Interpolation with unequal interval

Interpolation methods, Lagrange's interpolation formula, Lagrange's inverse interpolation formula, Newton's divided difference forumla, Cubic Spline

1. Numerical Methods - Interpolation
Unequal Intervals
N. B. Vyas
Department of Mathematics,
Atmiya Institute of Tech. and Science, Rajkot (Guj.)
[email protected]
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

2. Interpolation
To nd the value ofyfor anxbetween dierentx- values
x0; x1; : : : ; xnis called problem ofinterpolation.
To nd the value ofyfor anxwhich falls outside the range ofx
(x < x0orx > xn) is called the problem ofextrapolation.
Theorem byWeierstrassin 1885, \Every continuous
function in an interval (a,b) can be represented in that
interval to any desired accuracy by a polynomial."
Let us assign polynomialPnof degreen(or less) that assumes
the given data values
Pn(x0) =y0,Pn(x1) =y1,: : :,Pn(xn) =yn This polynomialPnis calledinterpolation polynomial. x0; x1; : : : ; xnis called thenodes(tabular points,pivotal
pointsorarguments).
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

3. Interpolation
To nd the value ofyfor anxbetween dierentx- values
x0; x1; : : : ; xnis called problem ofinterpolation.
To nd the value ofyfor anxwhich falls outside the range ofx
(x < x0orx > xn) is called the problem ofextrapolation.
Theorem byWeierstrassin 1885, \Every continuous
function in an interval (a,b) can be represented in that
interval to any desired accuracy by a polynomial."
Let us assign polynomialPnof degreen(or less) that assumes
the given data values
Pn(x0) =y0,Pn(x1) =y1,: : :,Pn(xn) =yn This polynomialPnis calledinterpolation polynomial. x0; x1; : : : ; xnis called thenodes(tabular points,pivotal
pointsorarguments).
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

4. Interpolation
To nd the value ofyfor anxbetween dierentx- values
x0; x1; : : : ; xnis called problem ofinterpolation.
To nd the value ofyfor anxwhich falls outside the range ofx
(x < x0orx > xn) is called the problem ofextrapolation.
Theorem byWeierstrassin 1885, \Every continuous
function in an interval (a,b) can be represented in that
interval to any desired accuracy by a polynomial."
Let us assign polynomialPnof degreen(or less) that assumes
the given data values
Pn(x0) =y0,Pn(x1) =y1,: : :,Pn(xn) =yn This polynomialPnis calledinterpolation polynomial. x0; x1; : : : ; xnis called thenodes(tabular points,pivotal
pointsorarguments).
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

5. Interpolation
To nd the value ofyfor anxbetween dierentx- values
x0; x1; : : : ; xnis called problem ofinterpolation.
To nd the value ofyfor anxwhich falls outside the range ofx
(x < x0orx > xn) is called the problem ofextrapolation.
Theorem byWeierstrassin 1885, \Every continuous
function in an interval (a,b) can be represented in that
interval to any desired accuracy by a polynomial."
Let us assign polynomialPnof degreen(or less) that assumes
the given data values
Pn(x0) =y0,Pn(x1) =y1,: : :,Pn(xn) =yn This polynomialPnis calledinterpolation polynomial. x0; x1; : : : ; xnis called thenodes(tabular points,pivotal
pointsorarguments).
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

6. Interpolation
To nd the value ofyfor anxbetween dierentx- values
x0; x1; : : : ; xnis called problem ofinterpolation.
To nd the value ofyfor anxwhich falls outside the range ofx
(x < x0orx > xn) is called the problem ofextrapolation.
Theorem byWeierstrassin 1885, \Every continuous
function in an interval (a,b) can be represented in that
interval to any desired accuracy by a polynomial."
Let us assign polynomialPnof degreen(or less) that assumes
the given data values
Pn(x0) =y0,Pn(x1) =y1,: : :,Pn(xn) =yn This polynomialPnis calledinterpolation polynomial. x0; x1; : : : ; xnis called thenodes(tabular points,pivotal
pointsorarguments).
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

