SUBROUTINE SB03MV( LTRAN, LUPPER, T, LDT, B, LDB, SCALE, X, LDX,
$ XNORM, INFO )
C
C PURPOSE
C
C To solve for the 2-by-2 symmetric matrix X in
C
C op(T)'*X*op(T) - X = SCALE*B,
C
C where T is 2-by-2, B is symmetric 2-by-2, and op(T) = T or T',
C where T' denotes the transpose of T.
C
C ARGUMENTS
C
C Mode Parameters
C
C LTRAN LOGICAL
C Specifies the form of op(T) to be used, as follows:
C = .FALSE.: op(T) = T,
C = .TRUE. : op(T) = T'.
C
C LUPPER LOGICAL
C Specifies which triangle of the matrix B is used, and
C which triangle of the matrix X is computed, as follows:
C = .TRUE. : The upper triangular part;
C = .FALSE.: The lower triangular part.
C
C Input/Output Parameters
C
C T (input) DOUBLE PRECISION array, dimension (LDT,2)
C The leading 2-by-2 part of this array must contain the
C matrix T.
C
C LDT INTEGER
C The leading dimension of array T. LDT >= 2.
C
C B (input) DOUBLE PRECISION array, dimension (LDB,2)
C On entry with LUPPER = .TRUE., the leading 2-by-2 upper
C triangular part of this array must contain the upper
C triangular part of the symmetric matrix B and the strictly
C lower triangular part of B is not referenced.
C On entry with LUPPER = .FALSE., the leading 2-by-2 lower
C triangular part of this array must contain the lower
C triangular part of the symmetric matrix B and the strictly
C upper triangular part of B is not referenced.
C
C LDB INTEGER
C The leading dimension of array B. LDB >= 2.
C
C SCALE (output) DOUBLE PRECISION
C The scale factor. SCALE is chosen less than or equal to 1
C to prevent the solution overflowing.
C
C X (output) DOUBLE PRECISION array, dimension (LDX,2)
C On exit with LUPPER = .TRUE., the leading 2-by-2 upper
C triangular part of this array contains the upper
C triangular part of the symmetric solution matrix X and the
C strictly lower triangular part of X is not referenced.
C On exit with LUPPER = .FALSE., the leading 2-by-2 lower
C triangular part of this array contains the lower
C triangular part of the symmetric solution matrix X and the
C strictly upper triangular part of X is not referenced.
C Note that X may be identified with B in the calling
C statement.
C
C LDX INTEGER
C The leading dimension of array X. LDX >= 2.
C
C XNORM (output) DOUBLE PRECISION
C The infinity-norm of the solution.
C
C Error Indicator
C
C INFO INTEGER
C = 0: successful exit;
C = 1: if T has almost reciprocal eigenvalues, so T
C is perturbed to get a nonsingular equation.
C
C NOTE: In the interests of speed, this routine does not
C check the inputs for errors.
C
C METHOD
C
C The equivalent linear algebraic system of equations is formed and
C solved using Gaussian elimination with complete pivoting.
C
C REFERENCES
C
C [1] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
C Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
C Ostrouchov, S., and Sorensen, D.
C LAPACK Users' Guide: Second Edition.
C SIAM, Philadelphia, 1995.
C
C NUMERICAL ASPECTS
C
C The algorithm is stable and reliable, since Gaussian elimination
C with complete pivoting is used.
C
C CONTRIBUTOR
C
C Release 3.0: V. Sima, Katholieke Univ. Leuven, Belgium, May 1997.
C Based on DLALD2 by P. Petkov, Tech. University of Sofia, September
C 1993.
C
C REVISIONS
C
C -
C
C KEYWORDS
C
C Discrete-time system, Lyapunov equation, matrix algebra.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO, ONE, FOUR
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, FOUR = 4.0D+0 )
C ..
C .. Scalar Arguments ..
LOGICAL LTRAN, LUPPER
INTEGER INFO, LDB, LDT, LDX
DOUBLE PRECISION SCALE, XNORM
C ..
C .. Array Arguments ..
DOUBLE PRECISION B( LDB, * ), T( LDT, * ), X( LDX, * )
C ..
C .. Local Scalars ..
INTEGER I, IP, IPSV, J, JP, JPSV, K
DOUBLE PRECISION EPS, SMIN, SMLNUM, TEMP, XMAX
C ..
C .. Local Arrays ..
INTEGER JPIV( 3 )
DOUBLE PRECISION BTMP( 3 ), T9( 3, 3 ), TMP( 3 )
C ..
C .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
C ..
C .. External Subroutines ..
EXTERNAL DSWAP
C ..
C .. Intrinsic Functions ..
INTRINSIC ABS, MAX
C ..
C .. Executable Statements ..
C
C Do not check the input parameters for errors.
C
INFO = 0
C
C Set constants to control overflow.
C
EPS = DLAMCH( 'P' )
SMLNUM = DLAMCH( 'S' ) / EPS
C
C Solve equivalent 3-by-3 system using complete pivoting.
C Set pivots less than SMIN to SMIN.
