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 the M selected eigenvalues pointed to by the logical
vector SELECT end up in the leading part of the matrix sequence
T22_k.
Given that N(k) = N(k+1) for all k where S(k) = -1, the T11_k are
void and the first M columns of the updated orthogonal
transformation matrix sequence Q_1, ..., Q_K span a periodic
deflating subspace corresponding to the same eigenvalues.
Specification
SUBROUTINE MB03KD( COMPQ, WHICHQ, STRONG, K, NC, KSCHUR, N, NI, S,
$ SELECT, T, LDT, IXT, Q, LDQ, IXQ, M, TOL,
$ IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ, STRONG
INTEGER INFO, K, KSCHUR, LDWORK, M, NC
DOUBLE PRECISION TOL
C .. Array Arguments ..
LOGICAL SELECT( * )
INTEGER IWORK( * ), IXQ( * ), IXT( * ), LDQ( * ),
$ LDT( * ), N( * ), NI( * ), S( * ), WHICHQ( * )
DOUBLE PRECISION DWORK( * ), Q( * ), T( * )
Arguments
Mode Parameters
COMPQ CHARACTER*1
Specifies whether to compute the orthogonal transformation
matrices Q_k, as follows:
= 'N': do not compute any of the matrices Q_k;
= 'I': each coefficient of Q is initialized internally to
the identity matrix, and the orthogonal matrices
Q_k are returned, where Q_k, k = 1, ..., K,
performed the reordering;
= 'U': each coefficient of Q must contain an orthogonal
matrix Q1_k on entry, and the products Q1_k*Q_k are
returned;
= '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;
= 1: the kth coefficient of Q is initialized to the
identity matrix, and the orthogonal matrix Q_k is
returned;
= 2: 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'.
STRONG CHARACTER*1
Specifies whether to perform the strong stability tests,
as follows:
= 'N': do not perform the strong stability tests;
= 'S': 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. All other T22 matrices are upper triangular.
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.
SELECT (input) LOGICAL array, dimension (NC)
SELECT specifies the eigenvalues in the selected cluster.
To select a real eigenvalue w(j), SELECT(j) must be set to
.TRUE.. To select a complex conjugate pair of eigenvalues
w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
either SELECT(j) or SELECT(j+1) or both must be set to
.TRUE.; a complex conjugate pair of eigenvalues must be
either both included in the cluster or both excluded.
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) = 2.
On exit, if COMPQ = 'I' or COMPQ = 'W' and WHICHQ(k) = 1,
Q_k contains the orthogonal matrix that performed the
reordering. If COMPQ = 'U', or COMPQ = 'W' and
WHICHQ(k) = 2, 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) >= max(1,N(k)), if COMPQ = 'I', or 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'.
M (output) INTEGER
The number of selected core eigenvalues which were
reordered to the top of T22_k.
Tolerances
TOL DOUBLE PRECISION
The tolerance parameter c. The weak and strong stability
tests performed for checking the reordering use a
threshold computed by the formula MAX(c*EPS*NRM, SMLNUM),
where NRM is the varying Frobenius norm of the matrices
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 value c 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 all blocks involved in reordering
have order 1;
LDWORK >= 25*K + MN, if there is at least a block of
order 2, but no adjacent blocks of
order 2 are involved in reordering;
LDWORK >= MAX(42*K + MN, 80*K - 48), if there is at least
a pair of adjacent blocks of order 2
involved in reordering;
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 = -i, the i-th argument had an illegal
value;
= 1: the reordering of T failed because some eigenvalues
are too close to separate (the problem is very ill-
conditioned); T may have been partially reordered.
Method
An adaptation of the LAPACK Library routine DTGSEN is used.Numerical Aspects
The implemented method is numerically backward stable.Further Comments
NoneExample
Program Text
* MB03KD EXAMPLE PROGRAM TEXT
* Copyright (c) 2002-2017 NICONET e.V.
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER KMAX, NMAX
PARAMETER ( KMAX = 6, NMAX = 50 )
INTEGER LDA1, LDA2, LDQ1, LDQ2, LDWORK, LIWORK
PARAMETER ( LDA1 = NMAX, LDA2 = NMAX, LDQ1 = NMAX,
$ LDQ2 = NMAX,
$ LDWORK = MAX( 42*KMAX + NMAX, 80*KMAX - 48,
$ KMAX + MAX( 2*NMAX, 8*KMAX ) ),
$ LIWORK = MAX( 4*KMAX, 2*KMAX + NMAX ) )
DOUBLE PRECISION HUND, ZERO
PARAMETER ( HUND = 1.0D2, ZERO = 0.0D0 )
*
* .. Local Scalars ..
CHARACTER COMPQ, DEFL, JOB, STRONG
INTEGER H, I, IHI, ILO, INFO, IWARN, J, K, L, M, N, P
DOUBLE PRECISION TOL
*
* .. Local Arrays ..
