**Purpose**

To find the eigenvalues of the generalized matrix product S(1) S(2) S(K) A(:,:,1) * A(:,:,2) * ... * A(:,:,K) where A(:,:,H) is upper Hessenberg and A(:,:,i), i <> H, is upper triangular, using a double-shift version of the periodic QZ method. In addition, A may be reduced to periodic Schur form: A(:,:,H) is upper quasi-triangular and all the other factors A(:,:,I) are upper triangular. Optionally, the 2-by-2 triangular matrices corresponding to 2-by-2 diagonal blocks in A(:,:,H) are so reduced that their product is a 2-by-2 diagonal matrix. If COMPQ = 'U' or COMPQ = 'I', then the orthogonal factors are computed and stored in the array Q so that for S(I) = 1, T Q(:,:,I)(in) A(:,:,I)(in) Q(:,:,MOD(I,K)+1)(in) T (1) = Q(:,:,I)(out) A(:,:,I)(out) Q(:,:,MOD(I,K)+1)(out), and for S(I) = -1, T Q(:,:,MOD(I,K)+1)(in) A(:,:,I)(in) Q(:,:,I)(in) T (2) = Q(:,:,MOD(I,K)+1)(out) A(:,:,I)(out) Q(:,:,I)(out). A partial generation of the orthogonal factors can be realized via the array QIND.

SUBROUTINE 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 ) C .. Scalar Arguments .. CHARACTER COMPQ, DEFL, JOB INTEGER H, IHI, ILO, INFO, IWARN, K, LDA1, LDA2, LDQ1, $ LDQ2, LDWORK, LIWORK, N C .. Array Arguments .. INTEGER IWORK(*), QIND(*), S(*), SCAL(*) DOUBLE PRECISION A(LDA1,LDA2,*), ALPHAI(*), ALPHAR(*), $ BETA(*), DWORK(*), Q(LDQ1,LDQ2,*)

**Mode Parameters**

JOB CHARACTER*1 Specifies the computation to be performed, as follows: = 'E': compute the eigenvalues only; A will not necessarily be put into periodic Schur form; = 'S': put A into periodic Schur form, and return the eigenvalues in ALPHAR, ALPHAI, BETA, and SCAL; = 'T': as JOB = 'S', but A is put into standardized periodic Schur form, that is, the general product of the 2-by-2 triangular matrices corresponding to a complex eigenvalue is diagonal. DEFL CHARACTER*1 Specifies the deflation strategy to be used, as follows: = 'C': apply a careful deflation strategy, that is, the criteria are based on the magnitudes of neighboring elements and infinite eigenvalues are only deflated at the top; this is the recommended option; = 'A': apply a more aggressive strategy, that is, elements on the subdiagonal or diagonal are set to zero as soon as they become smaller in magnitude than eps times the norm of the corresponding factor; this option is only recommended if balancing is applied beforehand and convergence problems are observed. COMPQ CHARACTER*1 Specifies whether or not the orthogonal transformations should be accumulated in the array Q, as follows: = 'N': do not modify Q; = 'U': modify (update) the array Q by the orthogonal transformations that are applied to the matrices in the array A to reduce them to periodic Schur form; = 'I': like COMPQ = 'U', except that each matrix in the array Q will be first initialized to the identity matrix; = 'P': use the parameters as encoded in QIND. QIND INTEGER array, dimension (K) If COMPQ = 'P', then this array describes the generation of the orthogonal factors as follows: If QIND(I) > 0, then the array Q(:,:,QIND(I)) is modified by the transformations corresponding to the i-th orthogonal factor in (1) and (2). If QIND(I) < 0, then the array Q(:,:,-QIND(I)) is initialized to the identity and modified by the transformations corresponding to the i-th orthogonal factor in (1) and (2). If QIND(I) = 0, then the transformations corresponding to the i-th orthogonal factor in (1), (2) are not applied.

