**Purpose**

To compute for the descriptor system (A-lambda E,B,C) the orthogonal transformation matrices Q and Z such that the transformed system (Q'*A*Z-lambda Q'*E*Z, Q'*B, C*Z) is in a SVD-like coordinate form with ( A11 A12 ) ( Er 0 ) Q'*A*Z = ( ) , Q'*E*Z = ( ) , ( A21 A22 ) ( 0 0 ) where Er is an upper triangular invertible matrix. Optionally, the A22 matrix can be further reduced to the form ( Ar X ) A22 = ( ) , ( 0 0 ) with Ar an upper triangular invertible matrix, and X either a full or a zero matrix. The left and/or right orthogonal transformations performed to reduce E and A22 can be optionally accumulated.

SUBROUTINE TG01FD( COMPQ, COMPZ, JOBA, L, N, M, P, A, LDA, E, LDE, $ B, LDB, C, LDC, Q, LDQ, Z, LDZ, RANKE, RNKA22, $ TOL, IWORK, DWORK, LDWORK, INFO ) C .. Scalar Arguments .. CHARACTER COMPQ, COMPZ, JOBA INTEGER INFO, L, LDA, LDB, LDC, LDE, LDQ, LDWORK, $ LDZ, M, N, P, RANKE, RNKA22 DOUBLE PRECISION TOL C .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ), $ DWORK( * ), E( LDE, * ), Q( LDQ, * ), $ Z( LDZ, * )

**Mode Parameters**

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. JOBA CHARACTER*1 = 'N': do not reduce A22; = 'R': reduce A22 to a SVD-like upper triangular form. = 'T': reduce A22 to an upper trapezoidal form.

L (input) INTEGER The number of rows of matrices A, B, and E. L >= 0. N (input) INTEGER The number of columns of matrices A, E, and C. N >= 0. M (input) INTEGER The number of columns of matrix B. M >= 0. P (input) INTEGER The number of rows of matrix C. P >= 0. A (input/output) DOUBLE PRECISION array, dimension (LDA,N) On entry, the leading L-by-N part of this array must contain the state dynamics matrix A. On exit, the leading L-by-N part of this array contains the transformed matrix Q'*A*Z. If JOBA = 'T', this matrix is in the form ( A11 * * ) Q'*A*Z = ( * Ar X ) , ( * 0 0 ) where A11 is a RANKE-by-RANKE matrix and Ar is a RNKA22-by-RNKA22 invertible upper triangular matrix. If JOBA = 'R' then A has the above form with X = 0. LDA INTEGER The leading dimension of array A. LDA >= MAX(1,L). E (input/output) DOUBLE PRECISION array, dimension (LDE,N) On entry, the leading L-by-N part of this array must contain the descriptor matrix E. On exit, the leading L-by-N part of this array contains the transformed matrix Q'*E*Z. ( Er 0 ) Q'*E*Z = ( ) , ( 0 0 ) where Er is a RANKE-by-RANKE upper triangular invertible matrix. LDE INTEGER The leading dimension of array E. LDE >= MAX(1,L). B (input/output) DOUBLE PRECISION array, dimension (LDB,M) On entry, the leading L-by-M part of this array must contain the input/state matrix B. On exit, the leading L-by-M part of this array contains the transformed matrix Q'*B. LDB INTEGER The leading dimension of array B. LDB >= MAX(1,L) if M > 0 or LDB >= 1 if M = 0. 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 array C. LDC >= MAX(1,P). Q (input/output) DOUBLE PRECISION array, dimension (LDQ,L) If COMPQ = 'N': Q is not referenced. If COMPQ = 'I': on entry, Q need not be set; on exit, the leading L-by-L part of this array contains the orthogonal matrix Q, where Q' is the product of Householder transformations which are applied to A, E, and B on the left. If COMPQ = 'U': on entry, the leading L-by-L part of this array must contain an orthogonal matrix Q1; on exit, the leading L-by-L part of this array contains the orthogonal matrix Q1*Q. LDQ INTEGER The leading dimension of array Q. LDQ >= 1, if COMPQ = 'N'; LDQ >= MAX(1,L), if COMPQ = 'U' or 'I'. 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 array Z. LDZ >= 1, if COMPZ = 'N'; LDZ >= MAX(1,N), if COMPZ = 'U' or 'I'. RANKE (output) INTEGER The estimated rank of matrix E, and thus also the order of the invertible upper triangular submatrix Er. RNKA22 (output) INTEGER If JOBA = 'R' or 'T', then RNKA22 is the estimated rank of matrix A22, and thus also the order of the invertible upper triangular submatrix Ar. If JOBA = 'N', then RNKA22 is not referenced.

