**Purpose**

To compute orthogonal transformation matrices Q and Z which reduce the regular pole pencil A-lambda*E of the descriptor system (A-lambda*E,B,C) to the form (if JOB = 'F') ( Af * ) ( Ef * ) Q'*A*Z = ( ) , Q'*E*Z = ( ) , (1) ( 0 Ai ) ( 0 Ei ) or to the form (if JOB = 'I') ( Ai * ) ( Ei * ) Q'*A*Z = ( ) , Q'*E*Z = ( ) , (2) ( 0 Af ) ( 0 Ef ) where the subpencil Af-lambda*Ef, with Ef nonsingular and upper triangular, contains the finite eigenvalues, and the subpencil Ai-lambda*Ei, with Ai nonsingular and upper triangular, contains the infinite eigenvalues. The subpencil Ai-lambda*Ei is in a staircase form (see METHOD). If JOBA = 'H', the submatrix Af is further reduced to an upper Hessenberg form.

SUBROUTINE TG01LD( JOB, JOBA, COMPQ, COMPZ, N, M, P, A, LDA, $ E, LDE, B, LDB, C, LDC, Q, LDQ, Z, LDZ, NF, ND, $ NIBLCK, IBLCK, TOL, IWORK, DWORK, LDWORK, $ INFO ) C .. Scalar Arguments .. CHARACTER COMPQ, COMPZ, JOB, JOBA INTEGER INFO, LDA, LDB, LDC, LDE, LDQ, LDWORK, LDZ, M, $ N, ND, NF, NIBLCK, P DOUBLE PRECISION TOL C .. Array Arguments .. INTEGER IBLCK( * ), IWORK(*) DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ), $ DWORK( * ), E( LDE, * ), Q( LDQ, * ), $ Z( LDZ, * )

**Mode Parameters**

JOB CHARACTER*1 = 'F': perform the finite-infinite separation; = 'I': perform the infinite-finite separation. JOBA CHARACTER*1 = 'H': reduce Af further to an upper Hessenberg form; = 'N': keep Af unreduced. COMPQ CHARACTER*1 = 'N': do not compute Q; = 'I': Q is initialized to the unit matrix, and the orthogonal matrix Q is returned; = 'U': Q must contain an orthogonal matrix Q1 on entry, and the product Q1*Q is returned. COMPZ CHARACTER*1 = 'N': do not compute Z; = 'I': Z is initialized to the unit matrix, and the orthogonal matrix Z is returned; = 'U': Z must contain an orthogonal matrix Z1 on entry, and the product Z1*Z is returned.