7. Interpolation
To nd the value ofyfor anxbetween dierentx- values
x0; x1; : : : ; xnis called problem ofinterpolation.
To nd the value ofyfor anxwhich falls outside the range ofx
(x < x0orx > xn) is called the problem ofextrapolation.
Theorem byWeierstrassin 1885, \Every continuous
function in an interval (a,b) can be represented in that
interval to any desired accuracy by a polynomial."
Let us assign polynomialPnof degreen(or less) that assumes
the given data values
Pn(x0) =y0,Pn(x1) =y1,: : :,Pn(xn) =yn This polynomialPnis calledinterpolation polynomial. x0; x1; : : : ; xnis called thenodes(tabular points,pivotal
pointsorarguments).
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

8. Interpolation
To nd the value ofyfor anxbetween dierentx- values
x0; x1; : : : ; xnis called problem ofinterpolation.
To nd the value ofyfor anxwhich falls outside the range ofx
(x < x0orx > xn) is called the problem ofextrapolation.
Theorem byWeierstrassin 1885, \Every continuous
function in an interval (a,b) can be represented in that
interval to any desired accuracy by a polynomial."
Let us assign polynomialPnof degreen(or less) that assumes
the given data values
Pn(x0) =y0,Pn(x1) =y1,: : :,Pn(xn) =yn This polynomialPnis calledinterpolation polynomial. x0; x1; : : : ; xnis called thenodes(tabular points,pivotal
pointsorarguments).
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

9. Interpolation with unequal intervals
Lagrange's interpolation formula with unequal intervals:
Lety=f(x) be continuous and dierentiable in the interval
(a; b).
Given the set ofn+ 1 values (x0; y0);(x1; y1); : : : ;(xn; yn) ofx
andy, where the values ofxneed not necessarily be equally
spaced.
It is required to ndPn(x), a polynomial of degreensuch thaty
andPn(x) agree at the tabulated points.
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

10. Interpolation with unequal intervals
Lagrange's interpolation formula with unequal intervals:
Lety=f(x) be continuous and dierentiable in the interval
(a; b).
Given the set ofn+ 1 values (x0; y0);(x1; y1); : : : ;(xn; yn) ofx
andy, where the values ofxneed not necessarily be equally
spaced.
It is required to ndPn(x), a polynomial of degreensuch thaty
andPn(x) agree at the tabulated points.
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

11. Interpolation with unequal intervals
Lagrange's interpolation formula with unequal intervals:
Lety=f(x) be continuous and dierentiable in the interval
(a; b).
Given the set ofn+ 1 values (x0; y0);(x1; y1); : : : ;(xn; yn) ofx
andy, where the values ofxneed not necessarily be equally
spaced.
It is required to ndPn(x), a polynomial of degreensuch thaty
andPn(x) agree at the tabulated points.
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

12. Interpolation with unequal intervals
Lagrange's interpolation formula with unequal intervals:
Lety=f(x) be continuous and dierentiable in the interval
(a; b).
Given the set ofn+ 1 values (x0; y0);(x1; y1); : : : ;(xn; yn) ofx
andy, where the values ofxneed not necessarily be equally
spaced.
It is required to ndPn(x), a polynomial of degreensuch thaty
andPn(x) agree at the tabulated points.
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

13. Lagrange's Interpolation
This polynomial is given by the following formula:
y=f(x)Pn(x) =
(xx1)(xx2): : :(xxn)
(x0x1)(x0x2): : :(x0xn)
y0
+
(xx0)(xx2): : :(xxn)
(x1x0)(x1x2): : :(x1xn)
y1+: : :+
(xx0)(xx1): : :(xxn1)
(xnx0)(xnx1): : :(xnxn1)
yn
NOTE:
The above formula can be used irrespective of whether the values
x0; x1; : : : ; xnare equally spaced or not.
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

14. Lagrange's Interpolation
This polynomial is given by the following formula:
y=f(x)Pn(x) =
(xx1)(xx2): : :(xxn)
(x0x1)(x0x2): : :(x0xn)
y0
+
(xx0)(xx2): : :(xxn)
(x1x0)(x1x2): : :(x1xn)
y1+: : :+
(xx0)(xx1): : :(xxn1)
(xnx0)(xnx1): : :(xnxn1)
yn
NOTE:
The above formula can be used irrespective of whether the values
x0; x1; : : : ; xnare equally spaced or not.
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

