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+#Example 9     Chapter 9       Page no.: 299
+#Newton-Gregory backward difference formula
+
+#Given Data
+mydata<- data.frame(t=c(10,20,30,40,50),
+                    s=c(0.1736, 0.3420, 0.5000, 0.6428, 0.7660),
+                    stringsAsFactors = TRUE
+)
+mydata
+
+theta<-mydata$t
+sin<- mydata$s
+s0<-sin[1]
+s1<-sin[2]
+s2<-sin[3]
+s3<-sin[4]
+s4<-sin[5]
+
+x0<-theta[1]
+x1<-theta[2]
+x2<-theta[3]
+x3<-theta[4]
+x4<-theta[5]
+
+h=x1-x0
+h
+
+x=25
+a1<- (x-x4)/h  #for backward interpolation ,start from bottom
+a2<- (x-x3)/h
+a3<- (x-x2)/h
+a4<- (x-x1)/h
+a5<- (x-x0)/h
+
+f1<-s1-s0
+f2<-s2-s1
+f3<-s3-s2
+f4<-s4-s3
+delta1<- f2-f1
+delta2<- f3-f2
+delta3<- f4-f3
+
+get1<-delta1-delta2
+get2<-delta2-delta3
+
+last<- get2-get1
+
+p1<- s4+ a1*(f4)/1
+p1
+p2<- p1+ a1*a2*(delta3)/2
+p2
+p3<- p2+ a1*a2*a3*(get2)
+p3
+p4<- p3+ a1*a2*a3*a4*(last)
+p4
+
+cat("using Newton-Gregory Backward difference formula value of sin(25) was obtained to be :",p4)
+
+#Differed value in textbook is due to approximation upto 4 decimal places.
+
+