N (input) INTEGER The number of rows of the matrix B, the number of columns of the matrix C and the order of the square matrices A and E. N >= 0. M (input) INTEGER The number of columns of the matrix B. M >= 0. P (input) INTEGER The number of rows of the matrix C. P >= 0. A (input/output) DOUBLE PRECISION array, dimension (LDA,N) On entry, the leading N-by-N part of this array must contain the N-by-N state matrix A. On exit, the leading N-by-N part of this array contains the transformed state matrix Q'*A*Z, ( Af * ) ( Ai * ) Q'*A*Z = ( ) , or Q'*A*Z = ( ) , ( 0 Ai ) ( 0 Af ) depending on JOB, with Af an NF-by-NF matrix, and Ai an (N-NF)-by-(N-NF) nonsingular and upper triangular matrix. If JOBA = 'H', Af is in an upper Hessenberg form. Otherwise, Af is unreduced. Ai has a block structure as in (3) or (4), where A0,0 is ND-by-ND and Ai,i , for i = 1, ..., NIBLCK, is IBLCK(i)-by-IBLCK(i). LDA INTEGER The leading dimension of the array A. LDA >= MAX(1,N). E (input/output) DOUBLE PRECISION array, dimension (LDE,N) On entry, the leading N-by-N part of this array must contain the N-by-N descriptor matrix E. On exit, the leading N-by-N part of this array contains the transformed descriptor matrix Q'*E*Z, ( Ef * ) ( Ei * ) Q'*E*Z = ( ) , or Q'*E*Z = ( ) , ( 0 Ei ) ( 0 Ef ) depending on JOB, with Ef an NF-by-NF nonsingular matrix, and Ei an (N-NF)-by-(N-NF) nilpotent matrix in an upper block triangular form, as in (3) or (4). LDE INTEGER The leading dimension of the array E. LDE >= MAX(1,N). B (input/output) DOUBLE PRECISION array, dimension (LDB,K), where K = M if JOB = 'F', and K = MAX(M,P) if JOB = 'I'. On entry, the leading N-by-M part of this array must contain the N-by-M input matrix B. On exit, the leading N-by-M part of this array contains the transformed input matrix Q'*B. LDB INTEGER The leading dimension of the array B. LDB >= MAX(1,N). C (input/output) DOUBLE PRECISION array, dimension (LDC,N) On entry, the leading P-by-N part of this array must contain the state/output matrix C. On exit, the leading P-by-N part of this array contains the transformed matrix C*Z. LDC INTEGER The leading dimension of the array C. LDC >= MAX(1,K), where K = P if JOB = 'F', and K = MAX(M,P) if JOB = 'I'. Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N) If COMPQ = 'N': Q is not referenced. If COMPQ = 'I': on entry, Q need not be set; on exit, the leading N-by-N part of this array contains the orthogonal matrix Q, where Q' is the product of Householder transformations applied to A, E, and B on the left. If COMPQ = 'U': on entry, the leading N-by-N part of this array must contain an orthogonal matrix Q1; on exit, the leading N-by-N part of this array contains the orthogonal matrix Q1*Q. LDQ INTEGER The leading dimension of the array Q. LDQ >= 1, if COMPQ = 'N'; LDQ >= MAX(1,N), if COMPQ = 'I' or 'U'. Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N) If COMPZ = 'N': Z is not referenced. If COMPZ = 'I': on entry, Z need not be set; on exit, the leading N-by-N part of this array contains the orthogonal matrix Z, which is the product of Householder transformations applied to A, E, and C on the right. If COMPZ = 'U': on entry, the leading N-by-N part of this array must contain an orthogonal matrix Z1; on exit, the leading N-by-N part of this array contains the orthogonal matrix Z1*Z. LDZ INTEGER The leading dimension of the array Z. LDZ >= 1, if COMPZ = 'N'; LDZ >= MAX(1,N), if COMPZ = 'I' or 'U'. NF (output) INTEGER. The order of the reduced matrices Af and Ef; also, the number of finite generalized eigenvalues of the pencil A-lambda*E. ND (output) INTEGER. The number of non-dynamic infinite eigenvalues of the pair (A,E). Note: N-ND is the rank of the matrix E. NIBLCK (output) INTEGER If ND > 0, the number of infinite blocks minus one. If ND = 0, then NIBLCK = 0. IBLCK (output) INTEGER array, dimension (N) IBLCK(i) contains the dimension of the i-th block in the staircase form (3) or (4), with i = 1,2, ..., NIBLCK.

TOL DOUBLE PRECISION A tolerance used in rank decisions to determine the effective rank, which is defined as the order of the largest leading (or trailing) triangular submatrix in the QR factorization with column pivoting whose estimated condition number is less than 1/TOL. If the user sets TOL <= 0, then an implicitly computed, default tolerance, TOLDEF = N**2*EPS, is used instead, where EPS is the machine precision (see LAPACK Library routine DLAMCH). TOL < 1.

IWORK INTEGER array, dimension (N) DWORK DOUBLE PRECISION array, dimension (LDWORK) On exit, if INFO = 0, DWORK(1) returns the optimal value of LDWORK. LDWORK INTEGER The length of the array DWORK. LDWORK >= 1, and if N > 0, LDWORK >= N + MAX(3*N,M,P). If LDWORK = -1, then 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 related to LDWORK is issued by XERBLA.

INFO INTEGER = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value; = 1: the pencil A-lambda*E is not regular.