15. Lagrange's Interpolation
This polynomial is given by the following formula:
y=f(x)Pn(x) =
(xx1)(xx2): : :(xxn)
(x0x1)(x0x2): : :(x0xn)
y0
+
(xx0)(xx2): : :(xxn)
(x1x0)(x1x2): : :(x1xn)
y1+: : :+
(xx0)(xx1): : :(xxn1)
(xnx0)(xnx1): : :(xnxn1)
yn
NOTE:
The above formula can be used irrespective of whether the values
x0; x1; : : : ; xnare equally spaced or not.
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

16. Lagrange's Interpolation
This polynomial is given by the following formula:
y=f(x)Pn(x) =
(xx1)(xx2): : :(xxn)
(x0x1)(x0x2): : :(x0xn)
y0
+
(xx0)(xx2): : :(xxn)
(x1x0)(x1x2): : :(x1xn)
y1+: : :+
(xx0)(xx1): : :(xxn1)
(xnx0)(xnx1): : :(xnxn1)
yn
NOTE:
The above formula can be used irrespective of whether the values
x0; x1; : : : ; xnare equally spaced or not.
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

17. Lagrange's Interpolation
This polynomial is given by the following formula:
y=f(x)Pn(x) =
(xx1)(xx2): : :(xxn)
(x0x1)(x0x2): : :(x0xn)
y0
+
(xx0)(xx2): : :(xxn)
(x1x0)(x1x2): : :(x1xn)
y1+: : :+
(xx0)(xx1): : :(xxn1)
(xnx0)(xnx1): : :(xnxn1)
yn
NOTE:
The above formula can be used irrespective of whether the values
x0; x1; : : : ; xnare equally spaced or not.
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

18. Lagrange's Inverse Interpolation
In the Lagrange's interpolation formulayis treated as dependent
variable and expressed as function of independent variablex.
Instead ifxis treated as dependent variable and expressed as the
function of independent variabley, then Lagrange's interpolation
formula becomes
x=g(y)Pn(y) =
(yy1)(yy2): : :(yyn)
(y0y1)(y0y2): : :(y0yn)
x0
+
(yy0)(yy2): : :(yyn)
(y1y0)(y1y2): : :(y1yn)
x1+: : :+
(yy0)(yy1): : :(yyn1)
(yny0)(yny1): : :(ynyn1)
xn
This relation is referred asLagrange's inverse interpolation
formula. N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

19. Lagrange's Inverse Interpolation
In the Lagrange's interpolation formulayis treated as dependent
variable and expressed as function of independent variablex.
Instead ifxis treated as dependent variable and expressed as the
function of independent variabley, then Lagrange's interpolation
formula becomes
x=g(y)Pn(y) =
(yy1)(yy2): : :(yyn)
(y0y1)(y0y2): : :(y0yn)
x0
+
(yy0)(yy2): : :(yyn)
(y1y0)(y1y2): : :(y1yn)
x1+: : :+
(yy0)(yy1): : :(yyn1)
(yny0)(yny1): : :(ynyn1)
xn
This relation is referred asLagrange's inverse interpolation
formula. N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

20. Lagrange's Inverse Interpolation
In the Lagrange's interpolation formulayis treated as dependent
variable and expressed as function of independent variablex.
Instead ifxis treated as dependent variable and expressed as the
function of independent variabley, then Lagrange's interpolation
formula becomes
x=g(y)Pn(y) =
(yy1)(yy2): : :(yyn)
(y0y1)(y0y2): : :(y0yn)
x0
+
(yy0)(yy2): : :(yyn)
(y1y0)(y1y2): : :(y1yn)
x1+: : :+
(yy0)(yy1): : :(yyn1)
(yny0)(yny1): : :(ynyn1)
xn
This relation is referred asLagrange's inverse interpolation
formula. N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

21. Lagrange's Inverse Interpolation
In the Lagrange's interpolation formulayis treated as dependent
variable and expressed as function of independent variablex.
Instead ifxis treated as dependent variable and expressed as the
function of independent variabley, then Lagrange's interpolation
formula becomes
x=g(y)Pn(y) =
(yy1)(yy2): : :(yyn)
(y0y1)(y0y2): : :(y0yn)
x0
+
(yy0)(yy2): : :(yyn)
(y1y0)(y1y2): : :(y1yn)
x1+: : :+
(yy0)(yy1): : :(yyn1)
(yny0)(yny1): : :(ynyn1)
xn
This relation is referred asLagrange's inverse interpolation
formula. N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

