matrix.f

CLM集合卡曼滤波数据同化算法

      if(incx.lt.0)ix = (-n+1)*incx + 1
      if(incy.lt.0)iy = (-n+1)*incy + 1
      do 10 i = 1,n
        sy(iy) = sy(iy) + sa*sx(ix)
        ix = ix + incx
        iy = iy + incy
   10 continue
      return
c
c        code for both increments equal to 1
c
c
c        clean-up loop
c
   20 m = mod(n,4)
      if( m .eq. 0 ) go to 40
      do 30 i = 1,m
        sy(i) = sy(i) + sa*sx(i)
   30 continue
      if( n .lt. 4 ) return
   40 mp1 = m + 1
      do 50 i = mp1,n,4
        sy(i) = sy(i) + sa*sx(i)
        sy(i + 1) = sy(i + 1) + sa*sx(i + 1)
        sy(i + 2) = sy(i + 2) + sa*sx(i + 2)
        sy(i + 3) = sy(i + 3) + sa*sx(i + 3)
   50 continue
      return
      end
!========================================================================

      real function sdot(n,sx,incx,sy,incy)
c
c     forms the dot product of two vectors.
c     uses unrolled loops for increments equal to one.
c     jack dongarra, linpack, 3/11/78.
c     modified 12/3/93, array(1) declarations changed to array(*)
c
      real sx(*),sy(*),stemp
      integer i,incx,incy,ix,iy,m,mp1,n
c
      stemp = 0.0e0
      sdot = 0.0e0
      if(n.le.0)return
      if(incx.eq.1.and.incy.eq.1)go to 20
c
c        code for unequal increments or equal increments
c          not equal to 1
c
      ix = 1
      iy = 1
      if(incx.lt.0)ix = (-n+1)*incx + 1
      if(incy.lt.0)iy = (-n+1)*incy + 1
      do 10 i = 1,n
        stemp = stemp + sx(ix)*sy(iy)
        ix = ix + incx
        iy = iy + incy
   10 continue
      sdot = stemp
      return
c
c        code for both increments equal to 1
c
c
c        clean-up loop
c
   20 m = mod(n,5)
      if( m .eq. 0 ) go to 40
      do 30 i = 1,m
        stemp = stemp + sx(i)*sy(i)
   30 continue
      if( n .lt. 5 ) go to 60
   40 mp1 = m + 1
      do 50 i = mp1,n,5
        stemp = stemp + sx(i)*sy(i) + sx(i + 1)*sy(i + 1) +
     *   sx(i + 2)*sy(i + 2) + sx(i + 3)*sy(i + 3) + sx(i + 4)*sy(i + 4)
   50 continue
   60 sdot = stemp
      return
      end
!===========================================================================
      SUBROUTINE SGEMV ( TRANS, M, N, ALPHA, A, LDA, X, INCX,
     $                   BETA, Y, INCY )
*     .. Scalar Arguments ..
      REAL               ALPHA, BETA
      INTEGER            INCX, INCY, LDA, M, N
      CHARACTER*1        TRANS
*     .. Array Arguments ..
      REAL               A( LDA, * ), X( * ), Y( * )
*     ..
*
*  Purpose
*  =======
*
*  SGEMV  performs one of the matrix-vector operations
*
*     y := alpha*A*x + beta*y,   or   y := alpha*A'*x + beta*y,
*
*  where alpha and beta are scalars, x and y are vectors and A is an
*  m by n matrix.
*
*  Parameters
*  ==========
*
*  TRANS  - CHARACTER*1.
*           On entry, TRANS specifies the operation to be performed as
*           follows:
*
*              TRANS = 'N' or 'n'   y := alpha*A*x + beta*y.
*
*              TRANS = 'T' or 't'   y := alpha*A'*x + beta*y.
*
*              TRANS = 'C' or 'c'   y := alpha*A'*x + beta*y.
*
*           Unchanged on exit.
*
*  M      - INTEGER.
*           On entry, M specifies the number of rows of the matrix A.
*           M must be at least zero.
*           Unchanged on exit.
*
*  N      - INTEGER.
*           On entry, N specifies the number of columns of the matrix A.
*           N must be at least zero.
*           Unchanged on exit.
*
*  ALPHA  - REAL            .
*           On entry, ALPHA specifies the scalar alpha.
*           Unchanged on exit.
*
*  A      - REAL             array of DIMENSION ( LDA, n ).
*           Before entry, the leading m by n part of the array A must
*           contain the matrix of coefficients.
*           Unchanged on exit.
*
*  LDA    - INTEGER.
*           On entry, LDA specifies the first dimension of A as declared
*           in the calling (sub) program. LDA must be at least
*           max( 1, m ).
*           Unchanged on exit.
