Polynomial evaluation and long division
Okumaya devam et...
← Previous revision | Revision as of 20:49, 5 May 2024 |
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| For this, a new sequence of constants is defined [[Recurrence relation|recursively]] as follows: | For this, a new sequence of constants is defined [[Recurrence relation|recursively]] as follows: |
| <math display="block">\begin{align} | {{NumBlk||<math display="block">\begin{align} |
| b_n & := a_n \\ | b_n & := a_n \\ |
| b_{n-1} & := a_{n-1} + b_n x_0 \\ | b_{n-1} & := a_{n-1} + b_n x_0 \\ |
| (1)\quad\quad\quad & ~~~ \vdots \\ | & ~~~ \vdots \\ |
| b_1 & := a_1 + b_2 x_0 \\ | b_1 & := a_1 + b_2 x_0 \\ |
| b_0 & := a_0 + b_1 x_0. | b_0 & := a_0 + b_1 x_0. |
| \end{align}</math> | \end{align}</math>|{{EquationRef|1}}}} |
| Then <math>b_0</math> is the value of <math>p(x_0)</math>. | Then <math>b_0</math> is the value of <math>p(x_0)</math>. |
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| Now, it can be proven that; | Now, it can be proven that; |
| <math display="block"> | {{NumBlk||<math display="block"> |
| (2)\quad\quad\quad p(x) = \left(b_1 + b_2 x + b_3 x^2 + b_4x^3 + \cdots + b_{n-1} x^{n-2} +b_nx^{n-1}\right) \left(x - x_0\right) + b_0 | p(x) = \left(b_1 + b_2 x + b_3 x^2 + b_4x^3 + \cdots + b_{n-1} x^{n-2} +b_nx^{n-1}\right) \left(x - x_0\right) + b_0 |
| </math> | </math>|{{EquationRef|2}}}} |
| This expression constitutes Horner's practical application, as it offers a very quick way of determining the outcome of; | This expression constitutes Horner's practical application, as it offers a very quick way of determining the outcome of; |
| <math display="block">p(x) / (x-x_0) </math> | <math display="block">p(x) / (x-x_0) </math> |
Okumaya devam et...