22. Lagrange's Inverse Interpolation
In the Lagrange's interpolation formulayis treated as dependent
variable and expressed as function of independent variablex.
Instead ifxis treated as dependent variable and expressed as the
function of independent variabley, then Lagrange's interpolation
formula becomes
x=g(y)Pn(y) =
(yy1)(yy2): : :(yyn)
(y0y1)(y0y2): : :(y0yn)
x0
+
(yy0)(yy2): : :(yyn)
(y1y0)(y1y2): : :(y1yn)
x1+: : :+
(yy0)(yy1): : :(yyn1)
(yny0)(yny1): : :(ynyn1)
xn
This relation is referred asLagrange's inverse interpolation
formula. N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

23. Lagrange's Inverse Interpolation
In the Lagrange's interpolation formulayis treated as dependent
variable and expressed as function of independent variablex.
Instead ifxis treated as dependent variable and expressed as the
function of independent variabley, then Lagrange's interpolation
formula becomes
x=g(y)Pn(y) =
(yy1)(yy2): : :(yyn)
(y0y1)(y0y2): : :(y0yn)
x0
+
(yy0)(yy2): : :(yyn)
(y1y0)(y1y2): : :(y1yn)
x1+: : :+
(yy0)(yy1): : :(yyn1)
(yny0)(yny1): : :(ynyn1)
xn
This relation is referred asLagrange's inverse interpolation
formula. N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

24. Example
Ex.
x 150 152 154 156
y=
p
x12.24712.32912.41012.490
Evaluate
p
155 using Lagrange's interpolation formula.
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

25. Example
Sol.: x0= 150,x1= 152,x2= 154 andx3= 156
y0= 12:247,y1= 12:329,y2= 12:410 andy3= 12:490By Lagrange's interpolation formula,
f(x)Pn(x) =
(xx1)(xx2)(xx3)
(x0x1)(x0x2)(x0x3)
y0
+
(xx0)(xx2)(xx3)
(x1x0)(x1x2)(x1x3)
y1+
(xx0)(xx1)(xx3)
(x2x0)(x2x1)(x2x3)
y2+
(xx0)(xx1)(xx2)
(x3x0)(x3x1)(x3x2)
y3forx= 155
)f(155) =
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

26. Example
Sol.: x0= 150,x1= 152,x2= 154 andx3= 156
y0= 12:247,y1= 12:329,y2= 12:410 andy3= 12:490By Lagrange's interpolation formula,
f(x)Pn(x) =
(xx1)(xx2)(xx3)
(x0x1)(x0x2)(x0x3)
y0
+
(xx0)(xx2)(xx3)
(x1x0)(x1x2)(x1x3)
y1+
(xx0)(xx1)(xx3)
(x2x0)(x2x1)(x2x3)
y2+
(xx0)(xx1)(xx2)
(x3x0)(x3x1)(x3x2)
y3forx= 155
)f(155) =
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

27. Example
Sol.: x0= 150,x1= 152,x2= 154 andx3= 156
y0= 12:247,y1= 12:329,y2= 12:410 andy3= 12:490By Lagrange's interpolation formula,
f(x)Pn(x) =
(xx1)(xx2)(xx3)
(x0x1)(x0x2)(x0x3)
y0
+
(xx0)(xx2)(xx3)
(x1x0)(x1x2)(x1x3)
y1+
(xx0)(xx1)(xx3)
(x2x0)(x2x1)(x2x3)
y2+
(xx0)(xx1)(xx2)
(x3x0)(x3x1)(x3x2)
y3forx= 155
)f(155) =
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

28. Example
Sol.: x0= 150,x1= 152,x2= 154 andx3= 156
y0= 12:247,y1= 12:329,y2= 12:410 andy3= 12:490By Lagrange's interpolation formula,
f(x)Pn(x) =
(xx1)(xx2)(xx3)
(x0x1)(x0x2)(x0x3)
y0
+
(xx0)(xx2)(xx3)
(x1x0)(x1x2)(x1x3)
y1+
(xx0)(xx1)(xx3)
(x2x0)(x2x1)(x2x3)
y2+
(xx0)(xx1)(xx2)
(x3x0)(x3x1)(x3x2)
y3forx= 155
)f(155) =
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

