The conventional algorithm for evaluating a polynomial a„x" + an-pP"-1+ ... + a,x + ao at x = c can be expressed in pseudocode by procedure polynomial(c, a,, a, ..., ɑn: real numbers) power := 1 Y := do for i = 1 to n power := power *c Y := Y + a; * power return y {y = ɑµc" + an-1c²-1+ ... + a,c + ao} where the final value of y is the value of the polynomial at x = c. a) Evaluate 3x² + x + 1 at x = 2 by working through each step of the algorithm showing the values assigned at each assignment step. pus

Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
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## Exploring Polynomial Evaluation Methods

In this lab, you will explore the conventional means to evaluate a polynomial and compare its efficiency to Horner’s method (see textbook, page 242).

### Objectives
1. **Implement each method as a C++ function**
2. **Time each method against a set of test polynomials to collect some empirical data.**
   - Recommend using C++’s high resolution clock, for an example see time point:  
   [https://www.geeksforgeeks.org/chrono-in-c/](https://www.geeksforgeeks.org/chrono-in-c/)
3. **Submit your code as a .cpp file and your findings as a PDF file.**

### Polynomial Evaluation

#### Conventional Algorithm
The **conventional algorithm** for evaluating a polynomial \( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \) at \( x = c \) can be expressed in pseudocode:

```plaintext
procedure polynomial(c, a_0, a_1, ..., a_n: real numbers)
    power := 1
    y := a_0
    for i := 1 to n
        power := power * c
        y := y + a_i * power
    return y  {y = a_n c^n + a_{n-1} c^{n-1} + \ldots + a_1 c + a_0}
```

**Note:** The final value of \( y \) is the value of the polynomial at \( x = c \).

**Example (a):** Evaluate \( 3x^2 + x + 1 \) at \( x = 2 \) by working through each step of the algorithm, showing the values assigned at each assignment step.

#### Horner’s Method
This exercise involves Horner’s method. This pseudocode shows how to use this method to find the value of \( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \) at \( x = c \).

```plaintext
procedure Horner(c, a_0, a_1, a_2, ..., a_n: real numbers)
    y := a_n
    for i := 1 to n
        y := y * c + a_{n
Transcribed Image Text:## Exploring Polynomial Evaluation Methods In this lab, you will explore the conventional means to evaluate a polynomial and compare its efficiency to Horner’s method (see textbook, page 242). ### Objectives 1. **Implement each method as a C++ function** 2. **Time each method against a set of test polynomials to collect some empirical data.** - Recommend using C++’s high resolution clock, for an example see time point: [https://www.geeksforgeeks.org/chrono-in-c/](https://www.geeksforgeeks.org/chrono-in-c/) 3. **Submit your code as a .cpp file and your findings as a PDF file.** ### Polynomial Evaluation #### Conventional Algorithm The **conventional algorithm** for evaluating a polynomial \( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \) at \( x = c \) can be expressed in pseudocode: ```plaintext procedure polynomial(c, a_0, a_1, ..., a_n: real numbers) power := 1 y := a_0 for i := 1 to n power := power * c y := y + a_i * power return y {y = a_n c^n + a_{n-1} c^{n-1} + \ldots + a_1 c + a_0} ``` **Note:** The final value of \( y \) is the value of the polynomial at \( x = c \). **Example (a):** Evaluate \( 3x^2 + x + 1 \) at \( x = 2 \) by working through each step of the algorithm, showing the values assigned at each assignment step. #### Horner’s Method This exercise involves Horner’s method. This pseudocode shows how to use this method to find the value of \( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \) at \( x = c \). ```plaintext procedure Horner(c, a_0, a_1, a_2, ..., a_n: real numbers) y := a_n for i := 1 to n y := y * c + a_{n
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