**Purpose**

To generate the benchmark examples for the numerical solution of continuous-time algebraic Riccati equations (CAREs) of the form 0 = Q + A'X + XA - XGX corresponding to the Hamiltonian matrix ( A G ) H = ( T ). ( Q -A ) A,G,Q,X are real N-by-N matrices, Q and G are symmetric and may be given in factored form -1 T T (I) G = B R B , (II) Q = C W C . Here, C is P-by-N, W P-by-P, B N-by-M, and R M-by-M, where W and R are symmetric. In linear-quadratic optimal control problems, usually W is positive semidefinite and R positive definite. The factorized form can be used if the CARE is solved using the deflating subspaces of the extended Hamiltonian pencil ( A 0 B ) ( I 0 0 ) ( T ) ( ) H - s K = ( Q A 0 ) - s ( 0 -I 0 ) , ( T ) ( ) ( 0 B R ) ( 0 0 0 ) where I and 0 denote the identity and zero matrix, respectively, of appropriate dimensions. NOTE: the formulation of the CARE and the related matrix (pencils) used here does not include CAREs as they arise in robust control (H_infinity optimization).

SUBROUTINE BB01AD(DEF, NR, DPAR, IPAR, BPAR, CHPAR, VEC, N, M, P, 1 A, LDA, B, LDB, C, LDC, G, LDG, Q, LDQ, X, LDX, 2 DWORK, LDWORK, INFO) C .. Scalar Arguments .. INTEGER INFO, LDA, LDB, LDC, LDG, LDQ, LDWORK, LDX, M, N, $ P CHARACTER DEF C .. Array Arguments .. INTEGER IPAR(4), NR(2) DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DPAR(*), DWORK(*), 1 G(*), Q(*), X(LDX,*) CHARACTER CHPAR*(*) LOGICAL BPAR(6), VEC(9)

**Mode Parameters**

DEF CHARACTER This parameter specifies if the default parameters are to be used or not. = 'N' or 'n' : The parameters given in the input vectors xPAR (x = 'D', 'I', 'B', 'CH') are used. = 'D' or 'd' : The default parameters for the example are used. This parameter is not meaningful if NR(1) = 1.

