most two non-zero super-diagonal
elements per column.
The factorization is obtained by Gaussian elimination with partial
pivoting and implicit row scaling.
The parameter LAMBDA is included in the routine so that SLAGTF may
be used, in conjunction with SLAGTS, to obtain eigenvectors of T by
inverse iteration.
Parameters
N is INTEGER
The order of the matrix T.
A
A is REAL array, dimension (N)
On entry, A must contain the diagonal elements of T.
On exit, A is overwritten by the n diagonal elements of the
upper triangular matrix U of the factorization of T.
LAMBDA
LAMBDA is REAL
On entry, the scalar lambda.
B
B is REAL array, dimension (N-1)
On entry, B must contain the (n-1) super-diagonal elements of
T.
On exit, B is overwritten by the (n-1) super-diagonal
elements of the matrix U of the factorization of T.
C
C is REAL array, dimension (N-1)
On entry, C must contain the (n-1) sub-diagonal elements of
T.
On exit, C is overwritten by the (n-1) sub-diagonal elements
of the matrix L of the factorization of T.
TOL
TOL is REAL
On entry, a relative tolerance used to indicate whether or
not the matrix (T - lambda*I) is nearly singular. TOL should
normally be chose as approximately the largest relative error
in the elements of T. For example, if the elements of T are
correct to about 4 significant figures, then TOL should be
set to about 5*10**(-4). If TOL is supplied as less than eps,
where eps is the relative machine precision, then the value
eps is used in place of TOL.
D
D is REAL array, dimension (N-2)
On exit, D is overwritten by the (n-2) second super-diagonal
elements of the matrix U of the factorization of T.
IN
IN is INTEGER array, dimension (N)
On exit, IN contains details of the permutation matrix P. If
an interchange occurred at the kth step of the elimination,
then IN(k) = 1, otherwise IN(k) = 0. The element IN(n)
returns the smallest positive integer j such that
abs( u(j,j) ) <= norm( (T - lambda*I)(j) )*TOL,
where norm( A(j) ) denotes the sum of the absolute values of
the jth row of the matrix A. If no such j exists then IN(n)
is returned as zero. If IN(n) is returned as positive, then a
diagonal element of U is small, indicating that
(T - lambda*I) is singular or nearly singular,
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -k, the kth argument had an illegal value
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
SLAMRG creates a permutation list to merge the entries of two independently sorted sets into a single set sorted in ascending order.
Purpose:
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.
Parameters
N1 is INTEGER
N2
N2 is INTEGER
These arguments contain the respective lengths of the two
sorted lists to be merged.
A
A is REAL array, dimension (N1+N2)
The first N1 elements of A contain a list of numbers which
are sorted in either ascending or descending order. Likewise
for the final N2 elements.
STRD1
STRD1 is INTEGER
STRD2
STRD2 is INTEGER
These are the strides to be taken through the array A.
Allowable strides are 1 and -1. They indicate whether a
subset of A is sorted in ascending (STRDx = 1) or descending
(STRDx = -1) order.
INDEX
INDEX is INTEGER array, dimension (N1+N2)
On exit this array will contain a permutation such that
if B( I ) = A( INDEX( I ) ) for I=1,N1+N2, then B will be
sorted in ascending order.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
SLARTGS generates a plane rotation designed to introduce a bulge in implicit QR iteration for the bidiagonal SVD problem.
Purpose:
SLARTGS generates a plane rotation designed to introduce a bulge in
Golub-Reinsch-style implicit QR iteration for the bidiagonal SVD
problem. X and Y are the top-row entries, and SIGMA is the shift.
The computed CS and SN define a plane rotation satisfying
[ CS SN ] . [ X^2 - SIGMA ] = [ R ],
[ -SN CS ] [ X * Y ] [ 0 ]
with R nonnegative. If X^2 - SIGMA and X * Y are 0, then the
rotation is by PI/2.
Parameters
X is REAL
The (1,1) entry of an upper bidiagonal matrix.
Y
Y is REAL
The (1,2) entry of an upper bidiagonal matrix.
SIGMA
SIGMA is REAL
The shift.
CS
CS is REAL
The cosine of the rotation.
SN
SN is REAL
The sine of the rotation.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
SLASQ1 computes the singular values of a real square bidiagonal matrix. Used by sbdsqr.
