Purpose
To reorder the diagonal blocks of the formal matrix product
T22_K^S(K) * T22_K-1^S(K-1) * ... * T22_1^S(1) (1)
of length K in the generalized periodic Schur form
[ T11_k T12_k T13_k ]
T_k = [ 0 T22_k T23_k ], k = 1, ..., K, (2)
[ 0 0 T33_k ]
where
- the submatrices T11_k are NI(k+1)-by-NI(k), if S(k) = 1, or
NI(k)-by-NI(k+1), if S(k) = -1, and contain dimension-induced
infinite eigenvalues,
- the submatrices T22_k are NC-by-NC and contain core eigenvalues,
which are generically neither zero nor infinite,
- the submatrices T33_k contain dimension-induced zero
eigenvalues,
such that pairs of adjacent diagonal blocks of sizes 1 and/or 2 in
the product (1) are swapped.
Optionally, the transformation matrices Q_1,...,Q_K from the
reduction into generalized periodic Schur form are updated with
respect to the performed reordering.
Specification
SUBROUTINE MB03KB( COMPQ, WHICHQ, WS, K, NC, KSCHUR, J1, N1, N2,
$ N, NI, S, T, LDT, IXT, Q, LDQ, IXQ, TOL, IWORK,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ
LOGICAL WS
INTEGER INFO, J1, K, KSCHUR, LDWORK, N1, N2, NC
C .. Array Arguments ..
INTEGER IWORK( * ), IXQ( * ), IXT( * ), LDQ( * ),
$ LDT( * ), N( * ), NI( * ), S( * ), WHICHQ( * )
DOUBLE PRECISION DWORK( * ), Q( * ), T( * ), TOL( * )
Arguments
Mode Parameters
COMPQ CHARACTER*1
= 'N': do not compute any of the matrices Q_k;
= 'U': each coefficient of Q must contain an orthogonal
matrix Q1_k on entry, and the products Q1_k*Q_k are
returned, where Q_k, k = 1, ..., K, performed the
reordering;
= 'W': the computation of each Q_k is specified
individually in the array WHICHQ.
WHICHQ INTEGER array, dimension (K)
If COMPQ = 'W', WHICHQ(k) specifies the computation of Q_k
as follows:
= 0: do not compute Q_k;
> 0: the kth coefficient of Q must contain an orthogonal
matrix Q1_k on entry, and the product Q1_k*Q_k is
returned.
This array is not referenced if COMPQ <> 'W'.
WS LOGICAL
= .FALSE. : do not perform the strong stability tests;
= .TRUE. : perform the strong stability tests; often,
this is not needed, and omitting them can save
some computations.
Input/Output Parameters
K (input) INTEGER
The period of the periodic matrix sequences T and Q (the
number of factors in the matrix product). K >= 2.
(For K = 1, a standard eigenvalue reordering problem is
obtained.)
NC (input) INTEGER
The number of core eigenvalues. 0 <= NC <= min(N).
KSCHUR (input) INTEGER
The index for which the matrix T22_kschur is upper quasi-
triangular.
J1 (input) INTEGER
The index of the first row and column of the first block
to swap in T22_k.
1 <= J1 <= NC-N1-N2+1.
N1 (input) INTEGER
The order of the first block to swap. N1 = 0, 1 or 2.
N2 (input) INTEGER
The order of the second block to swap. N2 = 0, 1 or 2.
N (input) INTEGER array, dimension (K)
The leading K elements of this array must contain the
dimensions of the factors of the formal matrix product T,
such that the k-th coefficient T_k is an N(k+1)-by-N(k)
matrix, if S(k) = 1, or an N(k)-by-N(k+1) matrix,
if S(k) = -1, k = 1, ..., K, where N(K+1) = N(1).
NI (input) INTEGER array, dimension (K)
The leading K elements of this array must contain the
dimensions of the factors of the matrix sequence T11_k.
N(k) >= NI(k) + NC >= 0.