TOL DOUBLE PRECISION The tolerance to be used in determining the rank of E and of A22. If the user sets TOL > 0, then the given value of TOL is used as a lower bound for the reciprocal condition numbers of leading submatrices of R or R22 in the QR decompositions E * P = Q * R of E or A22 * P22 = Q22 * R22 of A22. A submatrix whose estimated condition number is less than 1/TOL is considered to be of full rank. If the user sets TOL <= 0, then an implicitly computed, default tolerance, defined by TOLDEF = L*N*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 >= MAX( 1, N+P, MIN(L,N)+MAX(3*N-1,M,L) ). For optimal performance, LDWORK should be larger. 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.

The routine computes a truncated QR factorization with column pivoting of E, in the form ( E11 E12 ) E * P = Q * ( ) ( 0 E22 ) and finds the largest RANKE-by-RANKE leading submatrix E11 whose estimated condition number is less than 1/TOL. RANKE defines thus the rank of matrix E. Further E22, being negligible, is set to zero, and an orthogonal matrix Y is determined such that ( E11 E12 ) = ( Er 0 ) * Y . The overal transformation matrix Z results as Z = P * Y' and the resulting transformed matrices Q'*A*Z and Q'*E*Z have the form ( Er 0 ) ( A11 A12 ) E <- Q'* E * Z = ( ) , A <- Q' * A * Z = ( ) , ( 0 0 ) ( A21 A22 ) where Er is an upper triangular invertible matrix. If JOBA = 'R' the same reduction is performed on A22 to obtain it in the form ( Ar 0 ) A22 = ( ) , ( 0 0 ) with Ar an upper triangular invertible matrix. If JOBA = 'T' then A22 is row compressed using the QR factorization with column pivoting to the form ( Ar X ) A22 = ( ) ( 0 0 ) with Ar an upper triangular invertible matrix. The transformations are also applied to the rest of system matrices B <- Q' * B, C <- C * Z.