Purpose:
SLASQ1 computes the singular values of a real N-by-N bidiagonal
matrix with diagonal D and off-diagonal E. The singular values
are computed to high relative accuracy, in the absence of
denormalization, underflow and overflow. The algorithm was first
presented in
"Accurate singular values and differential qd algorithms" by K. V.
Fernando and B. N. Parlett, Numer. Math., Vol-67, No. 2, pp. 191-230,
1994,
and the present implementation is described in "An implementation of
the dqds Algorithm (Positive Case)", LAPACK Working Note.
Parameters
N is INTEGER
The number of rows and columns in the matrix. N >= 0.
D
D is REAL array, dimension (N)
On entry, D contains the diagonal elements of the
bidiagonal matrix whose SVD is desired. On normal exit,
D contains the singular values in decreasing order.
E
E is REAL array, dimension (N)
On entry, elements E(1:N-1) contain the off-diagonal elements
of the bidiagonal matrix whose SVD is desired.
On exit, E is overwritten.
WORK
WORK is REAL array, dimension (4*N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: the algorithm failed
= 1, a split was marked by a positive value in E
= 2, current block of Z not diagonalized after 100*N
iterations (in inner while loop) On exit D and E
represent a matrix with the same singular values
which the calling subroutine could use to finish the
computation, or even feed back into SLASQ1
= 3, termination criterion of outer while loop not met
(program created more than N unreduced blocks)
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
SLASQ2 computes all the eigenvalues of the symmetric positive definite tridiagonal matrix associated with the qd Array Z to high relative accuracy. Used by sbdsqr and sstegr.
Purpose:
SLASQ2 computes all the eigenvalues of the symmetric positive
definite tridiagonal matrix associated with the qd array Z to high
relative accuracy are computed to high relative accuracy, in the
absence of denormalization, underflow and overflow.
To see the relation of Z to the tridiagonal matrix, let L be a
unit lower bidiagonal matrix with subdiagonals Z(2,4,6,,..) and
let U be an upper bidiagonal matrix with 1's above and diagonal
Z(1,3,5,,..). The tridiagonal is L*U or, if you prefer, the
symmetric tridiagonal to which it is similar.
Note : SLASQ2 defines a logical variable, IEEE, which is true
on machines which follow ieee-754 floating-point standard in their
handling of infinities and NaNs, and false otherwise. This variable
is passed to SLASQ3.
Parameters
N is INTEGER
The number of rows and columns in the matrix. N >= 0.
Z
Z is REAL array, dimension ( 4*N )
On entry Z holds the qd array. On exit, entries 1 to N hold
the eigenvalues in decreasing order, Z( 2*N+1 ) holds the
trace, and Z( 2*N+2 ) holds the sum of the eigenvalues. If
N > 2, then Z( 2*N+3 ) holds the iteration count, Z( 2*N+4 )
holds NDIVS/NIN^2, and Z( 2*N+5 ) holds the percentage of
shifts that failed.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if the i-th argument is a scalar and had an illegal
value, then INFO = -i, if the i-th argument is an
array and the j-entry had an illegal value, then
INFO = -(i*100+j)
> 0: the algorithm failed
= 1, a split was marked by a positive value in E
= 2, current block of Z not diagonalized after 100*N
iterations (in inner while loop). On exit Z holds
a qd array with the same eigenvalues as the given Z.
= 3, termination criterion of outer while loop not met
(program created more than N unreduced blocks)
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
Further Details:
Local Variables: I0:N0 defines a current unreduced segment of Z.
The shifts are accumulated in SIGMA. Iteration count is in ITER.
Ping-pong is controlled by PP (alternates between 0 and 1).
SLASQ3 checks for deflation, computes a shift and calls dqds. Used by sbdsqr.
Purpose:
SLASQ3 checks for deflation, computes a shift (TAU) and calls dqds.
In case of failure it changes shifts, and tries again until output
is positive.
Parameters
I0 is INTEGER
First index.
N0
N0 is INTEGER
Last index.
Z
Z is REAL array, dimension ( 4*N0 )
Z holds the qd array.
PP
PP is INTEGER
PP=0 for ping, PP=1 for pong.
PP=2 indicates that flipping was applied to the Z array
and that the initial tests for deflation should not be
performed.
DMIN
DMIN is REAL
Minimum value of d.
SIGMA
SIGMA is REAL
Sum of shifts used in current segment.
DESIG
DESIG is REAL
Lower order part of SIGMA
QMAX
QMAX is REAL
Maximum value of q.
