text: Difference Operators
  We have already seen one difference operator called divided difference operator in the earlier section. We define few more difference operators and their properties in this section.
  Forward difference operator :
  
  Suppose that a fucntion f(x) is given at equally spaced discrete points say x0, x1, . . . xn as f0, f1, . . . fn respectively. Also let the constant difference between two consecutive points of x is called the interval of differencing or the step length denoted by h. Then the forward difference operator D is defined as
  
  Df(x) = f(x + h) - f(x) orDfi = fi+1 - fi
  where any typical fi = f(xi).
  Similarly the higher differences are defined as
  D2f(x)    =    D(Df(x))
  =    D(f(x + h) - f(x))
  =    f(x + 2h) - 2f(x + h) + f(x)
  =    fi+2 - 2fi+1 + fi
  or in general,                         Dn f(x) = Dn-1(Df(x)) = Dn-1f(x + h) - Dn-1f(x)
  Example :
  Let a function f(x) is given at the points (0, 7), (4, 43), (8,367) then find the forward difference of the function at x = 4.
  
  Df(x) = f(x + h) - f(x)
  
  ÞDf(4) = f(4 + h) - f(4)
  = f(8) - f(4) = 367 - 43 = 324
  Example :
  Find Df(x) for the function x2 + 2x + 3 with h = 2
  Df(x)  = f(x + h) - f(x)    =    (x + 2)2 + 2(x + 2) + 3 - x2 - 2x - 3
  =   x2 + 4x + 4 +  2x  + 4 +  x 2x     =   4x + 8
  D2f(x)  = 4(x + 2) + 8 - 4x - 8    = 4x + 8 - 4x = 8
  D3f(x)  = 8 - 8 = 0
  Properties :
  1. D(f(x) + g(x))  = Df(x)  + Dg(x)
  2. D(c f(x))  = c.Df(x)  where c is a constant
  3. If f(x) is a polynomial of degree n,
  4.  n f(x) = S ai xi then  Dnf(x) is constant, and is equal to ann!hn i=0
  
  Proof :
  
  Lets prove this by mathematical induction
  for n = 1,    f(x) = a0 + a1x
  Df(x) = a0 + a1(x + h) - a0 - a1x = a1h
  = a11!h1
  assume that the formula is true for all polynomials upto the degree n-1. Consider the nth degree polynomial
   n f(x) = S ai xi i=0
  
   ( n ) Dn f(x) = Dn S ai xi i=0
  
   ( n-1 ) + Dn(anxn) Dn f(x) = Dn S ai xi i=0
  
  = 0 + an Dn xn
  = 0 + an Dn-1((x + h)n - xn)
  = 0 + an Dn-1( nhxn-1 - g(x))
  
  where g(x) is a polynomial of degree less than (n-1) and hence Dn-1g(x) = 0
  ÞDn f(x) = an Dn-1( nhxn-1)
  = an(nh)(n-1)!hn-1
  = ann!hn
  
  Backward difference operator : Ñ
  
  The difference operator Ñ defined by
  Ñf(x) = f(x) - f(x-h)
  Ñfi = fi - fi-1
  is called backward difference operator.
  Ñ2fi = Ñ(Ñfi)  = Ñ(fi - fi-1)
  
  = fi - 2fi-1 + fi-2 and so on.
  
  Central difference operator : d
  The difference operator d  defined by
  
  df(x) = f(x + h/2) - f(x - h/2)
  
  dfi = fi + 1/2 - fi - 1/2
  is called the central difference operator.
  d2fi = d(fi + 1/2 - fi - 1/2)  = (fi+1 - fi) - (fi - fi-1)
  = fi+1 - 2fi + fi-1
  Shift operator :
  
  Efi = fi+1, E2fi = fi+2
  
  Averaging operator :
  
  mfi = 1/2 * (fi + 1/2 + fi - 1/2)
  
  Some relations between various operators :
  1. E = 1 + D
  
  2.
  
  Efi = fi+1 =  fi+1 - fi + fi = (1 + D)fi
  
  Þ E = 1 + D
  
   ¥ Enfi = S ( n i ) Di fi k=0
  
  for n = 1E1fi = fi+1 = (1 + D)fi
  
   ¥ RHS is S ( 1 i ) Di fi = D0 fi+D1 fi = (1+D)fi k=0
  
  Hence the result is true for n = 1, Assume now that the result is true for the value (n-1) then
  
   ¥ En-1fi = S ( n-1 i ) Di fi k=0
  
  consider
  
  Enfi = E En-1fi
  
   ¥ = (1 + D) S ( n-1 i ) Di fi k=0
   ¥ = S ( n-1 i ) Di fi k=0
  +
   ¥ S ( n-1 i ) Di+1 fi k=0
   ¥ = S ( n-1 i ) Di fi k=0
  +
   ¥ S ( n-1 j-1 ) Dj fi j=1
   ¥ = S [ ( n-1 i ) + ( n-1 i ) ] Di fi k=0
   ... ( n-1 i ) = 0 and j is arbitrary
   ¥ = S ( n i ) Di fi k=0
  
   ¥ ÞEnfi = S ( n i ) Di fi k=0
  
  Solution of Transcendental Equations | Solution of Linear System of Algebraic Equations | Interpolation & Curve Fitting
  Numerical Differentiation & Integration | Numerical Solution of Ordinary Differential Equations
  Numerical Solution of Partial Differential Equations
explanation: The document provides a detailed explanation of various difference operators used in numerical analysis, including the forward difference operator, backward difference operator, central difference operator, shift operator, and averaging operator. It includes definitions, properties, and examples for each operator. The document contains several examples that serve as both questions and answers, illustrating how to apply these operators to specific functions.
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distilabel_metadata.content: Difference Operators
  We have already seen one difference operator called divided difference operator in the earlier section. We define few more difference operators and their properties in this section.
  Forward difference operator :
  
