Horner's Rule

• A polynomial P = a_nxⁿ+a_n-1x^n-1 + ... + a₁x¹+a₀x⁰
  • A is an array of coefficients.
  • Or a list of P is sparse
  • x is a value.
• We already have a brute force polynomial evaluation algorithm

  • BRUTE_POLY(A,x)
    
     value ← 0
     for i ← n down to 0 do
         value ← value + A[i] * xi
     return value
    
    	

  • In the brute force version of xⁱ we have O(i)
  • In the non-brute force version we know O(lg(i))
  • So the algorithm is O(n²) or O(n lg(n))
• Horner's Method is ancient
  • Horner died in 1837
  • The method was used in China by 1261
• Observe

  P = anxn+an-1xn-1 + ... + a1x1+a0x0
    = (anx+an-1)xn-1 + ... + a1x1+a0x0
    = ((anx+an-1)x+ an-2)xn-2 + ... + a1x1+a0x0
    = [anx+an-1)x+ an-2)x + ... + a1)x + a0
  
  
    7x4+9x3+5x2+8x+4 = (((7x+ 4)x+5)x+ 8)x+4
      

• This leads to

• HORNER(A,n)
  
   value ← A[n]
   for i ← n-1 down to 0 do
       value ← value*x + A[i]
   return value
  
      

• Standard Questions.