**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.

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( * )

**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.

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.

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.

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.

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.

An adaptation of the LAPACK Library routine DTGSEN is used.

The implemented method is numerically backward stable.

None

**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

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 -2.0

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