*DECK CV REAL FUNCTION CV (XVAL, NDATA, NCONST, NORD, NBKPT, BKPT, W) C***BEGIN PROLOGUE CV C***PURPOSE Evaluate the variance function of the curve obtained C by the constrained B-spline fitting subprogram FC. C***LIBRARY SLATEC C***CATEGORY L7A3 C***TYPE SINGLE 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 CV( ) is a companion function subprogram for FC( ). The C documentation for FC( ) has complete usage instructions. C C CV( ) is used to evaluate the variance function of the curve C obtained by the constrained B-spline fitting subprogram, FC( ). 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 CV( ) is used after a call to FC( ). MODE, an input variable to C FC( ), is used to indicate if the variance function is desired. C In order to use CV( ), MODE must equal 2 or 4 on input to FC( ). C MODE is also used as an output flag from FC( ). Check to make C sure that MODE = 0 after calling FC( ), indicating a successful C constrained curve fit. The array SDDATA, as input to FC( ), must C also be defined with the standard deviation or uncertainty of the C Y values to use CV( ). C C To evaluate the variance function after calling FC( ) as stated C above, use CV( ) as shown here C C VAR=CV(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)/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 CV( ) are used in C FC( ) except the variable XVAL. Do not change the values of these C variables between the call to FC( ) and the use of CV( ). C C The following is a brief description of the variables C C XVAL The point where the variance is desired. C C NDATA The number of discrete (X,Y) pairs for which FC( ) C calculated a piece-wise polynomial curve. C C NCONST The number of conditions that constrained the B-spline in C FC( ). C C NORD The order of the B-spline used in FC( ). 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 real array of knots. Normally the problem data C interval will be included between the limits BKPT(NORD) C and BKPT(NBKPT-NORD+1). The additional end knots C BKPT(I),I=1,...,NORD-1 and I=NBKPT-NORD+2,...,NBKPT, are C required by FC( ) to compute the functions used to fit C the data. C C W(*) Real work array as used in FC( ). See FC( ) for the C required length of W(*). The contents of W(*) must not C be modified by the user if the variance function is C 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 BSPLVN, SDOT C***REVISION HISTORY (YYMMDD) C 780801 DATE WRITTEN C 890531 Changed all specific intrinsics to generic. (WRB) C 890531 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 CV DIMENSION BKPT(NBKPT), W(*), V(40) C***FIRST EXECUTABLE STATEMENT CV ZERO = 0. 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 (.NOT.(XVAL.GE.BKPT(ILEFT+1) .AND. ILEFT.LT.LAST-1)) GO TO 20 ILEFT = ILEFT + 1 GO TO 10 20 CALL BSPLVN(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) = SDOT(NORD,W(IP),1,V(NORD+1),1) IP = IP + MDW 30 CONTINUE CV = MAX(SDOT(NORD,V,1,V(NORD+1),1),ZERO) C C SCALE THE VARIANCE SO IT IS AN UNBIASED ESTIMATE. CV = CV/MAX(NDATA-N,1) RETURN END