  Suppose that a fucntion f(x) is given at equally spaced discrete points say x0, x1, . . . xn as f0, f1, . . . fn respectively. Also let the constant difference between two consecutive points of x is called the interval of differencing or the step length denoted by h. Then the forward difference operator D is defined as
  
  Df(x) = f(x + h) - f(x) orDfi = fi+1 - fi
  where any typical fi = f(xi).
  Similarly the higher differences are defined as
  D2f(x)    =    D(Df(x))
  =    D(f(x + h) - f(x))
  =    f(x + 2h) - 2f(x + h) + f(x)
  =    fi+2 - 2fi+1 + fi
  or in general,                         Dn f(x) = Dn-1(Df(x)) = Dn-1f(x + h) - Dn-1f(x)
  Example :
  Let a function f(x) is given at the points (0, 7), (4, 43), (8,367) then find the forward difference of the function at x = 4.
  
  Df(x) = f(x + h) - f(x)
  
  ÞDf(4) = f(4 + h) - f(4)
  = f(8) - f(4) = 367 - 43 = 324
  Example :
  Find Df(x) for the function x2 + 2x + 3 with h = 2
  Df(x)  = f(x + h) - f(x)    =    (x + 2)2 + 2(x + 2) + 3 - x2 - 2x - 3
  =   x2 + 4x + 4 +  2x  + 4 +  x 2x     =   4x + 8
  D2f(x)  = 4(x + 2) + 8 - 4x - 8    = 4x + 8 - 4x = 8
  D3f(x)  = 8 - 8 = 0
  Properties :
  1. D(f(x) + g(x))  = Df(x)  + Dg(x)
  2. D(c f(x))  = c.Df(x)  where c is a constant
  3. If f(x) is a polynomial of degree n,
  4.  n f(x) = S ai xi then  Dnf(x) is constant, and is equal to ann!hn i=0
  
  Proof :
  
  Lets prove this by mathematical induction
  for n = 1,    f(x) = a0 + a1x
  Df(x) = a0 + a1(x + h) - a0 - a1x = a1h
  = a11!h1
  assume that the formula is true for all polynomials upto the degree n-1. Consider the nth degree polynomial
   n f(x) = S ai xi i=0
  
   ( n ) Dn f(x) = Dn S ai xi i=0
  
   ( n-1 ) + Dn(anxn) Dn f(x) = Dn S ai xi i=0
  
  = 0 + an Dn xn
  = 0 + an Dn-1((x + h)n - xn)
  = 0 + an Dn-1( nhxn-1 - g(x))
  
  where g(x) is a polynomial of degree less than (n-1) and hence Dn-1g(x) = 0
  ÞDn f(x) = an Dn-1( nhxn-1)
  = an(nh)(n-1)!hn-1
  = ann!hn
  
  Backward difference operator : Ñ
  
  The difference operator Ñ defined by
  Ñf(x) = f(x) - f(x-h)
  Ñfi = fi - fi-1
  is called backward difference operator.
  Ñ2fi = Ñ(Ñfi)  = Ñ(fi - fi-1)
  
  = fi - 2fi-1 + fi-2 and so on.
  
  Central difference operator : d
  The difference operator d  defined by
  
  df(x) = f(x + h/2) - f(x - h/2)
  
  dfi = fi + 1/2 - fi - 1/2
  is called the central difference operator.
  d2fi = d(fi + 1/2 - fi - 1/2)  = (fi+1 - fi) - (fi - fi-1)
  = fi+1 - 2fi + fi-1
  Shift operator :
  
  Efi = fi+1, E2fi = fi+2
  
  Averaging operator :
  
  mfi = 1/2 * (fi + 1/2 + fi - 1/2)
  
  Some relations between various operators :
  1. E = 1 + D
  
  2.
  
  Efi = fi+1 =  fi+1 - fi + fi = (1 + D)fi
  
  Þ E = 1 + D
  
   ¥ Enfi = S ( n i ) Di fi k=0
  
  for n = 1E1fi = fi+1 = (1 + D)fi
  
   ¥ RHS is S ( 1 i ) Di fi = D0 fi+D1 fi = (1+D)fi k=0
  
  Hence the result is true for n = 1, Assume now that the result is true for the value (n-1) then
  
   ¥ En-1fi = S ( n-1 i ) Di fi k=0
  
  consider
  
  Enfi = E En-1fi
  
   ¥ = (1 + D) S ( n-1 i ) Di fi k=0
   ¥ = S ( n-1 i ) Di fi k=0
  +
   ¥ S ( n-1 i ) Di+1 fi k=0
   ¥ = S ( n-1 i ) Di fi k=0
  +
   ¥ S ( n-1 j-1 ) Dj fi j=1
   ¥ = S [ ( n-1 i ) + ( n-1 i ) ] Di fi k=0
   ... ( n-1 i ) = 0 and j is arbitrary
   ¥ = S ( n i ) Di fi k=0
  
   ¥ ÞEnfi = S ( n i ) Di fi k=0
  
  Solution of Transcendental Equations | Solution of Linear System of Algebraic Equations | Interpolation & Curve Fitting
  Numerical Differentiation & Integration | Numerical Solution of Ordinary Differential Equations
  Numerical Solution of Partial Differential Equations
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document provides a detailed explanation of various difference operators used in numerical analysis, including the forward difference operator, backward difference operator, central difference operator, shift operator, and averaging operator. It includes definitions, properties, and examples for each operator. The document contains several examples that serve as both questions and answers, illustrating how to apply these operators to specific functions.
  
  Label: multiple-question-and-answer