*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