The subroutine is based on the reduction algorithm of [1]. If JOB = 'F', the matrices Ai and Ei have the form ( A0,0 A0,k ... A0,1 ) ( 0 E0,k ... E0,1 ) Ai = ( 0 Ak,k ... Ak,1 ) , Ei = ( 0 0 ... Ek,1 ) ; (3) ( : : . : ) ( : : . : ) ( 0 0 ... A1,1 ) ( 0 0 ... 0 ) if JOB = 'I', the matrices Ai and Ei have the form ( A1,1 ... A1,k A1,0 ) ( 0 ... E1,k E1,0 ) Ai = ( : . : : ) , Ei = ( : . : : ) , (4) ( : ... Ak,k Ak,0 ) ( : ... 0 Ek,0 ) ( 0 ... 0 A0,0 ) ( 0 ... 0 0 ) where Ai,i , for i = 0, 1, ..., k, are nonsingular upper triangular matrices. A0,0 corresponds to the non-dynamic infinite modes of the system.

[1] Misra, P., Van Dooren, P., and Varga, A. Computation of structural invariants of generalized state-space systems. Automatica, 30, pp. 1921-1936, 1994.

The algorithm is numerically backward stable and requires 0( N**3 ) floating point operations.

The number of infinite poles is computed as NIBLCK NINFP = Sum IBLCK(i) = N - ND - NF. i=1 The multiplicities of infinite poles can be computed as follows: there are IBLCK(k)-IBLCK(k+1) infinite poles of multiplicity k, for k = 1, ..., NIBLCK, where IBLCK(NIBLCK+1) = 0. Note that each infinite pole of multiplicity k corresponds to an infinite eigenvalue of multiplicity k+1.

**Program Text**

* TG01LD EXAMPLE PROGRAM TEXT * Copyright (c) 2002-2017 NICONET e.V. * * .. Parameters .. INTEGER NIN, NOUT PARAMETER ( NIN = 5, NOUT = 6 ) INTEGER NMAX, MMAX, PMAX PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 ) INTEGER LDA, LDB, LDC, LDE, LDQ, LDZ PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX, $ LDE = NMAX, LDQ = NMAX, LDZ = NMAX ) INTEGER LDWORK PARAMETER ( LDWORK = NMAX+MAX( 3*NMAX, MMAX, PMAX ) ) * .. Local Scalars .. CHARACTER*1 COMPQ, COMPZ, JOB, JOBA INTEGER I, INFO, J, M, N, ND, NF, NIBLCK, P DOUBLE PRECISION TOL * .. Local Arrays .. INTEGER IBLCK(NMAX), IWORK(NMAX) DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX), $ DWORK(LDWORK), E(LDE,NMAX), Q(LDQ,NMAX), $ Z(LDZ,NMAX) * .. External Subroutines .. EXTERNAL TG01LD * .. Intrinsic Functions .. INTRINSIC MAX * .. Executable Statements .. * WRITE ( NOUT, FMT = 99999 ) * Skip the heading in the data file and read the data. READ ( NIN, FMT = '()' ) READ ( NIN, FMT = * ) N, M, P, JOB, JOBA, TOL COMPQ = 'I' COMPZ = 'I' IF ( N.LT.0 .OR. N.GT.NMAX ) THEN WRITE ( NOUT, FMT = 99988 ) N ELSE READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N ) READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,N ) IF ( M.LT.0 .OR. M.GT.MMAX ) THEN WRITE ( NOUT, FMT = 99987 ) M ELSE READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N ) IF ( P.LT.0 .OR. P.GT.PMAX ) THEN WRITE ( NOUT, FMT = 99986 ) P ELSE READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P ) * Find the reduced descriptor system * (A-lambda E,B,C). CALL TG01LD( JOB, JOBA, COMPQ, COMPZ, N, M, P, A, LDA, $ E, LDE, B, LDB, C, LDC, Q, LDQ, Z, LDZ, NF, $ ND, NIBLCK, IBLCK, TOL, IWORK, DWORK, $ LDWORK, INFO ) * IF ( INFO.NE.0 ) THEN WRITE ( NOUT, FMT = 99998 ) INFO ELSE WRITE ( NOUT, FMT = 99994 ) NF, ND WRITE ( NOUT, FMT = 99989 ) NIBLCK + 1 IF ( NIBLCK.GT.0 ) THEN WRITE ( NOUT, FMT = 99985 ) $ ( IBLCK(I), I = 1, NIBLCK ) END IF WRITE ( NOUT, FMT = 99997 ) DO 10 I = 1, N WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N ) 10 CONTINUE WRITE ( NOUT, FMT = 99996 ) DO 20 I = 1, N WRITE ( NOUT, FMT = 99995 ) ( E(I,J), J = 1,N ) 20 CONTINUE WRITE ( NOUT, FMT = 99993 ) DO 30 I = 1, N WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M ) 30 CONTINUE WRITE ( NOUT, FMT = 99992 ) DO 40 I = 1, P WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,N ) 40 CONTINUE WRITE ( NOUT, FMT = 99991 ) DO 50 I = 1, N WRITE ( NOUT, FMT = 99995 ) ( Q(I,J), J = 1,N ) 50 CONTINUE WRITE ( NOUT, FMT = 99990 ) DO 60 I = 1, N WRITE ( NOUT, FMT = 99995 ) ( Z(I,J), J = 1,N ) 60 CONTINUE END IF END IF END IF END IF STOP * 99999 FORMAT (' TG01LD EXAMPLE PROGRAM RESULTS',/1X) 99998 FORMAT (' INFO on exit from TG01LD = ',I2) 99997 FORMAT (/' The reduced state dynamics matrix Q''*A*Z is ') 99996 FORMAT (/' The reduced descriptor matrix Q''*E*Z is ') 99995 FORMAT (20(1X,F8.4)) 99994 FORMAT (' Order of reduced system =', I5/ $ ' Number of non-dynamic infinite eigenvalues =', I5) 99993 FORMAT (/' The reduced input/state matrix Q''*B is ') 99992 FORMAT (/' The reduced state/output matrix C*Z is ') 99991 FORMAT (/' The left transformation matrix Q is ') 99990 FORMAT (/' The right transformation matrix Z is ') 99989 FORMAT ( ' Number of infinite blocks = ',I5) 99988 FORMAT (/' N is out of range.',/' N = ',I5) 99987 FORMAT (/' M is out of range.',/' M = ',I5) 99986 FORMAT (/' P is out of range.',/' P = ',I5) 99985 FORMAT ( ' Dimension of the blocks'/20I5) END