*
*  X      - REAL             array of DIMENSION at least
*           ( 1 + ( n - 1 )*abs( INCX ) ) when TRANS = 'N' or 'n'
*           and at least
*           ( 1 + ( m - 1 )*abs( INCX ) ) otherwise.
*           Before entry, the incremented array X must contain the
*           vector x.
*           Unchanged on exit.
*
*  INCX   - INTEGER.
*           On entry, INCX specifies the increment for the elements of
*           X. INCX must not be zero.
*           Unchanged on exit.
*
*  BETA   - REAL            .
*           On entry, BETA specifies the scalar beta. When BETA is
*           supplied as zero then Y need not be set on input.
*           Unchanged on exit.
*
*  Y      - REAL             array of DIMENSION at least
*           ( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = 'N' or 'n'
*           and at least
*           ( 1 + ( n - 1 )*abs( INCY ) ) otherwise.
*           Before entry with BETA non-zero, the incremented array Y
*           must contain the vector y. On exit, Y is overwritten by the
*           updated vector y.
*
*  INCY   - INTEGER.
*           On entry, INCY specifies the increment for the elements of
*           Y. INCY must not be zero.
*           Unchanged on exit.
*
*
*  Level 2 Blas routine.
*
*  -- Written on 22-October-1986.
*     Jack Dongarra, Argonne National Lab.
*     Jeremy Du Croz, Nag Central Office.
*     Sven Hammarling, Nag Central Office.
*     Richard Hanson, Sandia National Labs.
*
*
*     .. Parameters ..
      REAL               ONE         , ZERO
      PARAMETER        ( ONE = 1.0E+0, ZERO = 0.0E+0 )
*     .. Local Scalars ..
      REAL               TEMP
      INTEGER            I, INFO, IX, IY, J, JX, JY, KX, KY, LENX, LENY
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      IF     ( .NOT.LSAME( TRANS, 'N' ).AND.
     $         .NOT.LSAME( TRANS, 'T' ).AND.
     $         .NOT.LSAME( TRANS, 'C' )      )THEN
         INFO = 1
      ELSE IF( M.LT.0 )THEN
         INFO = 2
      ELSE IF( N.LT.0 )THEN
         INFO = 3
      ELSE IF( LDA.LT.MAX( 1, M ) )THEN
         INFO = 6
      ELSE IF( INCX.EQ.0 )THEN
         INFO = 8
      ELSE IF( INCY.EQ.0 )THEN
         INFO = 11
      END IF
      IF( INFO.NE.0 )THEN
         CALL XERBLA( 'SGEMV ', INFO )
         RETURN
      END IF
*
*     Quick return if possible.
*
      IF( ( M.EQ.0 ).OR.( N.EQ.0 ).OR.
     $    ( ( ALPHA.EQ.ZERO ).AND.( BETA.EQ.ONE ) ) )
     $   RETURN
*
*     Set  LENX  and  LENY, the lengths of the vectors x and y, and set
*     up the start points in  X  and  Y.
*
      IF( LSAME( TRANS, 'N' ) )THEN
         LENX = N
         LENY = M
      ELSE
         LENX = M
         LENY = N
      END IF
      IF( INCX.GT.0 )THEN
         KX = 1
      ELSE
         KX = 1 - ( LENX - 1 )*INCX
      END IF
      IF( INCY.GT.0 )THEN
         KY = 1
      ELSE
         KY = 1 - ( LENY - 1 )*INCY
      END IF
*
*     Start the operations. In this version the elements of A are
*     accessed sequentially with one pass through A.
*
*     First form  y := beta*y.
*
      IF( BETA.NE.ONE )THEN
         IF( INCY.EQ.1 )THEN
            IF( BETA.EQ.ZERO )THEN
               DO 10, I = 1, LENY
                  Y( I ) = ZERO
   10          CONTINUE
            ELSE
               DO 20, I = 1, LENY
                  Y( I ) = BETA*Y( I )
   20          CONTINUE
            END IF
         ELSE
            IY = KY
            IF( BETA.EQ.ZERO )THEN
               DO 30, I = 1, LENY
                  Y( IY ) = ZERO
                  IY      = IY   + INCY
   30          CONTINUE
            ELSE
               DO 40, I = 1, LENY
                  Y( IY ) = BETA*Y( IY )
                  IY      = IY           + INCY
   40          CONTINUE
            END IF
         END IF
      END IF
      IF( ALPHA.EQ.ZERO )
     $   RETURN
      IF( LSAME( TRANS, 'N' ) )THEN
*
*        Form  y := alpha*A*x + y.