NR (input) INTEGER array, dimension (2) This array determines the example for which CAREX returns data. NR(1) is the group of examples. NR(1) = 1 : parameter-free problems of fixed size. NR(1) = 2 : parameter-dependent problems of fixed size. NR(1) = 3 : parameter-free problems of scalable size. NR(1) = 4 : parameter-dependent problems of scalable size. NR(2) is the number of the example in group NR(1). Let NEXi be the number of examples in group i. Currently, NEX1 = 6, NEX2 = 9, NEX3 = 2, NEX4 = 4. 1 <= NR(1) <= 4; 1 <= NR(2) <= NEXi , where i = NR(1). DPAR (input/output) DOUBLE PRECISION array, dimension (7) Double precision parameter vector. For explanation of the parameters see [1]. DPAR(1) : defines the parameters 'delta' for NR(1) = 3, 'q' for NR(1).NR(2) = 4.1, 'a' for NR(1).NR(2) = 4.2, and 'mu' for NR(1).NR(2) = 4.3. DPAR(2) : defines parameters 'r' for NR(1).NR(2) = 4.1, 'b' for NR(1).NR(2) = 4.2, and 'delta' for NR(1).NR(2) = 4.3. DPAR(3) : defines parameters 'c' for NR(1).NR(2) = 4.2 and 'kappa' for NR(1).NR(2) = 4.3. DPAR(j), j=4,5,6,7: These arguments are only used to generate Example 4.2 and define in consecutive order the intervals ['beta_1', 'beta_2'], ['gamma_1', 'gamma_2']. NOTE that if DEF = 'D' or 'd', the values of DPAR entries on input are ignored and, on output, they are overwritten with the default parameters. IPAR (input/output) INTEGER array, dimension (4) On input, IPAR(1) determines the actual state dimension, i.e., the order of the matrix A as follows, where NO = NR(1).NR(2). NR(1) = 1 or 2.1-2.8: IPAR(1) is ignored. NO = 2.9 : IPAR(1) = 1 generates the CARE for optimal state feedback (default); IPAR(1) = 2 generates the Kalman filter CARE. NO = 3.1 : IPAR(1) is the number of vehicles (parameter 'l' in the description in [1]). NO = 3.2, 4.1 or 4.2: IPAR(1) is the order of the matrix A. NO = 4.3 or 4.4 : IPAR(1) determines the dimension of the second-order system, i.e., the order of the stiffness matrix for Examples 4.3 and 4.4 (parameter 'l' in the description in [1]). The order of the output matrix A is N = 2*IPAR(1) for Example 4.3 and N = 2*IPAR(1)-1 for Examples 3.1 and 4.4. NOTE that IPAR(1) is overwritten for Examples 1.1-2.8. For the other examples, IPAR(1) is overwritten if the default parameters are to be used. On output, IPAR(1) contains the order of the matrix A. On input, IPAR(2) is the number of colums in the matrix B in (I) (in control problems, the number of inputs of the system). Currently, IPAR(2) is fixed or determined by IPAR(1) for all examples and thus is not referenced on input. On output, IPAR(2) is the number of columns of the matrix B from (I). NOTE that currently IPAR(2) is overwritten and that rank(G) <= IPAR(2). On input, IPAR(3) is the number of rows in the matrix C in (II) (in control problems, the number of outputs of the system). Currently, IPAR(3) is fixed or determined by IPAR(1) for all examples and thus is not referenced on input. On output, IPAR(3) contains the number of rows of the matrix C in (II). NOTE that currently IPAR(3) is overwritten and that rank(Q) <= IPAR(3). On input, if NR(1) = NR(2) = 4, and other data file than that used by default is desired, then IPAR(4) is the length of the character string in CHPAR specifying the file name. BPAR (input) BOOLEAN array, dimension (6) This array defines the form of the output of the examples and the storage mode of the matrices G and Q. BPAR(1) = .TRUE. : G is returned. BPAR(1) = .FALSE. : G is returned in factored form, i.e., B and R from (I) are returned. BPAR(2) = .TRUE. : The matrix returned in array G (i.e., G if BPAR(1) = .TRUE. and R if BPAR(1) = .FALSE.) is stored as full matrix. BPAR(2) = .FALSE. : The matrix returned in array G is provided in packed storage mode. BPAR(3) = .TRUE. : If BPAR(2) = .FALSE., the matrix returned in array G is stored in upper packed mode, i.e., the upper triangle of a symmetric n-by-n matrix is stored by columns, e.g., the matrix entry G(i,j) is stored in the array entry G(i+j*(j-1)/2) for i <= j. Otherwise, this entry is ignored. BPAR(3) = .FALSE. : If BPAR(2) = .FALSE., the matrix returned in array G is stored in lower packed mode, i.e., the lower triangle of a symmetric n-by-n matrix is stored by columns, e.g., the matrix entry G(i,j) is stored in the array entry G(i+(2*n-j)*(j-1)/2) for j <= i. Otherwise, this entry is ignored. BPAR(4) = .TRUE. : Q is returned. BPAR(4) = .FALSE. : Q is returned in factored form, i.e., C and W from (II) are returned. BPAR(5) = .TRUE. : The matrix returned in array Q (i.e., Q if BPAR(4) = .TRUE. and W if BPAR(4) = .FALSE.) is stored as full matrix. BPAR(5) = .FALSE. : The matrix returned in array Q is provided in packed storage mode. BPAR(6) = .TRUE. : If