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F 4.10.2 Systems with Tridiagonal Symmetric Strongly Nonsingular Matrices


      SUBROUTINE TRDSY (N,DM,DU,RS,X,MARK)
C
C*****************************************************************
C                                                                *
C     Solving a linear system of equations                       *
C                  A * X = RS                                    *
C     for a tridiagonal, symmetric, positive definite matrix A.  *
C     The matrix A is defined by the two N-vectors DM and DU.    *
C     The system of equations is given as follows:               *
C                                                                *
C     DM(1) * X(1) + DU(1) * X(2)                      = RS(1)   *
C                                                                *
C     DU(I-1) * X(I-1) + DM(I) * X(I) + DU(I) * X(I+1) = RS(I)   *
C            for I = 2, ... ,N-1, and                            *
C                                                                *
C     DU(N-1) * X(N-1) + DM(N) * X(N)                  = RS(N)   *
C                                                                *
C                                                                *
C                                                                *
C     INPUT PARAMETERS:                                          *
C     =================                                          *
C     N    : number of equations, N > 2                          *
C     DM   : N-vector DM(1:N); main diagonal of A                *
C            DM(1), DM(2), ... , DM(N)                           *
C     DU   : N-vector DU(1:N); co-diagonal of A                  *
C            DU(1), DU(2), ... , DU(N-1)                         *
C     RS   : N-vector X(1:N); the right hand side                *
C                                                                *
C                                                                *
C     OUTPUT PARAMETERS:                                         *
C     ==================                                         *
C     DM   :)                                                    *
C     DU   :) overwritten with auxiliary vectors                 *
C     RS   :)                                                    *
C     X    : N-vector X(1:N) containing the solution of the      *
C            system of equations                                 *
C     MARK : error parameter                                     *
C            MARK= 1 : ok                                        *
C            MARK= 0 : numerically the matrix A is not strongly  *
C                      nonsingular                               *
C            MARK=-1 : A is not positive definite.               *
C            MARK=-2 : condition N > 2 is not satisfied          *
C                                                                *
C     NOTE: If MARK = 1, then the determinant of A can be        *
C           calculated as:                                       *
C              DET A = DM(1) * DM(2) * ... * DM(N)               *
C                                                                *
C----------------------------------------------------------------*
C                                                                *
C  subroutines required: TRDSYP, TRDSYS, MACHPD                  *
C                                                                *
C*****************************************************************
C                                                                *
C  author     : Gisela Engeln-Muellges                           *
C  date       : 25.04.1988                                       *
C  source     : FORTRAN 77                                       *
C                                                                *
C*****************************************************************
C
      IMPLICIT DOUBLE PRECISION (A-H,O-Z)
      DOUBLE PRECISION DM(1:N),DU(1:N),RS(1:N),X(1:N)
      MARK = -2
      IF (N .LT. 3) RETURN
C
C  Factoring A
C
      CALL TRDSYP (N,DM,DU,MARK)
C
C  if MARK = 1 update and backsubstitute
C
      IF (MARK .EQ. 1) THEN
         CALL TRDSYS (N,DM,DU,RS,X)
      ENDIF
      RETURN
      END
C
C

