SUBROUTINE SB04MR( M, D, IPR, INFO )
C
C PURPOSE
C
C To solve a linear algebraic system of order M whose coefficient
C matrix has zeros below the second subdiagonal. The matrix is
C stored compactly, row-wise.
C
C ARGUMENTS
C
C Input/Output Parameters
C
C M (input) INTEGER
C The order of the system. M >= 0.
C Note that parameter M should have twice the value in the
C original problem (see SLICOT Library routine SB04MU).
C
C D (input/output) DOUBLE PRECISION array, dimension
C (M*(M+1)/2+3*M)
C On entry, the first M*(M+1)/2 + 2*M elements of this array
C must contain the coefficient matrix, stored compactly,
C row-wise, and the next M elements must contain the right
C hand side of the linear system, as set by SLICOT Library
C routine SB04MU.
C On exit, the content of this array is updated, the last M
C elements containing the solution with components
C interchanged (see IPR).
C
C IPR (output) INTEGER array, dimension (2*M)
C The leading M elements contain information about the
C row interchanges performed for solving the system.
C Specifically, the i-th component of the solution is
C specified by IPR(i).
C
C Error Indicator
C
C INFO INTEGER
C = 0: successful exit;
C = 1: if a singular matrix was encountered.
C
C METHOD
C
C Gaussian elimination with partial pivoting is used. The rows of
C the matrix are not actually permuted, only their indices are
C interchanged in array IPR.
C
C REFERENCES
C
C [1] Golub, G.H., Nash, S. and Van Loan, C.F.
C A Hessenberg-Schur method for the problem AX + XB = C.
C IEEE Trans. Auto. Contr., AC-24, pp. 909-913, 1979.
C
C NUMERICAL ASPECTS
C
C None.
C
C CONTRIBUTORS
C
C Release 3.0: V. Sima, Katholieke Univ. Leuven, Belgium, Sep. 1997.
C Supersedes Release 2.0 routine SB04AR by G. Golub, S. Nash, and
C C. Van Loan, Stanford University, California, United States of
C America, January 1982.
C
C REVISIONS
C
C -
C
C KEYWORDS
C
C Hessenberg form, orthogonal transformation, real Schur form,
C Sylvester equation.
C
C ******************************************************************
C
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
C .. Scalar Arguments ..
INTEGER INFO, M
C .. Array Arguments ..
INTEGER IPR(*)
DOUBLE PRECISION D(*)
C .. Local Scalars ..
INTEGER I, I1, I2, IPRM, IPRM1, J, K, L, M1, MPI, MPI1,
$ MPI2
DOUBLE PRECISION D1, D2, D3, DMAX
C .. External Subroutines ..
EXTERNAL DAXPY
C .. Intrinsic Functions ..
INTRINSIC ABS
C .. Executable Statements ..
C
INFO = 0
I2 = ( M*( M + 5 ) )/2
MPI = M
IPRM = I2
M1 = M
I1 = 1
C
DO 20 I = 1, M
MPI = MPI + 1
IPRM = IPRM + 1
IPR(MPI) = I1
IPR(I) = IPRM
I1 = I1 + M1
IF ( I.GE.3 ) M1 = M1 - 1
20 CONTINUE
C
M1 = M - 1
MPI1 = M + 1
C
C Reduce to upper triangular form.
C
DO 80 I = 1, M1
MPI = MPI1
MPI1 = MPI1 + 1
IPRM = IPR(MPI)
D1 = D(IPRM)
I1 = 2
IF ( I.EQ.M1 ) I1 = 1
MPI2 = MPI + I1
L = 0
DMAX = ABS( D1 )
C
DO 40 J = MPI1, MPI2
D2 = D(IPR(J))
D3 = ABS( D2 )
IF ( D3.GT.DMAX ) THEN
DMAX = D3
D1 = D2
L = J - MPI
END IF
40 CONTINUE
C
C Check singularity.
C
IF ( DMAX.EQ.ZERO ) THEN
INFO = 1
RETURN
END IF
C
IF ( L.GT.0 ) THEN
C
C Permute the row indices.
C
K = IPRM
J = MPI + L
IPRM = IPR(J)
IPR(J) = K
IPR(MPI) = IPRM
K = IPR(I)
I2 = I + L
IPR(I) = IPR(I2)
IPR(I2) = K
END IF
IPRM = IPRM + 1
C
C Annihilate the subdiagonal elements of the matrix.
C
I2 = I
D3 = D(IPR(I))
C
DO 60 J = MPI1, MPI2
I2 = I2 + 1
IPRM1 = IPR(J)
DMAX = -D(IPRM1)/D1
D(IPR(I2)) = D(IPR(I2)) + DMAX*D3
CALL DAXPY( M-I, DMAX, D(IPRM), 1, D(IPRM1+1), 1 )
60 CONTINUE
C
IPR(MPI1) = IPR(MPI1) + 1
IF ( I.NE.M1 ) IPR(MPI2) = IPR(MPI2) + 1
80 CONTINUE
C
MPI = M + M
IPRM = IPR(MPI)
C
C Check singularity.
C
IF ( D(IPRM).EQ.ZERO ) THEN
INFO = 1
RETURN
END IF
C
C Back substitution.
C
D(IPR(M)) = D(IPR(M))/D(IPRM)
C
DO 120 I = M1, 1, -1
MPI = MPI - 1
IPRM = IPR(MPI)
IPRM1 = IPRM
DMAX = ZERO
C
DO 100 K = I+1, M
IPRM1 = IPRM1 + 1
DMAX = DMAX + D(IPR(K))*D(IPRM1)
100 CONTINUE
C
D(IPR(I)) = ( D(IPR(I)) - DMAX )/D(IPRM)
120 CONTINUE
C
RETURN
C *** Last line of SB04MR ***
END