BPAR(5) = .FALSE., the matrix returned in array Q is stored in upper packed mode (see above). Otherwise, this entry is ignored. BPAR(6) = .FALSE. : If BPAR(5) = .FALSE., the matrix returned in array Q is stored in lower packed mode (see above). Otherwise, this entry is ignored. NOTE that there are no default values for BPAR. If all entries are declared to be .TRUE., then matrices G and Q are returned in conventional storage mode, i.e., as N-by-N arrays where the array element Z(I,J) contains the matrix entry Z_{i,j}. CHPAR (input/output) CHARACTER*255 On input, this is the name of a data file supplied by the user. In the current version, only Example 4.4 allows a user-defined data file. This file must contain consecutively DOUBLE PRECISION vectors mu, delta, gamma, and kappa. The length of these vectors is determined by the input value for IPAR(1). If on entry, IPAR(1) = L, then mu and delta must each contain L DOUBLE PRECISION values, and gamma and kappa must each contain L-1 DOUBLE PRECISION values. On output, this string contains short information about the chosen example. VEC (output) LOGICAL array, dimension (9) Flag vector which displays the availability of the output data: VEC(j), j=1,2,3, refer to N, M, and P, respectively, and are always .TRUE. VEC(4) refers to A and is always .TRUE. VEC(5) is .TRUE. if BPAR(1) = .FALSE., i.e., the factors B and R from (I) are returned. VEC(6) is .TRUE. if BPAR(4) = .FALSE., i.e., the factors C and W from (II) are returned. VEC(7) refers to G and is always .TRUE. VEC(8) refers to Q and is always .TRUE. VEC(9) refers to X and is .TRUE. if the exact solution matrix is available. NOTE that VEC(i) = .FALSE. for i = 1 to 9 if on exit INFO .NE. 0. N (output) INTEGER The order of the matrices A, X, G if BPAR(1) = .TRUE., and Q if BPAR(4) = .TRUE. M (output) INTEGER The number of columns in the matrix B (or the dimension of the control input space of the underlying dynamical system). P (output) INTEGER The number of rows in the matrix C (or the dimension of the output space of the underlying dynamical system). A (output) DOUBLE PRECISION array, dimension (LDA,N) The leading N-by-N part of this array contains the coefficient matrix A of the CARE. LDA INTEGER The leading dimension of array A. LDA >= N. B (output) DOUBLE PRECISION array, dimension (LDB,M) If (BPAR(1) = .FALSE.), then the leading N-by-M part of this array contains the matrix B of the factored form (I) of G. Otherwise, B is used as workspace. LDB INTEGER The leading dimension of array B. LDB >= N. C (output) DOUBLE PRECISION array, dimension (LDC,N) If (BPAR(4) = .FALSE.), then the leading P-by-N part of this array contains the matrix C of the factored form (II) of Q. Otherwise, C is used as workspace. LDC INTEGER The leading dimension of array C. LDC >= P, where P is the number of rows of the matrix C, i.e., the output value of IPAR(3). (For all examples, P <= N, where N equals the output value of the argument IPAR(1), i.e., LDC >= LDA is always safe.) G (output) DOUBLE PRECISION array, dimension (NG) If (BPAR(2) = .TRUE.) then NG = LDG*N. If (BPAR(2) = .FALSE.) then NG = N*(N+1)/2. If (BPAR(1) = .TRUE.), then array G contains the coefficient matrix G of the CARE. If (BPAR(1) = .FALSE.), then array G contains the 'control weighting matrix' R of G's factored form as in (I). (For all examples, M <= N.) The symmetric matrix contained in array G is stored according to BPAR(2) and BPAR(3). LDG INTEGER If conventional storage mode is used for G, i.e., BPAR(2) = .TRUE., then G is stored like a 2-dimensional array with leading dimension LDG. If packed symmetric storage mode is used, then LDG is not referenced. LDG >= N if BPAR(2) = .TRUE.. Q (output) DOUBLE PRECISION array, dimension (NQ) If (BPAR(5) = .TRUE.) then NQ = LDQ*N. If (BPAR(5) = .FALSE.) then NQ = N*(N+1)/2. If (BPAR(4) = .TRUE.), then array Q contains the coefficient matrix Q of the CARE. If (BPAR(4) = .FALSE.), then array Q contains the 'output weighting matrix' W of Q's factored form as in (II). The symmetric matrix contained in array Q is stored according to BPAR(5) and BPAR(6). LDQ INTEGER If conventional storage mode is used for Q, i.e., BPAR(5) = .TRUE., then Q is stored like a 2-dimensional array with leading dimension LDQ. If packed symmetric storage mode is used, then LDQ is not referenced. LDQ >= N if BPAR(5) = .TRUE.. X (output) DOUBLE PRECISION array, dimension (LDX,IPAR(1)) If an exact solution is available (NR = 1.1, 1.2, 2.1, 2.3-2.6, 3.2), then the leading N-by-N part of this array contains the solution matrix X in conventional storage mode. Otherwise, X is not referenced. LDX INTEGER The leading dimension of array X. LDX >= 1, and LDX >= N if NR = 1.1, 1.2, 2.1, 2.3-2.6, 3.2.