*
         JX = KX
         IF( INCY.EQ.1 )THEN
            DO 60, J = 1, N
               IF( X( JX ).NE.ZERO )THEN
                  TEMP = ALPHA*X( JX )
                  DO 50, I = 1, M
                     Y( I ) = Y( I ) + TEMP*A( I, J )
   50             CONTINUE
               END IF
               JX = JX + INCX
   60       CONTINUE
         ELSE
            DO 80, J = 1, N
               IF( X( JX ).NE.ZERO )THEN
                  TEMP = ALPHA*X( JX )
                  IY   = KY
                  DO 70, I = 1, M
                     Y( IY ) = Y( IY ) + TEMP*A( I, J )
                     IY      = IY      + INCY
   70             CONTINUE
               END IF
               JX = JX + INCX
   80       CONTINUE
         END IF
      ELSE
*
*        Form  y := alpha*A'*x + y.
*
         JY = KY
         IF( INCX.EQ.1 )THEN
            DO 100, J = 1, N
               TEMP = ZERO
               DO 90, I = 1, M
                  TEMP = TEMP + A( I, J )*X( I )
   90          CONTINUE
               Y( JY ) = Y( JY ) + ALPHA*TEMP
               JY      = JY      + INCY
  100       CONTINUE
         ELSE
            DO 120, J = 1, N
               TEMP = ZERO
               IX   = KX
               DO 110, I = 1, M
                  TEMP = TEMP + A( I, J )*X( IX )
                  IX   = IX   + INCX
  110          CONTINUE
               Y( JY ) = Y( JY ) + ALPHA*TEMP
               JY      = JY      + INCY
  120       CONTINUE
         END IF
      END IF
*
      RETURN
*
*     End of SGEMV .
*
      END
!==========================================================================

      SUBROUTINE SLAEBZ( IJOB, NITMAX, N, MMAX, MINP, NBMIN, ABSTOL,
     $                   RELTOL, PIVMIN, D, E, E2, NVAL, AB, C, MOUT,
     $                   NAB, WORK, IWORK, INFO )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      INTEGER            IJOB, INFO, MINP, MMAX, MOUT, N, NBMIN, NITMAX
      REAL               ABSTOL, PIVMIN, RELTOL
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * ), NAB( MMAX, * ), NVAL( * )
      REAL               AB( MMAX, * ), C( * ), D( * ), E( * ), E2( * ),
     $                   WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  SLAEBZ contains the iteration loops which compute and use the
*  function N(w), which is the count of eigenvalues of a symmetric
*  tridiagonal matrix T less than or equal to its argument  w.  It
*  performs a choice of two types of loops:
*
*  IJOB=1, followed by
*  IJOB=2: It takes as input a list of intervals and returns a list of
*          sufficiently small intervals whose union contains the same
*          eigenvalues as the union of the original intervals.
*          The input intervals are (AB(j,1),AB(j,2)], j=1,...,MINP.
*          The output interval (AB(j,1),AB(j,2)] will contain
*          eigenvalues NAB(j,1)+1,...,NAB(j,2), where 1 <= j <= MOUT.
*
*  IJOB=3: It performs a binary search in each input interval
*          (AB(j,1),AB(j,2)] for a point  w(j)  such that
*          N(w(j))=NVAL(j), and uses  C(j)  as the starting point of
*          the search.  If such a w(j) is found, then on output
*          AB(j,1)=AB(j,2)=w.  If no such w(j) is found, then on output
*          (AB(j,1),AB(j,2)] will be a small interval containing the
*          point where N(w) jumps through NVAL(j), unless that point
*          lies outside the initial interval.
*
*  Note that the intervals are in all cases half-open intervals,
*  i.e., of the form  (a,b] , which includes  b  but not  a .
*
*  To avoid underflow, the matrix should be scaled so that its largest
*  element is no greater than  overflow**(1/2) * underflow**(1/4)
*  in absolute value.  To assure the most accurate computation
*  of small eigenvalues, the matrix should be scaled to be
*  not much smaller than that, either.
*
*  See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal
*  Matrix", Report CS41, Computer Science Dept., Stanford
*  University, July 21, 1966
*
*  Note: the arguments are, in general, *not* checked for unreasonable
*  values.
*
*  Arguments
*  =========
*
*  IJOB    (input) INTEGER
*          Specifies what is to be done:
*          = 1:  Compute NAB for the initial intervals.
*          = 2:  Perform bisection iteration to find eigenvalues of T.
*          = 3:  Perform bisection iteration to invert N(w), i.e.,
*                to find a point which has a specified number of
*                eigenvalues of T to its left.
*          Other values will cause SLAEBZ to return with INFO=-1.
*
*  NITMAX  (input) INTEGER
*          The maximum number of "levels" of bisection to be
*          performed, i.e., an interval of width W will not be made
*          smaller than 2^(-NITMAX) * W.  If not all intervals
*          have converged after NITMAX iterations, then INFO is set
*          to the number of non-converged intervals.
*