K (input) INTEGER The number of factors. K >= 1. N (input) INTEGER The order of each factor in the array A. N >= 0. H (input) INTEGER Hessenberg index. The factor A(:,:,H) is on entry in upper Hessenberg form. 1 <= H <= K. ILO (input) INTEGER IHI (input) INTEGER It is assumed that each factor in A is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. 1 <= ILO <= IHI <= N, if N > 0; ILO = 1 and IHI = 0, if N = 0. S (input) INTEGER array, dimension (K) The leading K elements of this array must contain the signatures of the factors. Each entry in S must be either 1 or -1. A (input/output) DOUBLE PRECISION array, dimension (LDA1,LDA2,K) On entry, the leading N-by-N-by-K part of this array must contain the factors in upper Hessenberg-triangular form, that is, A(:,:,H) is upper Hessenberg and the other factors are upper triangular. On exit, if JOB = 'S' and INFO = 0, the leading N-by-N-by-K part of this array contains the factors of A in periodic Schur form, that is, A(:,:,H) is upper quasi triangular and the other factors are upper triangular. On exit, if JOB = 'T' and INFO = 0, the leading N-by-N-by-K part of this array contains the factors of A as for the option JOB = 'S', but the product of the triangular factors corresponding to a 2-by-2 block in A(:,:,H) is diagonal. On exit, if JOB = 'E', then the leading N-by-N-by-K part of this array contains meaningless elements in the off- diagonal blocks. LDA1 INTEGER The first leading dimension of the array A. LDA1 >= MAX(1,N). LDA2 INTEGER The second leading dimension of the array A. LDA2 >= MAX(1,N). Q (input/output) DOUBLE PRECISION array, dimension (LDQ1,LDQ2,K) On entry, if COMPQ = 'U', the leading N-by-N-by-K part of this array must contain the initial orthogonal factors as described in (1) and (2). On entry, if COMPQ = 'P', only parts of the leading N-by-N-by-K part of this array must contain some orthogonal factors as described by the parameters QIND. If COMPQ = 'I', this array should not set on entry. On exit, if COMPQ = 'U' or COMPQ = 'I', the leading N-by-N-by-K part of this array contains the modified orthogonal factors as described in (1) and (2). On exit, if COMPQ = 'P', only parts of the leading N-by-N-by-K part contain some modified orthogonal factors as described by the parameters QIND. This array is not referenced if COMPQ = 'N'. LDQ1 INTEGER The first leading dimension of the array Q. LDQ1 >= 1, and, if COMPQ <> 'N', LDQ1 >= MAX(1,N). LDQ2 INTEGER The second leading dimension of the array Q. LDQ2 >= 1, and, if COMPQ <> 'N', LDQ2 >= MAX(1,N). ALPHAR (output) DOUBLE PRECISION array, dimension (N) On exit, if INFO = 0, the leading N elements of this array contain the scaled real parts of the eigenvalues of the matrix product A. The i-th eigenvalue of A is given by (ALPHAR(I) + ALPHAI(I)*SQRT(-1))/BETA(I) * BASE**SCAL(I), where BASE is the machine base (often 2.0). Complex conjugate eigenvalues appear in consecutive locations. ALPHAI (output) DOUBLE PRECISION array, dimension (N) On exit, if INFO = 0, the leading N elements of this array contain the scaled imaginary parts of the eigenvalues of A. BETA (output) DOUBLE PRECISION array, dimension (N) On exit, if INFO = 0, the leading N elements of this array contain indicators for infinite eigenvalues. That is, if BETA(I) = 0.0, then the i-th eigenvalue is infinite. Otherwise BETA(I) is set to 1.0. SCAL (output) INTEGER array, dimension (N) On exit, if INFO = 0, the leading N elements of this array contain the scaling parameters for the eigenvalues of A.

IWORK INTEGER array, dimension (LIWORK) On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK, and, if IWARN > N, the nonzero absolute values in IWORK(2), ..., IWORK(N+1) are indices of the possibly inaccurate eigenvalues, as well as of the corresponding 1-by-1 or 2-by-2 diagonal blocks of the factors in the array A. The 2-by-2 blocks correspond to negative values in IWORK. One negative value is stored for each such eigenvalue pair. Its modulus indicates the starting index of a 2-by-2 block. This is also done for any value of IWARN, if a 2-by-2 block is found to have two real eigenvalues. On exit, if INFO = -22, IWORK(1) returns the minimum value of LIWORK. LIWORK INTEGER The length of the array IWORK. LIWORK >= 2*K+N. DWORK DOUBLE PRECISION array, dimension (LDWORK) On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK, and DWORK(2), ..., DWORK(1+K) contain the Frobenius norms of the factors of the formal matrix product used by the algorithm. On exit, if INFO = -24, DWORK(1) returns the minimum value of LDWORK. LDWORK INTEGER The length of the array DWORK. LDWORK >= K + MAX( 2*N,8*K ).

IWARN INTEGER = 0 : no warnings; = 1,..,N-1 : A is in periodic Schur form, but the algorithm was not able to reveal information about the eigenvalues from the 2-by-2 blocks. ALPHAR(i), ALPHAI(i), BETA(i) and SCAL(i), can be incorrect for i = 1, ..., IWARN+1; = N : some eigenvalues might be inaccurate; = N+1 : some eigenvalues might be inaccurate, and details can be found in IWORK.

INFO INTEGER = 0 : succesful exit; < 0 : if INFO = -i, the i-th argument had an illegal value; = 1,..,N : the periodic QZ iteration did not converge. A is not in periodic Schur form, but ALPHAR(i), ALPHAI(i), BETA(i) and SCAL(i), for i = INFO+1,...,N should be correct.

A modified version of the periodic QZ algorithm is used [1], [2].

[1] Bojanczyk, A., Golub, G. H. and Van Dooren, P. The periodic Schur decomposition: algorithms and applications. In F.T. Luk (editor), Advanced Signal Processing Algorithms, Architectures, and Implementations III, Proc. SPIE Conference, vol. 1770, pp. 31-42, 1992. [2] Kressner, D. An efficient and reliable implementation of the periodic QZ algorithm. IFAC Workshop on Periodic Control Systems (PSYCO 2001), Como (Italy), August 27-28 2001. Periodic Control Systems 2001 (IFAC Proceedings Volumes), Pergamon.