C
SMIN = MAX( ABS( T( 1, 1 ) ), ABS( T( 1, 2 ) ),
$ ABS( T( 2, 1 ) ), ABS( T( 2, 2 ) ) )
SMIN = MAX( EPS*SMIN, SMLNUM )
T9( 1, 1 ) = T( 1, 1 )*T( 1, 1 ) - ONE
T9( 2, 2 ) = T( 1, 1 )*T( 2, 2 ) + T( 1, 2 )*T( 2, 1 ) - ONE
T9( 3, 3 ) = T( 2, 2 )*T( 2, 2 ) - ONE
IF( LTRAN ) THEN
T9( 1, 2 ) = T( 1, 1 )*T( 1, 2 ) + T( 1, 1 )*T( 1, 2 )
T9( 1, 3 ) = T( 1, 2 )*T( 1, 2 )
T9( 2, 1 ) = T( 1, 1 )*T( 2, 1 )
T9( 2, 3 ) = T( 1, 2 )*T( 2, 2 )
T9( 3, 1 ) = T( 2, 1 )*T( 2, 1 )
T9( 3, 2 ) = T( 2, 1 )*T( 2, 2 ) + T( 2, 1 )*T( 2, 2 )
ELSE
T9( 1, 2 ) = T( 1, 1 )*T( 2, 1 ) + T( 1, 1 )*T( 2, 1 )
T9( 1, 3 ) = T( 2, 1 )*T( 2, 1 )
T9( 2, 1 ) = T( 1, 1 )*T( 1, 2 )
T9( 2, 3 ) = T( 2, 1 )*T( 2, 2 )
T9( 3, 1 ) = T( 1, 2 )*T( 1, 2 )
T9( 3, 2 ) = T( 1, 2 )*T( 2, 2 ) + T( 1, 2 )*T( 2, 2 )
END IF
BTMP( 1 ) = B( 1, 1 )
IF ( LUPPER ) THEN
BTMP( 2 ) = B( 1, 2 )
ELSE
BTMP( 2 ) = B( 2, 1 )
END IF
BTMP( 3 ) = B( 2, 2 )
C
C Perform elimination.
C
DO 50 I = 1, 2
XMAX = ZERO
C
DO 20 IP = I, 3
C
DO 10 JP = I, 3
IF( ABS( T9( IP, JP ) ).GE.XMAX ) THEN
XMAX = ABS( T9( IP, JP ) )
IPSV = IP
JPSV = JP
END IF
10 CONTINUE
C
20 CONTINUE
C
IF( IPSV.NE.I ) THEN
CALL DSWAP( 3, T9( IPSV, 1 ), 3, T9( I, 1 ), 3 )
TEMP = BTMP( I )
BTMP( I ) = BTMP( IPSV )
BTMP( IPSV ) = TEMP
END IF
IF( JPSV.NE.I )
$ CALL DSWAP( 3, T9( 1, JPSV ), 1, T9( 1, I ), 1 )
JPIV( I ) = JPSV
IF( ABS( T9( I, I ) ).LT.SMIN ) THEN
INFO = 1
T9( I, I ) = SMIN
END IF
C
DO 40 J = I + 1, 3
T9( J, I ) = T9( J, I ) / T9( I, I )
BTMP( J ) = BTMP( J ) - T9( J, I )*BTMP( I )
C
DO 30 K = I + 1, 3
T9( J, K ) = T9( J, K ) - T9( J, I )*T9( I, K )
30 CONTINUE
C
40 CONTINUE
C
50 CONTINUE
C
IF( ABS( T9( 3, 3 ) ).LT.SMIN )
$ T9( 3, 3 ) = SMIN
SCALE = ONE
IF( ( FOUR*SMLNUM )*ABS( BTMP( 1 ) ).GT.ABS( T9( 1, 1 ) ) .OR.
$ ( FOUR*SMLNUM )*ABS( BTMP( 2 ) ).GT.ABS( T9( 2, 2 ) ) .OR.
$ ( FOUR*SMLNUM )*ABS( BTMP( 3 ) ).GT.ABS( T9( 3, 3 ) ) ) THEN
SCALE = ( ONE / FOUR ) / MAX( ABS( BTMP( 1 ) ),
$ ABS( BTMP( 2 ) ), ABS( BTMP( 3 ) ) )
BTMP( 1 ) = BTMP( 1 )*SCALE
BTMP( 2 ) = BTMP( 2 )*SCALE
BTMP( 3 ) = BTMP( 3 )*SCALE
END IF
C
DO 70 I = 1, 3
K = 4 - I
TEMP = ONE / T9( K, K )
TMP( K ) = BTMP( K )*TEMP
C
DO 60 J = K + 1, 3
TMP( K ) = TMP( K ) - ( TEMP*T9( K, J ) )*TMP( J )
60 CONTINUE
C
70 CONTINUE
C
DO 80 I = 1, 2
IF( JPIV( 3-I ).NE.3-I ) THEN
TEMP = TMP( 3-I )
TMP( 3-I ) = TMP( JPIV( 3-I ) )
TMP( JPIV( 3-I ) ) = TEMP
END IF
80 CONTINUE
C
X( 1, 1 ) = TMP( 1 )
IF ( LUPPER ) THEN
X( 1, 2 ) = TMP( 2 )
ELSE
X( 2, 1 ) = TMP( 2 )
END IF
X( 2, 2 ) = TMP( 3 )
XNORM = MAX( ABS( TMP( 1 ) ) + ABS( TMP( 2 ) ),
$ ABS( TMP( 2 ) ) + ABS( TMP( 3 ) ) )
C
RETURN
C *** Last line of SB03MV ***
END