29. Example
Sol.: x0= 150,x1= 152,x2= 154 andx3= 156
y0= 12:247,y1= 12:329,y2= 12:410 andy3= 12:490By Lagrange's interpolation formula,
f(x)Pn(x) =
(xx1)(xx2)(xx3)
(x0x1)(x0x2)(x0x3)
y0
+
(xx0)(xx2)(xx3)
(x1x0)(x1x2)(x1x3)
y1+
(xx0)(xx1)(xx3)
(x2x0)(x2x1)(x2x3)
y2+
(xx0)(xx1)(xx2)
(x3x0)(x3x1)(x3x2)
y3forx= 155
)f(155) =
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

30. Example
Sol.: x0= 150,x1= 152,x2= 154 andx3= 156
y0= 12:247,y1= 12:329,y2= 12:410 andy3= 12:490By Lagrange's interpolation formula,
f(x)Pn(x) =
(xx1)(xx2)(xx3)
(x0x1)(x0x2)(x0x3)
y0
+
(xx0)(xx2)(xx3)
(x1x0)(x1x2)(x1x3)
y1+
(xx0)(xx1)(xx3)
(x2x0)(x2x1)(x2x3)
y2+
(xx0)(xx1)(xx2)
(x3x0)(x3x1)(x3x2)
y3forx= 155
)f(155) =
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

31. Example
Sol.: x0= 150,x1= 152,x2= 154 andx3= 156
y0= 12:247,y1= 12:329,y2= 12:410 andy3= 12:490By Lagrange's interpolation formula,
f(x)Pn(x) =
(xx1)(xx2)(xx3)
(x0x1)(x0x2)(x0x3)
y0
+
(xx0)(xx2)(xx3)
(x1x0)(x1x2)(x1x3)
y1+
(xx0)(xx1)(xx3)
(x2x0)(x2x1)(x2x3)
y2+
(xx0)(xx1)(xx2)
(x3x0)(x3x1)(x3x2)
y3forx= 155
)f(155) =
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

32. Example
Sol.: x0= 150,x1= 152,x2= 154 andx3= 156
y0= 12:247,y1= 12:329,y2= 12:410 andy3= 12:490By Lagrange's interpolation formula,
f(x)Pn(x) =
(xx1)(xx2)(xx3)
(x0x1)(x0x2)(x0x3)
y0
+
(xx0)(xx2)(xx3)
(x1x0)(x1x2)(x1x3)
y1+
(xx0)(xx1)(xx3)
(x2x0)(x2x1)(x2x3)
y2+
(xx0)(xx1)(xx2)
(x3x0)(x3x1)(x3x2)
y3forx= 155
)f(155) =
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

33. Example
Ex. f(0:4) for the table below by the Lagrange's
interpolation:
x 0.30.50.6
f(x)0.610.690.72
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

34. Example
Ex. f(x) for the following
data:
x0125
f(x)2312147
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

35. Example
Ex. xfory= 7 for the following data:
x134
y41219
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

36. Lagrange's Interpolation
Disadvantages:
In practice, we often do not know the degree of the interpolation
polynomial that will give the required accuracy, so we should be
prepared to increase the degree if necessary.
To increase the degree the addition of another interpolation point
leads to re-computation.
i.e. no previous work is useful.
E.g: In calculatingPk(x), no obvious advantage can be taken of
the fact that one already has calculatedPk1(x).
That means we need to calculate entirely new polynomial.
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

37. Lagrange's Interpolation
Disadvantages:
In practice, we often do not know the degree of the interpolation
polynomial that will give the required accuracy, so we should be
prepared to increase the degree if necessary.
To increase the degree the addition of another interpolation point
leads to re-computation.
i.e. no previous work is useful.
E.g: In calculatingPk(x), no obvious advantage can be taken of
the fact that one already has calculatedPk1(x).
That means we need to calculate entirely new polynomial.
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

38. Lagrange's Interpolation
Disadvantages:
In practice, we often do not know the degree of the interpolation
polynomial that will give the required accuracy, so we should be
prepared to increase the degree if necessary.
To increase the degree the addition of another interpolation point
leads to re-computation.
i.e. no previous work is useful.
E.g: In calculatingPk(x), no obvious advantage can be taken of
the fact that one already has calculatedPk1(x).
That means we need to calculate entirely new polynomial.
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

