INTERPOLATION AND APPROXIMATION
1 Interpolation with Unequal Intervals
Lagrange`s Interpolation formula
Inverse Interpolation by Lagrange`s Interpolation Polynomial
2 Divided Differences-Newton Divided Difference Interpolation
Newtons Divided Difference formula for unequal intervals
3 Interpolating with a cubic spline
Cubic spline interpolation
4 Interpolation with equals
Newtons` forward interpolation formula
Newtons` Backward interpolation formula
1 Interpolation with Unequal intervals
Interpolation is a process of estimating the value of a function at ana intermediate point when its value are known only at certain specified points.It is based on the following assumptions:
(i) Given equation is a polynomial or it can be represented by a polynomial with a good degree of approximation.
(ii) Function should vary in such a way that either it its increasing or decreasing in the given range without sudden jumps or falls of functional values in the given interval.
We shall discuss the concept of interpolation from a set of tabulated values of when the values of x are given intervals or at unequal intervals
.First we consider interpolation with unequal intervals.
Lagrange’s Interpolation formula
This is called the for Interpolation Lagrange’s. formula
1.Using Lagrange’s interpolation find the value of y-corresponding formula, to x=10 from the following data
Using x=10 and the given data, y (10) =14.67
2. Using Lagrange’s interpolation forfromula,the fi following data
X 3 7 9 10
Y 168 120 72 63
By Lagrange’s interpolation formula
Using x=6 and the given data, y(6)=147
Using x=5 and the given data, y(5)=32.93
1.Find the polynomial degree 3 fitting the following data
X -1 0 2 3
Y -2 -1 1 4
Inverse Interpolation by Lagrange’s interpolating
Lagrange’s interpolation formula can be used is not in the table. The process of finding such of x is called inverse interpolation.
If x is the dependent variable and y is the independent variable, we can write a formula for x as a function of y.
2 Divided Difference –Newton Divided Difference Interpolation Formula
If the values of x are given at unequal intervals, it is convenient to introduce the idea of divided differences. The divided difference are the differences of y=f(x) defined, taking into consideration the changes in the values of the argument .Using divided differences of the function y=f(x),we establish Newton’s which divided is used for different interpolation which the values of x are at unequal intervals and also for fitting an approximate
curve for the given data.
Let the function assume the values Corresponding to the arguments x1 , x2 , x3 ,..... xn respectively, where the intervals need to be equal.
The first divided difference of f(x) for the arguments is defined by
3 Interpolating with a cubic spline
cubic spline Interpolation
We consider the problem of interpolation between given data points (xi, yi), i=0, 1, 2, 3….n where
by means of a smooth polynomial curve.
By means of method of least squares, we can fit is the new technique recently developed to fit a smooth curve passing he given set of points.
Definition of Cubic spline
A cubic spline s(x) is defined by the following properties.
S(x),s’(x),s”(x) are continuous on [a,b]
S(x) is a cubic polynomial in each sub-interval(xi,xi+1),i=0,1,2,3…..n=1
Conditions for fitting spline fit
The conditions for a cubic spline fit are that we pass a set of cubic through the points, using a new cubic in each interval. Further it is required both the slope and the curvature be the same for the pair of cubic that join at each point.
Natural cubic spline
A cubic spline s(x) such that s(x) is linear in the intervals
i.e. s1=0 and sn=0 is called a natural cubic spline
where s1 =second derivative at (x1 , y1 )
sn Second derivative at (xn , yn )
The three alternative choices used are
with the assumption, for a set of data that are fit by a single cubic equation their cubic splines will all be this same cubic
1.Fit a natural cubic spline to the following data
x 1 2 3
y -8 -1 18
And compute (i) y(1.5) (ii) y’(1)
For equally spaced intervals,the relation on s1 , s2 & s3 is given by
2.The following values of x and y are given ,obtain the natural cubic spline which agree with y(x) at the set of data points
X 2 3 4
Y 11 49 123
Hence compute (i) y(2.5) and (ii) y’ (2)
1.Fit the following data by a cubic spline curve
x 0 1 2 3 4
y -8 -7 0 19 56
Using the end condition that s1 & s5 are linear extrapolations.
4 Interpolation with equal intervals
(Newton’s Forward and Backward Difference formulas)
If a function y=f(x) is not known explicitly the value of y can be obtained when s set of
This formula is called Newton Backward interpolation formula.
We can also use this formula to extrapolate the values of y, a short distance ahead of yn.
1.Using Newton’s Forward interpolation formula
X 1.0 1.1 1.2 1.3 1.4
F(x) 0.841 0.891 0.932 0.964 0.985
Forward difference table:
4.Find the cubic polynomial which takes the value y(0)=1,y(1)=0,y(2)=1,y(3)=10.Hence or otherwise ,obtain y(4).Ans:y(4)=33
Exercise: 1.Given the data
x 0 1 2 3 4
y 2 3 12 35 78
Find the cubic function of x,using Newton’s backward interpolation formula
4.Find the polynomial which passes through the point newton's (7,3)(8,1)(9,1)(10,9) using interpolation formula.