LOGICAL SELECT( NMAX )
INTEGER IWORK( LIWORK ), IXQ( KMAX ), IXT( KMAX ),
$ LDQ( KMAX ), LDT( KMAX ), ND( KMAX ),
$ NI( KMAX ), QIND( KMAX ), S( KMAX ),
$ SCAL( NMAX )
DOUBLE PRECISION A( LDA1, LDA2, KMAX ), ALPHAI( NMAX ),
$ ALPHAR( NMAX ), BETA( NMAX ), DWORK( LDWORK),
$ Q( LDQ1, LDQ2, KMAX ), QK( NMAX*NMAX*KMAX ),
$ T( NMAX*NMAX*KMAX )
*
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
*
* .. External Subroutines ..
EXTERNAL DLACPY, MB03BD, MB03KD
*
* .. Intrinsic Functions ..
INTRINSIC INT, MAX
*
* .. Executable Statements ..
*
WRITE( NOUT, FMT = 99999 )
* Skip the heading in the data file and read in the data.
READ( NIN, FMT = * )
READ( NIN, FMT = * ) JOB, DEFL, COMPQ, STRONG, K, N, H, ILO, IHI
IF( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE( NOUT, FMT = 99998 ) N
ELSE
TOL = HUND
READ( NIN, FMT = * ) ( S( I ), I = 1, K )
READ( NIN, FMT = * ) ( ( ( A( I, J, L ), J = 1, N ),
$ I = 1, N ), L = 1, K )
IF( LSAME( COMPQ, 'U' ) )
$ READ( NIN, FMT = * ) ( ( ( Q( I, J, L ), J = 1, N ),
$ I = 1, N ), L = 1, K )
IF( LSAME( COMPQ, 'P' ) ) THEN
READ( NIN, FMT = * ) ( QIND( I ), I = 1, K )
DO 10 L = 1, K
IF( QIND( L ).GT.0 )
$ READ( NIN, FMT = * ) ( ( Q( I, J, QIND( L ) ),
$ J = 1, N ), I = 1, N )
10 CONTINUE
END IF
IF( LSAME( JOB, 'E' ) )
$ JOB = 'S'
* Compute the eigenvalues and the transformed matrices.
CALL MB03BD( JOB, DEFL, COMPQ, QIND, K, N, H, ILO, IHI, S, A,
$ LDA1, LDA2, Q, LDQ1, LDQ2, ALPHAR, ALPHAI, BETA,
$ SCAL, IWORK, LIWORK, DWORK, LDWORK, IWARN, INFO )
*
IF( INFO.NE.0 ) THEN
WRITE( NOUT, FMT = 99997 ) INFO
ELSE IF( IWARN.EQ.0 ) THEN
* Prepare the data for calling MB03KD, which uses different
* data structures and reverse ordering of the factors.
DO 20 L = 1, K
ND( L ) = MAX( 1, N )
NI( L ) = 0
LDT( L ) = MAX( 1, N )
IXT( L ) = ( L - 1 )*LDT( L )*N + 1
LDQ( L ) = MAX( 1, N )
IXQ( L ) = IXT( L )
IF( L.LE.INT( K/2 ) ) THEN
I = S( K - L + 1 )
S( K - L + 1 ) = S( L )
S( L ) = I
END IF
20 CONTINUE
DO 30 L = 1, K
CALL DLACPY( 'Full', N, N, A( 1, 1, K-L+1 ), LDA1,
$ T( IXT( L ) ), LDT( L ) )
30 CONTINUE
IF( LSAME( COMPQ, 'U' ) .OR. LSAME( COMPQ, 'I' ) ) THEN
COMPQ = 'U'
DO 40 L = 1, K
CALL DLACPY( 'Full', N, N, Q( 1, 1, K-L+1 ), LDQ1,
$ QK( IXQ( L ) ), LDQ( L ) )
40 CONTINUE
ELSE IF( LSAME( COMPQ, 'P' ) ) THEN
COMPQ = 'W'
DO 50 L = 1, K
IF( QIND( L ).LT.0 )
$ QIND( L ) = 2
P = QIND( L )
IF( P.NE.0 )
$ CALL DLACPY( 'Full', N, N, Q( 1, 1, K-P+1 ), LDQ1,
$ QK( IXQ( P ) ), LDQ( P ) )
50 CONTINUE
END IF
* Select eigenvalues with negative real part.
DO 60 I = 1, N
SELECT( I ) = ALPHAR( I ).LT.ZERO
60 CONTINUE
WRITE( NOUT, FMT = 99996 )
WRITE( NOUT, FMT = 99995 ) ( ALPHAR( I ), I = 1, N )
WRITE( NOUT, FMT = 99994 )
WRITE( NOUT, FMT = 99995 ) ( ALPHAI( I ), I = 1, N )
WRITE( NOUT, FMT = 99993 )
WRITE( NOUT, FMT = 99995 ) ( BETA( I ), I = 1, N )
WRITE( NOUT, FMT = 99992 )
WRITE( NOUT, FMT = 99991 ) ( SCAL( I ), I = 1, N )
* Compute the transformed matrices, after reordering the
* eigenvalues.