TG01LD EXAMPLE PROGRAM DATA 4 2 2 F N 0.0 -1 0 0 3 0 0 1 2 1 1 0 4 0 0 0 0 1 2 0 0 0 1 0 1 3 9 6 3 0 0 2 0 1 0 0 0 0 1 1 1 -1 0 1 0 0 1 -1 1

TG01LD EXAMPLE PROGRAM RESULTS Order of reduced system = 3 Number of non-dynamic infinite eigenvalues = 1 Number of infinite blocks = 1 The reduced state dynamics matrix Q'*A*Z is 2.4497 -1.3995 0.2397 -4.0023 -0.0680 -0.0030 0.1739 -1.6225 0.3707 0.0161 -0.9482 0.1049 0.0000 0.0000 0.0000 2.2913 The reduced descriptor matrix Q'*E*Z is 9.9139 4.7725 -3.4725 -2.3836 0.0000 -1.2024 2.0137 0.7926 0.0000 0.0000 0.2929 -0.9914 0.0000 0.0000 0.0000 0.0000 The reduced input/state matrix Q'*B is -0.2157 -0.9705 0.3015 0.9516 0.7595 0.0991 1.1339 0.3780 The reduced state/output matrix C*Z is 0.5345 -1.1134 0.3758 0.5774 -1.0690 0.2784 -1.2026 0.5774 The left transformation matrix Q is -0.2157 -0.5088 0.6109 0.5669 -0.1078 -0.2544 -0.7760 0.5669 -0.9705 0.1413 -0.0495 -0.1890 0.0000 0.8102 0.1486 0.5669 The right transformation matrix Z is -0.5345 0.6263 0.4617 -0.3299 -0.8018 -0.5219 -0.2792 -0.0825 0.0000 -0.4871 0.8375 0.2474 -0.2673 0.3132 -0.0859 0.9073