39. Lagrange's Interpolation
Disadvantages:
In practice, we often do not know the degree of the interpolation
polynomial that will give the required accuracy, so we should be
prepared to increase the degree if necessary.
To increase the degree the addition of another interpolation point
leads to re-computation.
i.e. no previous work is useful.
E.g: In calculatingPk(x), no obvious advantage can be taken of
the fact that one already has calculatedPk1(x).
That means we need to calculate entirely new polynomial.
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

40. Divided Difference
Letf(x0); f(x1); : : : ; f(xn) be the values of a functionf
corresponding to the argumentsx0; x1; : : : ; xnwhere the intervals
x1x0; x2x1; : : : ; xnxn1are not necessarily equally spaced.
Then therst divided dierenceofffor the arguments
x0; x1; : : : ; xnare dened by ,
f(x0; x1) =
f(x1)f(x0)
x1x0
f(x1; x2) =
f(x2)f(x1)
x2x1
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

41. Divided Difference
Letf(x0); f(x1); : : : ; f(xn) be the values of a functionf
corresponding to the argumentsx0; x1; : : : ; xnwhere the intervals
x1x0; x2x1; : : : ; xnxn1are not necessarily equally spaced.
Then therst divided dierenceofffor the arguments
x0; x1; : : : ; xnare dened by ,
f(x0; x1) =
f(x1)f(x0)
x1x0
f(x1; x2) =
f(x2)f(x1)
x2x1
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

42. Divided Difference
Letf(x0); f(x1); : : : ; f(xn) be the values of a functionf
corresponding to the argumentsx0; x1; : : : ; xnwhere the intervals
x1x0; x2x1; : : : ; xnxn1are not necessarily equally spaced.
Then therst divided dierenceofffor the arguments
x0; x1; : : : ; xnare dened by ,
f(x0; x1) =
f(x1)f(x0)
x1x0
f(x1; x2) =
f(x2)f(x1)
x2x1
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

43. Divided Difference
Letf(x0); f(x1); : : : ; f(xn) be the values of a functionf
corresponding to the argumentsx0; x1; : : : ; xnwhere the intervals
x1x0; x2x1; : : : ; xnxn1are not necessarily equally spaced.
Then therst divided dierenceofffor the arguments
x0; x1; : : : ; xnare dened by ,
f(x0; x1) =
f(x1)f(x0)
x1x0
f(x1; x2) =
f(x2)f(x1)
x2x1
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

44. Divided Difference
Thesecond divided dierenceofffor three arguments
x0; x1; x2is dened by
f(x0; x1; x2) =
f(x1; x2)f(x0; x1)
x2x0
and similarly the divided dierence of ordernis dened by
f(x0; x1; : : : ; xn) =
f(x1; x2; : : : ; xn)f(x0; x1; : : : ; xn1)
xnx0
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

45. Divided Difference
Thesecond divided dierenceofffor three arguments
x0; x1; x2is dened by
f(x0; x1; x2) =
f(x1; x2)f(x0; x1)
x2x0
and similarly the divided dierence of ordernis dened by
f(x0; x1; : : : ; xn) =
f(x1; x2; : : : ; xn)f(x0; x1; : : : ; xn1)
xnx0
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

46. Divided Difference
Properties:
The divided dierences are symmetrical in all their arguments;
that is, the value of any divided dierence is independent of the
order of the arguments.
The divided dierence operator is linear. Then
th
order divided dierences of a polynomial of degreenare
constant, equal to the coecient ofx
n
.
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

47. Divided Difference
Properties:
The divided dierences are symmetrical in all their arguments;
that is, the value of any divided dierence is independent of the
order of the arguments.
The divided dierence operator is linear. Then
th
order divided dierences of a polynomial of degreenare
constant, equal to the coecient ofx
n
.
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

48. Divided Difference
Properties:
The divided dierences are symmetrical in all their arguments;
that is, the value of any divided dierence is independent of the
order of the arguments.
The divided dierence operator is linear. Then
th
order divided dierences of a polynomial of degreenare
constant, equal to the coecient ofx
n
.
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