The implemented method is numerically backward stable. 3 The algorithm requires 0(K N ) floating point operations.

None

**Program Text**

* MB03BD 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 = KMAX + MAX( 2*NMAX, 8*KMAX ), $ LIWORK = 2*KMAX + NMAX ) * * .. Local Scalars .. CHARACTER COMPQ, DEFL, JOB INTEGER H, I, IHI, ILO, INFO, IWARN, J, K, L, N * * .. Local Arrays .. INTEGER IWORK( LIWORK ), QIND( KMAX ), S( KMAX ), $ SCAL( NMAX ) DOUBLE PRECISION A( LDA1, LDA2, KMAX ), ALPHAI( NMAX ), $ ALPHAR( NMAX ), BETA( NMAX ), DWORK( LDWORK), $ Q( LDQ1, LDQ2, KMAX ) * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * * .. External Subroutines .. EXTERNAL MB03BD * * .. Intrinsic Functions .. INTRINSIC 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, K, N, H, ILO, IHI IF( N.LT.0 .OR. N.GT.NMAX ) THEN WRITE( NOUT, FMT = 99998 ) N ELSE 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 * Compute the eigenvalues and the transformed matrices, if * required. 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 IF( LSAME( JOB, 'S' ) .OR. LSAME( JOB, 'T' ) ) THEN WRITE( NOUT, FMT = 99996 ) DO 30 L = 1, K WRITE( NOUT, FMT = 99988 ) L DO 20 I = 1, N WRITE( NOUT, FMT = 99995 ) ( A( I, J, L ), J = 1, N $ ) 20 CONTINUE 30 CONTINUE END IF IF( LSAME( COMPQ, 'U' ) .OR. LSAME( COMPQ, 'I' ) ) THEN WRITE( NOUT, FMT = 99994 ) DO 50 L = 1, K WRITE( NOUT, FMT = 99988 ) L DO 40 I = 1, N WRITE( NOUT, FMT = 99995 ) ( Q( I, J, L ), J = 1, N $ ) 40 CONTINUE 50 CONTINUE ELSE IF( LSAME( COMPQ, 'P' ) ) THEN WRITE( NOUT, FMT = 99994 ) DO 70 L = 1, K IF( QIND( L ).GT.0 ) THEN WRITE( NOUT, FMT = 99988 ) QIND( L ) DO 60 I = 1, N WRITE( NOUT, FMT = 99995 ) $ ( Q( I, J, QIND( L ) ), J = 1, N ) 60 CONTINUE END IF 70 CONTINUE END IF WRITE( NOUT, FMT = 99993 ) WRITE( NOUT, FMT = 99995 ) ( ALPHAR( I ), I = 1, N ) WRITE( NOUT, FMT = 99992 ) WRITE( NOUT, FMT = 99995 ) ( ALPHAI( I ), I = 1, N ) WRITE( NOUT, FMT = 99991 ) WRITE( NOUT, FMT = 99995 ) ( BETA( I ), I = 1, N ) WRITE( NOUT, FMT = 99990 ) WRITE( NOUT, FMT = 99989 ) ( SCAL( I ), I = 1, N ) ELSE WRITE( NOUT, FMT = 99987 ) IWARN END IF END IF STOP * 99999 FORMAT( 'MB03BD EXAMPLE PROGRAM RESULTS', 1X ) 99998 FORMAT( 'N is out of range.', /, 'N = ', I5 ) 99997 FORMAT( 'INFO on exit from MB03BD = ', I2 ) 99996 FORMAT( 'The matrix A on exit is ' ) 99995 FORMAT( 50( 1X, F8.4 ) ) 99994 FORMAT( 'The matrix Q on exit is ' ) 99993 FORMAT( 'The vector ALPHAR is ' ) 99992 FORMAT( 'The vector ALPHAI is ' ) 99991 FORMAT( 'The vector BETA is ' ) 99990 FORMAT( 'The vector SCAL is ' ) 99989 FORMAT( 50( 1X, I8 ) ) 99988 FORMAT( 'The factor ', I2, ' is ' ) 99987 FORMAT( 'IWARN on exit from MB03BD = ', I2 ) END

MB03BD EXAMPLE PROGRAM DATA S C I 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

MB03BD EXAMPLE PROGRAM RESULTS The matrix A on exit is The factor 1 is -2.1306 0.8205 0.7462 0.0000 2.8786 1.0564 0.0000 0.0000 1.9566 The factor 2 is -4.0763 -1.0376 -2.6948 -1.9525 1.8283 2.2987 0.0000 0.0000 1.8990 The factor 3 is 3.3463 -2.3239 -0.5623 0.0000 1.0778 -0.0646 0.0000 0.0000 -2.2180 The matrix Q on exit is The factor 1 is 0.2594 0.7715 -0.5809 -0.9552 0.1162 -0.2723 -0.1426 0.6255 0.7671 The factor 2 is -0.1766 0.8037 -0.5683 -0.9636 -0.0234 0.2664 0.2008 0.5946 0.7785 The factor 3 is 0.6295 0.7315 0.2619 -0.7394 0.4605 0.4911 0.2386 -0.5028 0.8308 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