NFAIL
NFAIL is INTEGER
Increment NFAIL by 1 each time the shift was too big.
ITER
ITER is INTEGER
Increment ITER by 1 for each iteration.
NDIV
NDIV is INTEGER
Increment NDIV by 1 for each division.
IEEE
IEEE is LOGICAL
Flag for IEEE or non IEEE arithmetic (passed to SLASQ5).
TTYPE
TTYPE is INTEGER
Shift type.
DMIN1
DMIN1 is REAL
DMIN2
DMIN2 is REAL
DN
DN is REAL
DN1
DN1 is REAL
DN2
DN2 is REAL
G
G is REAL
TAU
TAU is REAL
These are passed as arguments in order to save their values
between calls to SLASQ3.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
SLASQ4 computes an approximation to the smallest eigenvalue using values of d from the previous transform. Used by sbdsqr.
Purpose:
SLASQ4 computes an approximation TAU to the smallest eigenvalue
using values of d from the previous transform.
Parameters
I0 is INTEGER
First index.
N0
N0 is INTEGER
Last index.
Z
Z is REAL array, dimension ( 4*N0 )
Z holds the qd array.
PP
PP is INTEGER
PP=0 for ping, PP=1 for pong.
N0IN
N0IN is INTEGER
The value of N0 at start of EIGTEST.
DMIN
DMIN is REAL
Minimum value of d.
DMIN1
DMIN1 is REAL
Minimum value of d, excluding D( N0 ).
DMIN2
DMIN2 is REAL
Minimum value of d, excluding D( N0 ) and D( N0-1 ).
DN
DN is REAL
d(N)
DN1
DN1 is REAL
d(N-1)
DN2
DN2 is REAL
d(N-2)
TAU
TAU is REAL
This is the shift.
TTYPE
TTYPE is INTEGER
Shift type.
G
G is REAL
G is passed as an argument in order to save its value between
calls to SLASQ4.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
Further Details:
CNST1 = 9/16
SLASQ5 computes one dqds transform in ping-pong form. Used by sbdsqr and sstegr.
Purpose:
SLASQ5 computes one dqds transform in ping-pong form, one
version for IEEE machines another for non IEEE machines.
Parameters
I0 is INTEGER
First index.
N0
N0 is INTEGER
Last index.
Z
Z is REAL array, dimension ( 4*N )
Z holds the qd array. EMIN is stored in Z(4*N0) to avoid
an extra argument.
PP
PP is INTEGER
PP=0 for ping, PP=1 for pong.
TAU
TAU is REAL
This is the shift.
SIGMA
SIGMA is REAL
This is the accumulated shift up to this step.
DMIN
DMIN is REAL
Minimum value of d.
DMIN1
DMIN1 is REAL
Minimum value of d, excluding D( N0 ).
DMIN2
DMIN2 is REAL
Minimum value of d, excluding D( N0 ) and D( N0-1 ).
DN
DN is REAL
d(N0), the last value of d.
DNM1
DNM1 is REAL
d(N0-1).
DNM2
DNM2 is REAL
d(N0-2).
IEEE
IEEE is LOGICAL
Flag for IEEE or non IEEE arithmetic.
EPS
EPS is REAL
This is the value of epsilon used.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
SLASQ6 computes one dqd transform in ping-pong form. Used by sbdsqr and sstegr.
Purpose:
SLASQ6 computes one dqd (shift equal to zero) transform in
ping-pong form, with protection against underflow and overflow.
Parameters
I0 is INTEGER
First index.
N0
N0 is INTEGER
Last index.
Z
Z is REAL array, dimension ( 4*N )
Z holds the qd array. EMIN is stored in Z(4*N0) to avoid
an extra argument.
PP
PP is INTEGER
PP=0 for ping, PP=1 for pong.
DMIN
DMIN is REAL
Minimum value of d.
DMIN1
DMIN1 is REAL
Minimum value of d, excluding D( N0 ).
DMIN2
DMIN2 is REAL
Minimum value of d, excluding D( N0 ) and D( N0-1 ).
DN
DN is REAL
d(N0), the last value of d.
DNM1
DNM1 is REAL
d(N0-1).
DNM2
DNM2 is REAL
d(N0-2).
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
SLASRT sorts numbers in increasing or decreasing order.
Purpose:
Sort the numbers in D in increasing order (if ID = 'I') or
in decreasing order (if ID = 'D' ).
