OpenLibm/slatec/dcv.f

*DECK DCV
      DOUBLE PRECISION FUNCTION DCV (XVAL, NDATA, NCONST, NORD, NBKPT,
     +   BKPT, W)
C***BEGIN PROLOGUE  DCV
C***PURPOSE  Evaluate the variance function of the curve obtained
C            by the constrained B-spline fitting subprogram DFC.
C***LIBRARY   SLATEC
C***CATEGORY  L7A3
C***TYPE      DOUBLE PRECISION (CV-S, DCV-D)
C***KEYWORDS  ANALYSIS OF COVARIANCE, B-SPLINE,
C             CONSTRAINED LEAST SQUARES, CURVE FITTING
C***AUTHOR  Hanson, R. J., (SNLA)
C***DESCRIPTION
C
C     DCV( ) is a companion function subprogram for DFC( ).  The
C     documentation for DFC( ) has complete usage instructions.
C
C     DCV( ) is used to evaluate the variance function of the curve
C     obtained by the constrained B-spline fitting subprogram, DFC( ).
C     The variance function defines the square of the probable error
C     of the fitted curve at any point, XVAL.  One can use the square
C     root of this variance function to determine a probable error band
C     around the fitted curve.
C
C     DCV( ) is used after a call to DFC( ).  MODE, an input variable to
C     DFC( ), is used to indicate if the variance function is desired.
C     In order to use DCV( ), MODE must equal 2 or 4 on input to DFC( ).
C     MODE is also used as an output flag from DFC( ).  Check to make
C     sure that MODE = 0 after calling DFC( ), indicating a successful
C     constrained curve fit.  The array SDDATA, as input to DFC( ), must
C     also be defined with the standard deviation or uncertainty of the
C     Y values to use DCV( ).
C
C     To evaluate the variance function after calling DFC( ) as stated
C     above, use DCV( ) as shown here
C
C          VAR=DCV(XVAL,NDATA,NCONST,NORD,NBKPT,BKPT,W)
C
C     The variance function is given by
C
C      VAR=(transpose of B(XVAL))*C*B(XVAL)/DBLE(MAX(NDATA-N,1))
C
C     where N = NBKPT - NORD.
C
C     The vector B(XVAL) is the B-spline basis function values at
C     X=XVAL.  The covariance matrix, C, of the solution coefficients
C     accounts only for the least squares equations and the explicitly
C     stated equality constraints.  This fact must be considered when
C     interpreting the variance function from a data fitting problem
C     that has inequality constraints on the fitted curve.
C
C     All the variables in the calling sequence for DCV( ) are used in
C     DFC( ) except the variable XVAL.  Do not change the values of
C     these variables between the call to DFC( ) and the use of DCV( ).
C
C     The following is a brief description of the variables
C
C     XVAL    The point where the variance is desired, a double
C             precision variable.
C
C     NDATA   The number of discrete (X,Y) pairs for which DFC( )
C             calculated a piece-wise polynomial curve.
C
C     NCONST  The number of conditions that constrained the B-spline in
C             DFC( ).
C
C     NORD    The order of the B-spline used in DFC( ).
C             The value of NORD must satisfy 1 < NORD < 20 .
C
C             (The order of the spline is one more than the degree of
C             the piece-wise polynomial defined on each interval.  This
C             is consistent with the B-spline package convention.  For
C             example, NORD=4 when we are using piece-wise cubics.)
C
C     NBKPT   The number of knots in the array BKPT(*).
C             The value of NBKPT must satisfy NBKPT .GE. 2*NORD.
C
C     BKPT(*) The double precision array of knots.  Normally the problem
C             data interval will be included between the limits
C             BKPT(NORD) and BKPT(NBKPT-NORD+1).  The additional end
C             knots BKPT(I),I=1,...,NORD-1 and I=NBKPT-NORD+2,...,NBKPT,
C             are required by DFC( ) to compute the functions used to
C             fit the data.
C
C     W(*)    Double precision work array as used in DFC( ).  See DFC( )
C             for the required length of W(*).  The contents of W(*)
C             must not be modified by the user if the variance function
C             is desired.
C
C***REFERENCES  R. J. Hanson, Constrained least squares curve fitting
C                 to discrete data using B-splines, a users guide,
C                 Report SAND78-1291, Sandia Laboratories, December
C                 1978.
C***ROUTINES CALLED  DDOT, DFSPVN
C***REVISION HISTORY  (YYMMDD)
C   780801  DATE WRITTEN
C   890531  Changed all specific intrinsics to generic.  (WRB)
C   890831  Modified array declarations.  (WRB)
C   890911  Removed unnecessary intrinsics.  (WRB)
C   891006  Cosmetic changes to prologue.  (WRB)
C   891006  REVISION DATE from Version 3.2
C   891214  Prologue converted to Version 4.0 format.  (BAB)
C   920501  Reformatted the REFERENCES section.  (WRB)
C***END PROLOGUE  DCV
      INTEGER I, ILEFT, IP, IS, LAST, MDG, MDW, N, NBKPT, NCONST,
     *     NDATA, NORD
      DOUBLE PRECISION BKPT, DDOT, V, W, XVAL, ZERO
      DIMENSION BKPT(*),W(*),V(40)
C***FIRST EXECUTABLE STATEMENT  DCV
      ZERO = 0.0D0
      MDG = NBKPT - NORD + 3
      MDW = NBKPT - NORD + 1 + NCONST
      IS = MDG*(NORD + 1) + 2*MAX(NDATA,NBKPT) + NBKPT + NORD**2
      LAST = NBKPT - NORD + 1
      ILEFT = NORD
   10 IF (XVAL .LT. BKPT(ILEFT+1) .OR. ILEFT .GE. LAST - 1) GO TO 20
         ILEFT = ILEFT + 1
      GO TO 10
   20 CONTINUE
      CALL DFSPVN(BKPT,NORD,1,XVAL,ILEFT,V(NORD+1))
      ILEFT = ILEFT - NORD + 1
      IP = MDW*(ILEFT - 1) + ILEFT + IS
      N = NBKPT - NORD
      DO 30 I = 1, NORD
         V(I) = DDOT(NORD,W(IP),1,V(NORD+1),1)
         IP = IP + MDW
   30 CONTINUE
      DCV = MAX(DDOT(NORD,V,1,V(NORD+1),1),ZERO)
C
C     SCALE THE VARIANCE SO IT IS AN UNBIASED ESTIMATE.
      DCV = DCV/MAX(NDATA-N,1)
      RETURN
      END