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

None

**Program Text**

* TG01FD EXAMPLE PROGRAM TEXT * Copyright (c) 2002-2017 NICONET e.V. * * .. Parameters .. INTEGER NIN, NOUT PARAMETER ( NIN = 5, NOUT = 6 ) INTEGER LMAX, NMAX, MMAX, PMAX PARAMETER ( LMAX = 20, NMAX = 20, MMAX = 20, PMAX = 20 ) INTEGER LDA, LDB, LDC, LDE, LDQ, LDZ PARAMETER ( LDA = LMAX, LDB = LMAX, LDC = PMAX, $ LDE = LMAX, LDQ = LMAX, LDZ = NMAX ) INTEGER LDWORK PARAMETER ( LDWORK = MAX( 1, PMAX, $ MIN(LMAX,NMAX)+MAX( 3*NMAX, MMAX, LMAX ) ) ) * .. Local Scalars .. CHARACTER*1 COMPQ, COMPZ, JOBA INTEGER I, INFO, J, L, M, N, P, RANKE, RNKA22 DOUBLE PRECISION TOL * .. Local Arrays .. INTEGER IWORK(NMAX) DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX), $ DWORK(LDWORK), E(LDE,NMAX), Q(LDQ,LMAX), $ Z(LDZ,NMAX) * .. External Subroutines .. EXTERNAL TG01FD * .. Intrinsic Functions .. INTRINSIC MAX, MIN * .. Executable Statements .. * WRITE ( NOUT, FMT = 99999 ) * Skip the heading in the data file and read the data. READ ( NIN, FMT = '()' ) READ ( NIN, FMT = * ) L, N, M, P, TOL COMPQ = 'I' COMPZ = 'I' JOBA = 'R' IF ( L.LT.0 .OR. L.GT.LMAX ) THEN WRITE ( NOUT, FMT = 99989 ) L ELSE 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,L ) READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,L ) 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,L ) 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 transformed descriptor system * (A-lambda E,B,C). CALL TG01FD( COMPQ, COMPZ, JOBA, L, N, M, P, A, LDA, $ E, LDE, B, LDB, C, LDC, Q, LDQ, Z, LDZ, $ RANKE, RNKA22, TOL, IWORK, DWORK, LDWORK, $ INFO ) * IF ( INFO.NE.0 ) THEN WRITE ( NOUT, FMT = 99998 ) INFO ELSE WRITE ( NOUT, FMT = 99994 ) RANKE, RNKA22 WRITE ( NOUT, FMT = 99997 ) DO 10 I = 1, L WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N ) 10 CONTINUE WRITE ( NOUT, FMT = 99996 ) DO 20 I = 1, L WRITE ( NOUT, FMT = 99995 ) ( E(I,J), J = 1,N ) 20 CONTINUE WRITE ( NOUT, FMT = 99993 ) DO 30 I = 1, L 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, L WRITE ( NOUT, FMT = 99995 ) ( Q(I,J), J = 1,L ) 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 END IF STOP * 99999 FORMAT (' TG01FD EXAMPLE PROGRAM RESULTS',/1X) 99998 FORMAT (' INFO on exit from TG01FD = ',I2) 99997 FORMAT (/' The transformed state dynamics matrix Q''*A*Z is ') 99996 FORMAT (/' The transformed descriptor matrix Q''*E*Z is ') 99995 FORMAT (20(1X,F8.4)) 99994 FORMAT (' Rank of matrix E =', I5/ $ ' Rank of matrix A22 =', I5) 99993 FORMAT (/' The transformed input/state matrix Q''*B is ') 99992 FORMAT (/' The transformed state/output matrix C*Z is ') 99991 FORMAT (/' The left transformation matrix Q is ') 99990 FORMAT (/' The right transformation matrix Z is ') 99989 FORMAT (/' L is out of range.',/' L = ',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) END

TG01FD EXAMPLE PROGRAM DATA 4 4 2 2 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

TG01FD EXAMPLE PROGRAM RESULTS Rank of matrix E = 3 Rank of matrix A22 = 1 The transformed state dynamics matrix Q'*A*Z is 2.0278 0.1078 3.9062 -2.1571 -0.0980 0.2544 1.6053 -0.1269 0.2713 0.7760 -0.3692 -0.4853 0.0690 -0.5669 -2.1974 0.3086 The transformed descriptor matrix Q'*E*Z is 10.1587 5.8230 1.3021 0.0000 0.0000 -2.4684 -0.1896 0.0000 0.0000 0.0000 1.0338 0.0000 0.0000 0.0000 0.0000 0.0000 The transformed input/state matrix Q'*B is -0.2157 -0.9705 0.3015 0.9516 0.7595 0.0991 1.1339 0.3780 The transformed state/output matrix C*Z is 0.3651 -1.0000 -0.4472 -0.8165 -1.0954 1.0000 -0.8944 0.0000 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 -0.3651 0.0000 0.4472 0.8165 -0.9129 0.0000 0.0000 -0.4082 0.0000 -1.0000 0.0000 0.0000 -0.1826 0.0000 -0.8944 0.4082