49. Newton's Divided Difference Interpolation
An interpolation formula which has the property that a
polynomial of higher degree may be derived from it by simply
adding new terms.
Newton's general interpolation formula is one such formula and
terms in it are called divided dierences.
Letf(x0); f(x1); : : : ; f(xn) be the values of a functionf
corresponding to the argumentsx0; x1; : : : ; xnwhere the intervals
x1x0; x2x1; : : : ; xnxn1are not necessarily equally spaced.
By the denition of divided dierence,
f(x; x0) =
f(x)f(x0)
xx0
)
f(x) =f(x0) + (xx0)f(x; x0) (1)
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

50. Newton's Divided Difference Interpolation
An interpolation formula which has the property that a
polynomial of higher degree may be derived from it by simply
adding new terms.
Newton's general interpolation formula is one such formula and
terms in it are called divided dierences.
Letf(x0); f(x1); : : : ; f(xn) be the values of a functionf
corresponding to the argumentsx0; x1; : : : ; xnwhere the intervals
x1x0; x2x1; : : : ; xnxn1are not necessarily equally spaced.
By the denition of divided dierence,
f(x; x0) =
f(x)f(x0)
xx0
)
f(x) =f(x0) + (xx0)f(x; x0) (1)
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

51. Newton's Divided Difference Interpolation
An interpolation formula which has the property that a
polynomial of higher degree may be derived from it by simply
adding new terms.
Newton's general interpolation formula is one such formula and
terms in it are called divided dierences.
Letf(x0); f(x1); : : : ; f(xn) be the values of a functionf
corresponding to the argumentsx0; x1; : : : ; xnwhere the intervals
x1x0; x2x1; : : : ; xnxn1are not necessarily equally spaced.
By the denition of divided dierence,
f(x; x0) =
f(x)f(x0)
xx0
)
f(x) =f(x0) + (xx0)f(x; x0) (1)
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

52. Newton's Divided Difference Interpolation
An interpolation formula which has the property that a
polynomial of higher degree may be derived from it by simply
adding new terms.
Newton's general interpolation formula is one such formula and
terms in it are called divided dierences.
Letf(x0); f(x1); : : : ; f(xn) be the values of a functionf
corresponding to the argumentsx0; x1; : : : ; xnwhere the intervals
x1x0; x2x1; : : : ; xnxn1are not necessarily equally spaced.
By the denition of divided dierence,
f(x; x0) =
f(x)f(x0)
xx0
)
f(x) =f(x0) + (xx0)f(x; x0) (1)
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

53. Newton's Divided Difference Interpolation
An interpolation formula which has the property that a
polynomial of higher degree may be derived from it by simply
adding new terms.
Newton's general interpolation formula is one such formula and
terms in it are called divided dierences.
Letf(x0); f(x1); : : : ; f(xn) be the values of a functionf
corresponding to the argumentsx0; x1; : : : ; xnwhere the intervals
x1x0; x2x1; : : : ; xnxn1are not necessarily equally spaced.
By the denition of divided dierence,
f(x; x0) =
f(x)f(x0)
xx0
)
f(x) =f(x0) + (xx0)f(x; x0) (1)
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

54. Newton's Divided Difference Interpolation
Further
f(x; x0; x1) =
f(x; x0)f(x0; x1)
xx1
which yields
f(x; x0) =f(x0; x1) + (xx1)f(x; x0; x1) (2)
Similarly
f(x; x0; x1) =f(x0; x1; x2) + (xx2)f(x; x0; x1; x2) (3)
and in general
f(x; x0; :::; xn1) =f(x0; x1; :::; xn) + (xxn)f(x; x0; x1; :::; xn) (4)
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

55. Newton's Divided Difference Interpolation
Further
f(x; x0; x1) =
f(x; x0)f(x0; x1)
xx1
which yields
f(x; x0) =f(x0; x1) + (xx1)f(x; x0; x1) (2)
Similarly
f(x; x0; x1) =f(x0; x1; x2) + (xx2)f(x; x0; x1; x2) (3)
and in general
f(x; x0; :::; xn1) =f(x0; x1; :::; xn) + (xxn)f(x; x0; x1; :::; xn) (4)
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

56. Newton's Divided Difference Interpolation
Further
f(x; x0; x1) =
f(x; x0)f(x0; x1)
xx1
which yields
f(x; x0) =f(x0; x1) + (xx1)f(x; x0; x1) (2)
Similarly
f(x; x0; x1) =f(x0; x1; x2) + (xx2)f(x; x0; x1; x2) (3)
and in general
f(x; x0; :::; xn1) =f(x0; x1; :::; xn) + (xxn)f(x; x0; x1; :::; xn) (4)
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