1.Apply Lagrange’s formula inversely to obtai f(0)=-4,f(1)=1,f(3)=29,f(4)=52.
2.Using Lagrange’s formula of interpolation ,
X 7 8 9 10
Y 3 1 1 9
3.Use Lagrange’s formula to find the value of
x 3 7 9 10
y 168 120 72 63
4.If log(300)=2.4771,log(304)=2.4829,log(305)=2.4843,log(307)=2.4871find log(301) Ans: 2.8746
1. State the Lagrange’s. interpolation formula
Let y = f(x) be a function which takes the values y0, y1,……n y-corresponding to x=x0,x1,……nx
Then Lagrange’s interpolation formula is
Y = f(x) = --- --
2. What is the assumption we make when Lagran
Lagrange’s interpolation formula can be used whether the values of x, the independent variable are equally spaced or not whether the difference of y become smaller or not.
3. When Newton’s backward interpolation formu
The formula is used mainly to interpolate the values of y near the end of a set of tabular values and also for extrapolation the values of y a short distance ahead of y0
4. What are the errors in Trapezoidal rule of numerical integration?
The error in the Trapezoidal rule is
E< ---- y”
5.Why Simpson’s one third rule is called a c
Since the end point ordinates y0 and yn are included in the Simpson closed formula.
6. What are the advantages of Lagrange’sformula? formul
The forward and backward interpolation formulae of Newton can be used only when the values of theindependent variable x are equally spaced and can also be used when the differences
of the dependent variable y become smaller ultimately. But Lagrange’s interpolati be used whether the values of x, the independent variable are equally spaced or not and whether
the difference of y become smaller or not.
7. When do we apply Lagrange’s interpolation?
Lagrange’s formulainterpolcationbe used when the val not. It is mainly used when the values are unevenly spaced.
8. When do we apply Lagrange’s interpolation?
Lagrange’s interpolation formula spacedcanor be us not. It is mainly used when the values are unevenly spaced.
9. What are the disadvantages in practice in ap
1. It takes time.
2. It is laborious
10. When Newton’s backwardformulaisused. interpolation
The formula is used mainly to interpolate
11. When Newton’s forward interpolation formu
The formula is used mainly tobeginniginterpolateofsetof tabular values.
12. When do we use Newton’s divided differenc
Sol: This is used when the data are unequally spaced.
13. Write Forward difference operator.
Let y = f (x) be a function of x and let of the values of y. corresponding to of the values of x. Here, the independent variable (or argument), x
proceeds at equally spaced intervals and h (constant),the difference between two consecutive values of x is called the interval of differencing. Now the forward difference operator is defined as
These are called first differences.
14.Write Backward difference operator.
The backward difference operator is defined as
These are called first differences
1.Using Newton’s divided difference and hence formula, find f(4).
X 0 1 2 5
f(x) 2 3 12 147
2.Find the cubic polynomial which takes the following values:
X 0 1 2 3
f(x) 1 2 1 10
3.The following values of x and y are given:
X 1 2 3 4
f(x) 1 2 5 11
Find the cubic splines and evaluate y(1.5) and y’(3).
4.Find the rate of growth of the population in 1941 and 1971 from the table below.
Year X 1931 1941 1951 1961 1971
Population 40.62 60.8 79.95 103.56 132.65
5.Derive Newton’s backward difference formula
6.Using Lagrange’s interpolation formula fin
7.Using Newton’s divided difference formula
X 0 1 2 4 5
f(x) 1 14 15 5 6
8.Obtain the cubic spline approximation for the function y=f(x) from the following data, given that y0” 3=”=0y.
X -1 0 1 2
Y -1 1 3 35
9.The following table gives the values of density of saturated water for various temperatures of saturated steam.
Temperature0 C 0 100 150 200 250 300
Density hg/m3 958 917 865 799 712
Find by interpolation, the density when the temperature is 2750 .
10.Use Lagrange’s10656 method,giventhatlog 10to654 =2find.8156, log log 10 658 =2.8182 , log 10 659 =2.8189 and log 10 661 =2.8202.
11.Find f’(x) at x=1.5 and x=4.0 from the fol differentiation.
x 1.5 2.0 2.5 3.0 3.5 4.0
Y=f(x) 3.375 7.0 13.625 24.0 38.875 59.0
12.If f(0)=1,f(1)=2,f(2)=33 and f(3)=244. Find a cubic spline approximation, assuming M(0)=M(3)=0.Also find f(2.5).
13.Fit a set of 2 cubic splines to a half ellipse described by f(x)= [25-4x2]1/2. Choose the three data points (n=2) as (-2.5,0), (0,1.67) and (2.5 , 0) and use the free boundary conditions.
14.Find the value of y at x=21 and x=28 from the data given below
x 20 23 26 29
y 0.3420 0.3907 0.4384 0.4848
15. The population of a town is as follows:
x year 1941 1951 1961 1971 1981 1991
y population 20 24 29 36 46 51
Estimate the population increase during the period 1946 to1976.