DWORK DOUBLE PRECISION array, dimension (LDWORK) LDWORK INTEGER The length of the array DWORK. LDWORK >= N*MAX(4,N).

INFO INTEGER = 0 : successful exit; < 0 : if INFO = -i, the i-th argument had an illegal value; = 1 : data file could not be opened or had wrong format; = 2 : division by zero; = 3 : G can not be computed as in (I) due to a singular R matrix.

[1] Abels, J. and Benner, P. CAREX - A Collection of Benchmark Examples for Continuous-Time Algebraic Riccati Equations (Version 2.0). SLICOT Working Note 1999-14, November 1999. Available from http://www.win.tue.nl/niconet/NIC2/reports.html. This is an updated and extended version of [2] Benner, P., Laub, A.J., and Mehrmann, V. A Collection of Benchmark Examples for the Numerical Solution of Algebraic Riccati Equations I: Continuous-Time Case. Technical Report SPC 95_22, Fak. f. Mathematik, TU Chemnitz-Zwickau (Germany), October 1995.

If the original data as taken from the literature is given via matrices G and Q, but factored forms are requested as output, then these factors are obtained from Cholesky or LDL' decompositions of G and Q, i.e., the output data will be corrupted by roundoff errors.

Some benchmark examples read data from the data files provided with the collection.

**Program Text**

* BB01AD EXAMPLE PROGRAM TEXT * * Copyright (c) 2002-2017 NICONET e.V. * * .. Parameters .. INTEGER NIN, NOUT PARAMETER ( NIN = 5, NOUT = 6 ) INTEGER MMAX, NMAX, PMAX PARAMETER ( MMAX = 100, NMAX = 100, PMAX = 100 ) INTEGER LDA, LDB, LDC, LDG, LDQ, LDX PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX, $ LDG = NMAX, LDQ = NMAX, LDX = NMAX ) INTEGER LDWORK PARAMETER ( LDWORK = NMAX*MAX( 4, NMAX ) ) * .. Local Scalars .. CHARACTER DEF INTEGER I, INFO, ISYMM, J, LBPAR, LDPAR, LIPAR, M, N, P * .. Local Arrays .. DOUBLE PRECISION A(LDA, NMAX), B(LDB,MMAX), C(LDC, NMAX), $ DPAR(7), DWORK(LDWORK), G(LDG, NMAX), $ Q(LDQ, NMAX), X(LDX, NMAX) INTEGER IPAR(4), NR(2) LOGICAL BPAR(6), VEC(9) CHARACTER CHPAR*255 * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. External Subroutines .. EXTERNAL BB01AD, MA02DD * .. 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 = * ) DEF READ( NIN, FMT = * ) ( NR(I), I = 1, 2 ) IF( LSAME( DEF, 'N' ) ) THEN READ( NIN, FMT = * ) LBPAR IF( LBPAR.GT.0 ) READ( NIN, FMT = * ) ( BPAR(I), I = 1, LBPAR ) READ( NIN, FMT = * ) LDPAR IF( LDPAR.GT.0 ) READ( NIN, FMT = * ) ( DPAR(I), I = 1, LDPAR ) READ( NIN, FMT = * ) LIPAR IF( LIPAR.GT.0 ) READ( NIN, FMT = * ) ( IPAR(I), I = 1, LIPAR ) END IF * Generate benchmark example CALL BB01AD( DEF, NR, DPAR, IPAR, BPAR, CHPAR, VEC, N, M, P, A, $ LDA, B, LDB, C, LDC, G, LDG, Q, LDQ, X, LDX, DWORK, $ LDWORK, INFO ) * IF( INFO.NE.0 ) THEN WRITE( NOUT, FMT = 99998 ) INFO ELSE WRITE( NOUT, FMT = * ) CHPAR(1:70) WRITE( NOUT, FMT = 99997 ) N WRITE( NOUT, FMT = 99996 ) M WRITE( NOUT, FMT = 99995 ) P WRITE( NOUT, FMT = 99994 ) DO 10 I = 1, N WRITE( NOUT, FMT = 99979 ) ( A(I,J), J = 1, N ) 10 CONTINUE IF( VEC(5) ) THEN WRITE( NOUT, FMT = 99993 ) DO 20 I = 1, N WRITE( NOUT, FMT = 99979 ) ( B(I,J), J = 1, M ) 20 CONTINUE ELSE WRITE( NOUT, FMT = 99992 ) END IF