Use Quick Sort, reverting to Insertion sort on arrays of
size <= 20. Dimension of STACK limits N to about 2**32.
Parameters
ID is CHARACTER*1
= 'I': sort D in increasing order;
= 'D': sort D in decreasing order.
N
N is INTEGER
The length of the array D.
D
D is REAL array, dimension (N)
On entry, the array to be sorted.
On exit, D has been sorted into increasing order
(D(1) <= ... <= D(N) ) or into decreasing order
(D(1) >= ... >= D(N) ), depending on ID.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
SPTTRF
Purpose:
SPTTRF computes the L*D*L**T factorization of a real symmetric
positive definite tridiagonal matrix A. The factorization may also
be regarded as having the form A = U**T*D*U.
Parameters
N is INTEGER
The order of the matrix A. N >= 0.
D
D is REAL array, dimension (N)
On entry, the n diagonal elements of the tridiagonal matrix
A. On exit, the n diagonal elements of the diagonal matrix
D from the L*D*L**T factorization of A.
E
E is REAL array, dimension (N-1)
On entry, the (n-1) subdiagonal elements of the tridiagonal
matrix A. On exit, the (n-1) subdiagonal elements of the
unit bidiagonal factor L from the L*D*L**T factorization of A.
E can also be regarded as the superdiagonal of the unit
bidiagonal factor U from the U**T*D*U factorization of A.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -k, the k-th argument had an illegal value
> 0: if INFO = k, the leading minor of order k is not
positive definite; if k < N, the factorization could not
be completed, while if k = N, the factorization was
completed, but D(N) <= 0.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
SSTEBZ
Purpose:
SSTEBZ computes the eigenvalues of a symmetric tridiagonal
matrix T. 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.
Parameters
RANGE is CHARACTER*1
= 'A': ("All") all eigenvalues will be found.
= 'V': ("Value") all eigenvalues in the half-open interval
(VL, VU] will be found.
= 'I': ("Index") the IL-th through IU-th eigenvalues (of the
entire matrix) will be found.
ORDER
ORDER is CHARACTER*1
= 'B': ("By Block") the eigenvalues will be grouped by
split-off block (see IBLOCK, ISPLIT) and
ordered from smallest to largest within
the block.
= 'E': ("Entire matrix")
the eigenvalues for the entire matrix
will be ordered from smallest to
largest.
N
N is INTEGER
The order of the tridiagonal matrix T. N >= 0.
VL
VL is REAL
If RANGE='V', the lower bound of the interval to
be searched for eigenvalues. Eigenvalues less than or equal
to VL, or greater than VU, will not be returned. VL < VU.
Not referenced if RANGE = 'A' or 'I'.
VU
VU is REAL
If RANGE='V', the upper bound of the interval to
be searched for eigenvalues. Eigenvalues less than or equal
to VL, or greater than VU, will not be returned. VL < VU.
Not referenced if RANGE = 'A' or 'I'.
IL
IL is INTEGER
If RANGE='I', the index of the
smallest eigenvalue to be returned.
1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
Not referenced if RANGE = 'A' or 'V'.
IU
IU is INTEGER
If RANGE='I', the index of the
largest eigenvalue to be returned.
1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
Not referenced if RANGE = 'A' or 'V'.
ABSTOL
ABSTOL is REAL
The absolute tolerance for the eigenvalues. An eigenvalue
(or cluster) is considered to be located if it has been
determined to lie in an interval whose width is ABSTOL or
less. If ABSTOL is less than or equal to zero, then ULP*|T|
will be used, where |T| means the 1-norm of T.
Eigenvalues will be computed most accurately when ABSTOL is
set to twice the underflow threshold 2*SLAMCH('S'), not zero.
D
D is REAL array, dimension (N)
The n diagonal elements of the tridiagonal matrix T.
E
E is REAL array, dimension (N-1)
The (n-1) off-diagonal elements of the tridiagonal matrix T.
M
M is INTEGER
The actual number of eigenvalues found. 0 <= M <= N.
(See also the description of INFO=2,3.)
NSPLIT
NSPLIT is INTEGER
The number of diagonal blocks in the matrix T.
1 <= NSPLIT <= N.
W
W is REAL array, dimension (N)
On exit, the first M elements of W will contain the
eigenvalues. (SSTEBZ may use the remaining N-M elements as
workspace.)