S (input) INTEGER array, dimension (K)
The leading K elements of this array must contain the
signatures (exponents) of the factors in the K-periodic
matrix sequence. Each entry in S must be either 1 or -1;
the value S(k) = -1 corresponds to using the inverse of
the factor T_k.
T (input/output) DOUBLE PRECISION array, dimension (*)
On entry, this array must contain at position IXT(k) the
matrix T_k, which is at least N(k+1)-by-N(k), if S(k) = 1,
or at least N(k)-by-N(k+1), if S(k) = -1, in periodic
Schur form.
On exit, the matrices T_k are overwritten by the reordered
periodic Schur form.
LDT INTEGER array, dimension (K)
The leading dimensions of the matrices T_k in the one-
dimensional array T.
LDT(k) >= max(1,N(k+1)), if S(k) = 1,
LDT(k) >= max(1,N(k)), if S(k) = -1.
IXT INTEGER array, dimension (K)
Start indices of the matrices T_k in the one-dimensional
array T.
Q (input/output) DOUBLE PRECISION array, dimension (*)
On entry, this array must contain at position IXQ(k) a
matrix Q_k of size at least N(k)-by-N(k), provided that
COMPQ = 'U', or COMPQ = 'W' and WHICHQ(k) > 0.
On exit, if COMPQ = 'U', or COMPQ = 'W' and WHICHQ(k) > 0,
Q_k is post-multiplied with the orthogonal matrix that
performed the reordering.
This array is not referenced if COMPQ = 'N'.
LDQ INTEGER array, dimension (K)
The leading dimensions of the matrices Q_k in the one-
dimensional array Q. LDQ(k) >= 1, and
LDQ(k) >= max(1,N(k)), if COMPQ = 'U', or COMPQ = 'W' and
WHICHQ(k) > 0;
This array is not referenced if COMPQ = 'N'.
IXQ INTEGER array, dimension (K)
Start indices of the matrices Q_k in the one-dimensional
array Q.
This array is not referenced if COMPQ = 'N'.
Tolerances
TOL DOUBLE PRECISION array, dimension (3)
This array contains tolerance parameters. The weak and
strong stability tests use a threshold computed by the
formula MAX( c*EPS*NRM, SMLNUM ), where c is a constant,
NRM is the Frobenius norm of the matrix formed by
concatenating K pairs of adjacent diagonal blocks of sizes
1 and/or 2 in the T22_k submatrices from (2), which are
swapped, and EPS and SMLNUM are the machine precision and
safe minimum divided by EPS, respectively (see LAPACK
Library routine DLAMCH). The norm NRM is computed by this
routine; the other values are stored in the array TOL.
TOL(1), TOL(2), and TOL(3) contain c, EPS, and SMLNUM,
respectively. TOL(1) should normally be at least 10.
Workspace
IWORK INTEGER array, dimension (4*K)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= 10*K + MN, if N1 = 1, N2 = 1;
LDWORK >= 25*K + MN, if N1 = 1, N2 = 2;
LDWORK >= MAX(23*K + MN, 25*K - 12), if N1 = 2, N2 = 1;
LDWORK >= MAX(42*K + MN, 80*K - 48), if N1 = 2, N2 = 2;
where MN = MXN, if MXN > 10, and MN = 0, otherwise, with
MXN = MAX(N(k),k=1,...,K).
If LDWORK = -1 a workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -22, then LDWORK is too small; appropriate
value for LDWORK is returned in DWORK(1); the other
arguments are not tested, for efficiency;
= 1: the swap was rejected from stability reasons; the
blocks are not swapped and T and Q are unchanged.
Method
The algorithm described in [1] is used. Both weak and strong stability tests are performed.References
[1] Granat, R., Kagstrom, B. and Kressner, D.
Computing periodic deflating subspaces associated with a
specified set of eigenvalues.
BIT Numerical Mathematics, vol. 47, 763-791, 2007.
Numerical Aspects
The implemented method is numerically backward stable.
3
The algorithm requires 0(K NC ) floating point operations.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
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