      SUBROUTINE TRDSYP (N,DM,DU,MARK)
C
C*****************************************************************
C                                                                *
C     Factoring a tridiagonal, symmetric, and positive definite  *
C     matrix A, that is given by the two N-vectors DM and DU,    *
C     into the product A = R(TRANSP) * D * R  for a unit upper   *
C     triangular matrix R by applying the Cholesky-method for    *
C     tridiagonal matrices. The form of the system matrix A is   *
C     identical with the one desribed in SUBROUTINE TRDSY.       *
C                                                                *
C                                                                *
C     INPUT PARAMETERS:                                          *
C     =================                                          *
C     N    : number of equations, N > 2                          *
C     DM   : N-vector DM(1:N); main diagonal of A                *
C            DM(1), DM(2), ... , DM(N)                           *
C     DU   : N-vector DU(1:N); co-diagonal of A                  *
C            DU(1), DU(2), ... , DU(N-1);                        *
C            due to symmetry of A its lower and upper            *
C            co-diagonals coincide                               *
C                                                                *
C                                                                *
C     OUTPUT PARAMETERS:                                         *
C     ==================                                         *
C     DM   :) overwritten with auxiliary vectors containing the  *
C     DU   :) factors of A. The co-diagonal of the unit upper    *
C             triangular bidiagonal matrix R is stored in DU,    *
C             while the diagonal matrix D is stored in DM.       *
C     MARK : error parameter                                     *
C            MARK= 1 : ok                                        *
C            MARK= 0 : numerically the matrix A is not strongly  *
C                      nonsingular.                              *
C            MARK=-1 : A is not positive definite.               *
C            MARK=-2 : condition N > 2 is not met.               *
C                                                                *
C----------------------------------------------------------------*
C                                                                *
C  subroutines required: MACHPD                                  *
C                                                                *
C*****************************************************************
C                                                                *
C  author     : Gisela Engeln-Muellges                           *
C  date       : 25.04.1988                                       *
C  source     : FORTRAN 77                                       *
C                                                                *
C*****************************************************************
C
      IMPLICIT DOUBLE PRECISION (A-H,O-Z)
      DOUBLE PRECISION DM(1:N),DU(1:N)
C
C   calculating the machine constant
C
      FMACHP = 1.0D0
   10 FMACHP = 0.5D0 * FMACHP
      IF (MACHPD(1.0D0+FMACHP) .EQ. 1) GOTO 10
      FMACHP = FMACHP * 2.0D0
C
C   determining the relative error bound
C
      EPS = 4.0D0 * FMACHP
C
C   checking whether N > 2
C
      MARK = -2
      IF (N .LT. 3) RETURN
      DU(N) = 0.0D0
C
C   testing for a positive definite, strong nonsingular matrix A
C   for N=1
C
      ROW = DABS(DM(1)) + DABS(DU(1))
      IF (ROW .EQ. 0.0D0) THEN
         MARK = 0
         RETURN
      ENDIF
      D = 1.0D0/ROW
      IF (DM(1) .LT. 0.0D0) THEN
         MARK = -1
         RETURN
      ELSEIF (DABS(DM(1))*D .LE. EPS) THEN
         MARK = 0
         RETURN
      ENDIF
C
C   factoring A while checking for a positive definite, strong
C   nonsingular matrix A
C
      DUMMY = DU(1)
      DU(1) = DU(1)/DM(1)
      DO 20 I=2,N,1
         ROW = (DABS (DM(I)) + DABS(DU(I)) + DABS(DUMMY))
         IF (ROW .EQ. 0.0D0) THEN
            MARK = 0
            RETURN
         ENDIF
         D = 1.0D0/ROW
         DM(I) = DM(I) - DUMMY * DU(I-1)
         IF (DM(I) .LT. 0.0D0) THEN
            MARK = -1
            RETURN
         ELSEIF (DABS(DM(I))*D .LE. EPS) THEN
            MARK = 0
            RETURN
         ENDIF
         IF (I .LT. N) THEN
            DUMMY = DU(I)
            DU(I) = DU(I)/DM(I)
         ENDIF
   20 CONTINUE
      MARK=1
      RETURN
      END
C
C

      SUBROUTINE TRDSYS (N,DM,DU,RS,X)
C
C*****************************************************************
C                                                                *
C     Solving a linear system of equations                       *
C                  A * X = RS                                    *
C     for a tridiagonal, symmetric, and positive definite matrix *
C     A, whose tridiagonal factors have been calculated by the   *
C     SUBROUTINE TRDSYP.                                         *
C     Here the factoring matrices D and R are                    *
C     used as input matrices and they are stored in the two      *
C     N-vectors DM and DU, respectively.                         *
C                                                                *
C                                                                *
C     INPUT PARAMETERS:                                          *
C     =================                                          *
C     N    : number of equations, N > 2                          *
C     DM   : N-vector DM(1:N); the diagonal matrix D             *
C     DU   : N-vector DU(1:N); the upper co-diagonal entries of R*
C     RS   : N-vector RS(1:N); the right hand side               *
C                                                                *
C                                                                *
C     OUTPUT PARAMETER:                                          *
C     =================                                          *
C     X    : N-vector X(1:N) containing the solution of the      *
C            system of equations                                 *
C                                                                *
C----------------------------------------------------------------*
C                                                                *
C  subroutines required: TRDSYP, TRDSYS                          *
C                                                                *
C*****************************************************************
C                                                                *
C  author     : Gisela Engeln-Muellges                           *
C  date       : 25.04.1988                                       *
C  source     : FORTRAN 77                                       *
C                                                                *
C*****************************************************************
C
      IMPLICIT DOUBLE PRECISION (A-H,O-Z)
      DOUBLE PRECISION DM(1:N),DU(1:N),RS(1:N),X(1:N)
C
C  updating
C
      DUMMY = RS(1)
      RS(1) = DUMMY/DM(1)
      DO 10 I=2,N,1
         DUMMY = RS(I) - DU(I-1) * DUMMY
         RS(I) = DUMMY/DM(I)
   10 CONTINUE
C
C  backsubstitution
C
      X(N) = RS(N)
      DO 20 I=N-1,1,-1
         X(I) = RS(I) - DU(I) * X(I+1)
   20 CONTINUE
      RETURN
      END


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