IBLOCK
IBLOCK is INTEGER array, dimension (N)
At each row/column j where E(j) is zero or small, the
matrix T is considered to split into a block diagonal
matrix. On exit, if INFO = 0, IBLOCK(i) specifies to which
block (from 1 to the number of blocks) the eigenvalue W(i)
belongs. (SSTEBZ may use the remaining N-M elements as
workspace.)
ISPLIT
ISPLIT is INTEGER array, dimension (N)
The splitting points, at which T breaks up into submatrices.
The first submatrix consists of rows/columns 1 to ISPLIT(1),
the second of rows/columns ISPLIT(1)+1 through ISPLIT(2),
etc., and the NSPLIT-th consists of rows/columns
ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.
(Only the first NSPLIT elements will actually be used, but
since the user cannot know a priori what value NSPLIT will
have, N words must be reserved for ISPLIT.)
WORK
WORK is REAL array, dimension (4*N)
IWORK
IWORK is INTEGER array, dimension (3*N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: some or all of the eigenvalues failed to converge or
were not computed:
=1 or 3: Bisection failed to converge for some
eigenvalues; these eigenvalues are flagged by a
negative block number. The effect is that the
eigenvalues may not be as accurate as the
absolute and relative tolerances. This is
generally caused by unexpectedly inaccurate
arithmetic.
=2 or 3: RANGE='I' only: Not all of the eigenvalues
IL:IU were found.
Effect: M < IU+1-IL
Cause: non-monotonic arithmetic, causing the
Sturm sequence to be non-monotonic.
Cure: recalculate, using RANGE='A', and pick
out eigenvalues IL:IU. In some cases,
increasing the PARAMETER "FUDGE" may
make things work.
= 4: RANGE='I', and the Gershgorin interval
initially used was too small. No eigenvalues
were computed.
Probable cause: your machine has sloppy
floating-point arithmetic.
Cure: Increase the PARAMETER "FUDGE",
recompile, and try again.
Internal Parameters:
RELFAC REAL, default = 2.0e0
The relative tolerance. An interval (a,b] lies within
"relative tolerance" if b-a < RELFAC*ulp*max(|a|,|b|),
where "ulp" is the machine precision (distance from 1 to
the next larger floating point number.)
FUDGE REAL, default = 2
A "fudge factor" to widen the Gershgorin intervals. Ideally,
a value of 1 should work, but on machines with sloppy
arithmetic, this needs to be larger. The default for
publicly released versions should be large enough to handle
the worst machine around. Note that this has no effect
on accuracy of the solution.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
SSTEDC
Purpose:
SSTEDC computes all eigenvalues and, optionally, eigenvectors of a
symmetric tridiagonal matrix using the divide and conquer method.
The eigenvectors of a full or band real symmetric matrix can also be
found if SSYTRD or SSPTRD or SSBTRD has been used to reduce this
matrix to tridiagonal form.
This code makes very mild assumptions about floating point
arithmetic. It will work on machines with a guard digit in
add/subtract, or on those binary machines without guard digits
which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
It could conceivably fail on hexadecimal or decimal machines
without guard digits, but we know of none. See SLAED3 for details.
Parameters
COMPZ is CHARACTER*1
= 'N': Compute eigenvalues only.
= 'I': Compute eigenvectors of tridiagonal matrix also.
= 'V': Compute eigenvectors of original dense symmetric
matrix also. On entry, Z contains the orthogonal
matrix used to reduce the original matrix to
tridiagonal form.
N
N is INTEGER
The dimension of the symmetric tridiagonal matrix. N >= 0.
D
D is REAL array, dimension (N)
On entry, the diagonal elements of the tridiagonal matrix.
On exit, if INFO = 0, the eigenvalues in ascending order.
E
E is REAL array, dimension (N-1)
On entry, the subdiagonal elements of the tridiagonal matrix.
On exit, E has been destroyed.
Z
Z is REAL array, dimension (LDZ,N)
On entry, if COMPZ = 'V', then Z contains the orthogonal
matrix used in the reduction to tridiagonal form.
On exit, if INFO = 0, then if COMPZ = 'V', Z contains the
orthonormal eigenvectors of the original symmetric matrix,
and if COMPZ = 'I', Z contains the orthonormal eigenvectors
of the symmetric tridiagonal matrix.
If COMPZ = 'N', then Z is not referenced.
LDZ
LDZ is INTEGER
The leading dimension of the array Z. LDZ >= 1.