IF( VEC(6) ) THEN WRITE( NOUT,FMT = 99991 ) DO 30 I = 1, P WRITE( NOUT, FMT = 99979 ) ( C(I,J), J = 1, N ) 30 CONTINUE ELSE WRITE( NOUT, FMT = 99990 ) END IF IF( .NOT.VEC(5) ) THEN WRITE( NOUT, FMT = 99989 ) IF( .NOT.BPAR(2) ) THEN ISYMM = ( N * ( N + 1 ) ) / 2 CALL DCOPY( ISYMM, G, 1, DWORK, 1 ) IF( BPAR(3) ) THEN CALL MA02DD( 'Unpack', 'Upper', N, G, LDG, DWORK ) ELSE CALL MA02DD( 'Unpack', 'Lower', N, G, LDG, DWORK ) END IF END IF DO 40 I = 1, N WRITE( NOUT, FMT = 99979 ) ( G(I,J), J = 1, N ) 40 CONTINUE ELSE WRITE( NOUT, FMT = 99988 ) END IF IF( .NOT.VEC(6) ) THEN IF( .NOT. BPAR(5) ) THEN ISYMM = ( N * ( N + 1 ) ) / 2 CALL DCOPY( ISYMM, Q, 1, DWORK, 1 ) IF( BPAR(6) ) THEN CALL MA02DD( 'Unpack', 'Upper', N, Q, LDQ, DWORK ) ELSE CALL MA02DD( 'Unpack', 'Lower', N, Q, LDQ, DWORK ) END IF END IF WRITE( NOUT, FMT = 99987 ) DO 50 I = 1, N WRITE( NOUT, FMT = 99979 ) ( Q(I,J), J = 1, N ) 50 CONTINUE ELSE WRITE( NOUT, FMT = 99986 ) END IF IF( VEC(6) ) THEN IF( .NOT.BPAR(5) ) THEN ISYMM = ( P * ( P + 1 ) ) / 2 CALL DCOPY( ISYMM, Q, 1, DWORK, 1 ) IF( BPAR(6) ) THEN CALL MA02DD( 'Unpack', 'Upper', P, Q, LDQ, DWORK ) ELSE CALL MA02DD( 'Unpack', 'Lower', P, Q, LDQ, DWORK ) END IF END IF WRITE( NOUT, FMT = 99985 ) DO 60 I = 1, N WRITE( NOUT, FMT = 99979 ) ( Q(I,J), J = 1, N ) 60 CONTINUE ELSE WRITE( NOUT, FMT = 99984 ) END IF IF( VEC(5) ) THEN IF( .NOT.BPAR(2) ) THEN ISYMM = ( M * ( M + 1 ) ) / 2 CALL DCOPY( ISYMM, G, 1, DWORK, 1 ) IF( BPAR(3) ) THEN CALL MA02DD( 'Unpack', 'Upper', M, G, LDG, DWORK ) ELSE CALL MA02DD( 'Unpack', 'Lower', M, G, LDG, DWORK ) END IF END IF WRITE( NOUT, FMT = 99983 ) DO 70 I = 1, N WRITE( NOUT, FMT = 99979 ) ( G(I,J), J = 1, N ) 70 CONTINUE ELSE WRITE( NOUT, FMT = 99982 ) END IF IF( VEC(9) ) THEN WRITE( NOUT, FMT = 99981 ) DO 80 I = 1, N WRITE( NOUT, FMT = 99979 ) ( X(I,J), J = 1, N ) 80 CONTINUE ELSE WRITE( NOUT, FMT = 99980 ) END IF END IF STOP * 99999 FORMAT (' BB01AD EXAMPLE PROGRAM RESULTS', /1X) 99998 FORMAT (' INFO on exit from BB03AD = ', I3) 99997 FORMAT (/' Order of matrix A: N = ', I3) 99996 FORMAT (' Number of columns in matrix B: M = ', I3) 99995 FORMAT (' Number of rows in matrix C: P = ', I3) 99994 FORMAT (' A = ') 99993 FORMAT (' B = ') 99992 FORMAT (' B is not provided.') 99991 FORMAT (' C = ') 99990 FORMAT (' C is not provided.') 99989 FORMAT (' G = ') 99988 FORMAT (' G is not provided.') 99987 FORMAT (' Q = ') 99986 FORMAT (' Q is not provided.') 99985 FORMAT (' W = ') 99984 FORMAT (' W is not provided.') 99983 FORMAT (' R = ') 99982 FORMAT (' R is not provided.') 99981 FORMAT (' X = ') 99980 FORMAT (' X is not provided.') 99979 FORMAT (20(1X,F8.4)) * END

BB01AD EXAMPLE PROGRAM DATA N 2 3 6 .T. .T. .T. .F. .F. .T. 1 .1234 0

BB01AD EXAMPLE PROGRAM RESULTS Kenney/Laub/Wette 1989, Ex.2: ARE ill conditioned for EPS -> oo Order of matrix A: N = 2 Number of columns in matrix B: M = 1 Number of rows in matrix C: P = 2 A = 0.0000 0.1234 0.0000 0.0000 B is not provided. C = 1.0000 0.0000 0.0000 1.0000 G = 0.0000 0.0000 0.0000 1.0000 Q is not provided. W = 1.0000 0.0000 0.0000 1.0000 R is not provided. X = 9.0486 1.0000 1.0000 1.1166