#This code calculates the coefficients of the "starting" spline which is of degree 2k-2. 
# Yd is a matrix of dimension 2x(K/2) containing the derivatives of the (K-1)-integral of a two sided 
#Brownian motion (Y) + t^{2K} at the boundary points -C and C. It starts with the (K-2)th 
#derivative of Y at -C and C, (K-4)th,...,0.

StartingSpline <- function(K,C,Yd){
C <- 2
K <- 4
Yd <- rbind(-1,2)
if((K-2*floor(K/2))!=0){
print("Enter please an even K !")	
}

#This part gives the coefficients when K=2.	

if(K==2){
Coef <- c(6*C^2,(Yd[2,1]-Yd[1,1])/(2*C),(Yd[2,1]+Yd[1,1])/2 - 6*C^4)
}

#This part of the code calculates the coefficients when K > 2 (and even).	

if(K > 2){
d <- 2*K-2
a.d <- (factorial(2*K)/factorial(2))*C^2  
Coef  <- a.d
for(i in (d-1):0){
   p <- 2*K-i	
      if(p <= K){
        if((p-2*floor(p/2))!=0){
	     Coef <- c(Coef,0)	
         }
          else{	
          Coef <- c(Coef,(factorial(2*K)/factorial(2*i))*C^{2*i}-sum(Coef[Coef!=0]*(1/factorial(2*((i-1):1)))*C^{2*((i-1):1)}))
          }
       }
  
      if(p > K){
	     if((p-2*floor(p/2))!=0){
	     Coef <- c(Coef, (Yd[2,(p-K+1)/2]-Yd[1,(p-K+1)/2])/(2*C))
        }
        else{ 
	   i <- p/2
          Coef <- c(Coef,(Yd[2,(p-K)/2]+Yd[1,(p-K)/2])/2 - sum(Coef[2*((i-1):1)]*(1/factorial(2*((i-1):1)))*C^{2*((i-1):1)}))
        }

       }
    } 

}
 	Coef <- Coef/factorial((2*K-2):0)
  	Coef
}