If eigenvectors are desired, then LDZ >= max(1,N).
WORK
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK.
If COMPZ = 'N' or N <= 1 then LWORK must be at least 1.
If COMPZ = 'V' and N > 1 then LWORK must be at least
( 1 + 3*N + 2*N*lg N + 4*N**2 ),
where lg( N ) = smallest integer k such
that 2**k >= N.
If COMPZ = 'I' and N > 1 then LWORK must be at least
( 1 + 4*N + N**2 ).
Note that for COMPZ = 'I' or 'V', then if N is less than or
equal to the minimum divide size, usually 25, then LWORK need
only be max(1,2*(N-1)).
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
IWORK
IWORK is INTEGER array, dimension (MAX(1,LIWORK))
On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
LIWORK
LIWORK is INTEGER
The dimension of the array IWORK.
If COMPZ = 'N' or N <= 1 then LIWORK must be at least 1.
If COMPZ = 'V' and N > 1 then LIWORK must be at least
( 6 + 6*N + 5*N*lg N ).
If COMPZ = 'I' and N > 1 then LIWORK must be at least
( 3 + 5*N ).
Note that for COMPZ = 'I' or 'V', then if N is less than or
equal to the minimum divide size, usually 25, then LIWORK
need only be 1.
If LIWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the IWORK array,
returns this value as the first entry of the IWORK array, and
no error message related to LIWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: The algorithm failed to compute an eigenvalue while
working on the submatrix lying in rows and columns
INFO/(N+1) through mod(INFO,N+1).
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
Contributors:
SSTEQR
Purpose:
SSTEQR computes all eigenvalues and, optionally, eigenvectors of a
symmetric tridiagonal matrix using the implicit QL or QR method.
The eigenvectors of a full or band symmetric matrix can also be found
if SSYTRD or SSPTRD or SSBTRD has been used to reduce this matrix to
tridiagonal form.
Parameters
COMPZ is CHARACTER*1
= 'N': Compute eigenvalues only.
= 'V': Compute eigenvalues and eigenvectors of the original
symmetric matrix. On entry, Z must contain the
orthogonal matrix used to reduce the original matrix
to tridiagonal form.
= 'I': Compute eigenvalues and eigenvectors of the
tridiagonal matrix. Z is initialized to the identity
matrix.
N
N is INTEGER
The order of the matrix. N >= 0.
D
D is REAL array, dimension (N)
On entry, the diagonal elements of the tridiagonal matrix.
On exit, if INFO = 0, the eigenvalues in ascending order.
E
E is REAL array, dimension (N-1)
On entry, the (n-1) subdiagonal elements of the tridiagonal
matrix.
On exit, E has been destroyed.
Z
Z is REAL array, dimension (LDZ, N)
On entry, if COMPZ = 'V', then Z contains the orthogonal
matrix used in the reduction to tridiagonal form.
On exit, if INFO = 0, then if COMPZ = 'V', Z contains the
orthonormal eigenvectors of the original symmetric matrix,
and if COMPZ = 'I', Z contains the orthonormal eigenvectors
of the symmetric tridiagonal matrix.
If COMPZ = 'N', then Z is not referenced.
LDZ
LDZ is INTEGER
The leading dimension of the array Z. LDZ >= 1, and if
eigenvectors are desired, then LDZ >= max(1,N).
WORK
WORK is REAL array, dimension (max(1,2*N-2))
If COMPZ = 'N', then WORK is not referenced.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: the algorithm has failed to find all the eigenvalues in
a total of 30*N iterations; if INFO = i, then i
elements of E have not converged to zero; on exit, D
and E contain the elements of a symmetric tridiagonal
matrix which is orthogonally similar to the original
matrix.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
SSTERF
Purpose:
SSTERF computes all eigenvalues of a symmetric tridiagonal matrix
using the Pal-Walker-Kahan variant of the QL or QR algorithm.
Parameters
N is INTEGER
The order of the matrix. N >= 0.
D
D is REAL array, dimension (N)
On entry, the n diagonal elements of the tridiagonal matrix.
On exit, if INFO = 0, the eigenvalues in ascending order.
E
E is REAL array, dimension (N-1)
On entry, the (n-1) subdiagonal elements of the tridiagonal
matrix.
On exit, E has been destroyed.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: the algorithm failed to find all of the eigenvalues in
a total of 30*N iterations; if INFO = i, then i
elements of E have not converged to zero.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
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