57. Newton's Divided Difference Interpolation
Further
f(x; x0; x1) =
f(x; x0)f(x0; x1)
xx1
which yields
f(x; x0) =f(x0; x1) + (xx1)f(x; x0; x1) (2)
Similarly
f(x; x0; x1) =f(x0; x1; x2) + (xx2)f(x; x0; x1; x2) (3)
and in general
f(x; x0; :::; xn1) =f(x0; x1; :::; xn) + (xxn)f(x; x0; x1; :::; xn) (4)
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

58. Newton's Divided Difference Interpolation
multiplying equation (2) by (xx0)
, (3) by (xx0) (xx1)
and so on,
and nally the last term (4) by
(xx0) (xx1):::(xxn1) and
adding (1), (2) , (3) up to (4)
we obtain
f(x) =
f(x0) + (xx0)f(x0; x1) + (xx0) (xx1)f(x0; x1; x2) +:::+
(xx0) (xx1):::(xxn1)f(x0; x1; :::; xn)
This formula is calledNewton's divided dierenceformula.
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

59. Newton's Divided Difference Interpolation
multiplying equation (2) by (xx0)
, (3) by (xx0) (xx1)
and so on,
and nally the last term (4) by
(xx0) (xx1):::(xxn1) and
adding (1), (2) , (3) up to (4)
we obtain
f(x) =
f(x0) + (xx0)f(x0; x1) + (xx0) (xx1)f(x0; x1; x2) +:::+
(xx0) (xx1):::(xxn1)f(x0; x1; :::; xn)
This formula is calledNewton's divided dierenceformula.
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

60. Newton's Divided Difference Interpolation
multiplying equation (2) by (xx0)
, (3) by (xx0) (xx1)
and so on,
and nally the last term (4) by
(xx0) (xx1):::(xxn1) and
adding (1), (2) , (3) up to (4)
we obtain
f(x) =
f(x0) + (xx0)f(x0; x1) + (xx0) (xx1)f(x0; x1; x2) +:::+
(xx0) (xx1):::(xxn1)f(x0; x1; :::; xn)
This formula is calledNewton's divided dierenceformula.
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

61. Newton's Divided Difference Interpolation
multiplying equation (2) by (xx0)
, (3) by (xx0) (xx1)
and so on,
and nally the last term (4) by
(xx0) (xx1):::(xxn1) and
adding (1), (2) , (3) up to (4)
we obtain
f(x) =
f(x0) + (xx0)f(x0; x1) + (xx0) (xx1)f(x0; x1; x2) +:::+
(xx0) (xx1):::(xxn1)f(x0; x1; :::; xn)
This formula is calledNewton's divided dierenceformula.
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

62. Newton's Divided Difference Interpolation
multiplying equation (2) by (xx0)
, (3) by (xx0) (xx1)
and so on,
and nally the last term (4) by
(xx0) (xx1):::(xxn1) and
adding (1), (2) , (3) up to (4)
we obtain
f(x) =
f(x0) + (xx0)f(x0; x1) + (xx0) (xx1)f(x0; x1; x2) +:::+
(xx0) (xx1):::(xxn1)f(x0; x1; :::; xn)
This formula is calledNewton's divided dierenceformula.
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

63. Newton's Divided Difference Interpolation
The divided dierence upto third order
x y 1
st
order 2
nd
order 3
rd
order
dierence dierence dierence
x0y0
[x0; x1]
x1y1 [x0; x1; x2]
[x1; x2] [ x0; x1; x2; x3]
x2y2 [x1; x2; x3]
[x2; x3]
x3y3
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

64. Example
Ex.
x01234
y139-81
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

65. Example
Sol.
By Newton's divided dierence formula
f(x) =
f(x0) + (xx0)f(x0; x1) + (xx0) (xx1)f(x0; x1; x2) +:::
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

66. Example
Sol.
By Newton's divided dierence formula
f(x) =
f(x0) + (xx0)f(x0; x1) + (xx0) (xx1)f(x0; x1; x2) +:::
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals

67. Example
Ex.
(4;1245);(1;33);(0;5);(2;9);(5;1335)
N. B. Vyas Numerical Methods - Interpolation Unequal Intervals