1. State the Lagrange’s. interpolation formula
Sol:
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
Sol:
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
Sol:
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?
Sol:
The error in the Trapezoidal rule is
E< ---- y”
5.Why Simpson’s one third rule is called a c
Sol:
Since the end point ordinates y0 and yn are included in the Simpson closed formula.
6. What are the advantages of Lagrange’sformula? formul
Sol:
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?
Sol:
Lagrange’s formula interpolcation be used when the val not. It is mainly used when the values are unevenly spaced.
8. When do we apply Lagrange’s interpolation?
Sol:
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
Sol:
1. It takes time.
2. It is laborious
10. When Newton’s backwardformulaisused. interpolation
Sol:
The formula is used mainly to interpolate
values.
11. When Newton’s forward interpolation formu
Sol:
The formula is used mainly to beginnig interpolate of set of 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.
Sol:
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.
Sol:
The backward difference operator is defined as
These are called first differences
Part B
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
Y
5.Derive Newton’s backward difference formula
6.Using Lagrange’s interpolation formula fin
(0,-12),(1,0),(3,6),(4,12).
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
(thousands)
Estimate the population increase during the period 1946 to1976.
Tutorial Problems
Tutorial 1
1.Apply Lagrange’s formula inversely to obtai f(0)=-4,f(1)=1,f(3)=29,f(4)=52.
Ans: 0.8225
2.Using Lagrange’s formula of interpolation ,
X 7 8 9 10
Y 3 1 1 9
Ans: 3.625
3.Use Lagrange’s formula to find the value of
x 3 7 9 10
y 168 120 72 63
Ans: 147
4.If log(300)=2.4771,log(304)=2.4829,log(305)=2.4843,log(307)=2.4871find log(301) Ans: 2.8746
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