CALL MB03KD( COMPQ, QIND, STRONG, K, N, H, ND, NI, S,
$ SELECT, T, LDT, IXT, QK, LDQ, IXQ, M, TOL,
$ IWORK, DWORK, LDWORK, INFO )
IF( INFO.NE.0 ) THEN
WRITE( NOUT, FMT = 99990 ) INFO
ELSE
WRITE( NOUT, FMT = 99989 )
DO 80 L = 1, K
P = K - L + 1
WRITE( NOUT, FMT = 99988 ) L
DO 70 I = 1, N
WRITE( NOUT, FMT = 99995 )
$ ( T( IXT( P ) + I - 1 + ( J - 1 )*LDT( P ) ),
$ J = 1, N )
70 CONTINUE
80 CONTINUE
IF( LSAME( COMPQ, 'U' ) .OR. LSAME( COMPQ, 'I' ) ) THEN
WRITE( NOUT, FMT = 99987 )
DO 100 L = 1, K
P = K - L + 1
WRITE( NOUT, FMT = 99988 ) L
DO 90 I = 1, N
WRITE( NOUT, FMT = 99995 )
$ ( QK( IXQ( P ) + I - 1 +
$ ( J - 1 )*LDQ( P ) ), J = 1, N )
90 CONTINUE
100 CONTINUE
ELSE IF( LSAME( COMPQ, 'W' ) ) THEN
WRITE( NOUT, FMT = 99987 )
DO 120 L = 1, K
IF( QIND( L ).GT.0 ) THEN
P = K - QIND( L ) + 1
WRITE( NOUT, FMT = 99988 ) QIND( L )
DO 110 I = 1, N
WRITE( NOUT, FMT = 99995 )
$ ( QK( IXQ( P ) + I - 1 +
$ ( J - 1 )*LDQ( P ) ), J = 1, N )
110 CONTINUE
END IF
120 CONTINUE
END IF
END IF
ELSE
WRITE( NOUT, FMT = 99979 ) IWARN
END IF
END IF
STOP
*
99999 FORMAT( 'MB03KD EXAMPLE PROGRAM RESULTS', 1X )
99998 FORMAT( 'N is out of range.', /, 'N = ', I5 )
99997 FORMAT( 'INFO on exit from MB03BD = ', I2 )
99996 FORMAT( 'The vector ALPHAR is ' )
99995 FORMAT( 50( 1X, F8.4 ) )
99994 FORMAT( 'The vector ALPHAI is ' )
99993 FORMAT( 'The vector BETA is ' )
99992 FORMAT( 'The vector SCAL is ' )
99991 FORMAT( 50( 1X, I5 ) )
99990 FORMAT( 'INFO on exit from MB03KD = ', I2 )
99989 FORMAT( 'The matrix A on exit is ' )
99988 FORMAT( 'The factor ', I2, ' is ' )
99987 FORMAT( 'The matrix Q on exit is ' )
99986 FORMAT( 'LDT', 3I5 )
99985 FORMAT( 'IXT', 3I5 )
99984 FORMAT( 'LDQ', 3I5 )
99983 FORMAT( 'IXQ', 3I5 )
99982 FORMAT( 'ND' , 3I5 )
99981 FORMAT( 'NI' , 3I5)
99980 FORMAT( 'SELECT', 3L5 )
99979 FORMAT( 'IWARN on exit from MB03BD = ', I2 )
END
Program Data
MB03KD EXAMPLE PROGRAM DATA S C I N 3 3 2 1 3 -1 1 -1 2.0 0.0 1.0 0.0 -2.0 -1.0 0.0 0.0 3.0 1.0 2.0 0.0 4.0 -1.0 3.0 0.0 3.0 1.0 1.0 0.0 1.0 0.0 4.0 -1.0 0.0 0.0 -2.0Program Results
MB03KD EXAMPLE PROGRAM RESULTS
The vector ALPHAR is
0.3230 0.3230 -0.8752
The vector ALPHAI is
0.5694 -0.5694 0.0000
The vector BETA is
1.0000 1.0000 1.0000
The vector SCAL is
0 0 -1
The matrix A on exit is
The factor 1 is
2.5997 -0.0087 1.6898
0.0000 1.9846 0.1942
0.0000 0.0000 2.3259
The factor 2 is
-2.0990 -1.0831 -2.5601
0.0000 3.4838 0.2950
0.0000 3.4552 -2.1690
The factor 3 is
1.8451 0.9260 1.2717
0.0000 1.3976 -2.3544
0.0000 0.0000 -3.1023
The matrix Q on exit is
The factor 1 is
-0.2052 0.4647 -0.8614
0.2033 0.8811 0.4270
-0.9574 0.0875 0.2753
The factor 2 is
-0.7743 -0.1384 0.6176
0.6070 -0.4386 0.6627
0.1791 0.8880 0.4236
The factor 3 is
-0.6714 0.7225 -0.1651
-0.3658 -0.5168 -0.7740
-0.6446 -0.4593 0.6112